Abstract
This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin â with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by shortlived attempts to include the matter field described by SchrĂ¶dingerâs or Diracâs equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included.
Introduction
Preface
This historical review of classical unified field theories consists of two parts. In the first, the development of unified field theory between 1914 and 1933, i.e., during the years Einstein^{Footnote 1} lived and worked in Berlin, will be covered. In the second, the very active period after 1933 until the 1960s to 1970s will be reviewed. In the first version of Part I presented here, in view of the immense amount of material, neither all shades of unified field theory nor all the contributions from the various scientific schools will be discussed with the same intensity; I apologise for the shortcoming and promise to improve on it with the next version. At least, even if I do not discuss them all in detail, as many references as are necessary for a first acquaintance with the field are listed here; completeness may be reached only (if at all) by later updates. Although I also tried to take into account the published correspondence between the main figures, my presentation, again, is far from exhaustive in this context. Eventually, unpublished correspondence will have to be worked in, and this may change some of the conclusions. Purposely I included mathematicians and also theoretical physicists of lesser rank than those who are known to be responsible for big advances. My aim is to describe the field in its full variety as it presented itself to the reader at the time.
The review is written such that physicists should be able to follow the technical aspects of the papers (cf. Section 2), while historians of science without prior knowledge of the mathematics of general relativity at least might gain an insight into the development of concepts, methods, and scientific communities involved. I should hope that readers find more than one opportunity for further indepth studies concerning the many questions left open.
I profited from earlier reviews of the field, or of parts of it, by Pauli^{Footnote 2} ([246], Section V); Ludwig [212]; Whittaker ([414], pp. 188â196); Lichnerowicz [209]; Tonnelat ([356], pp. 1â14); Jordan ([176], Section III); Schmutzer ([290], Section X); Treder ([183], pp. 30â43); Bergmann ([12], pp. 62â73); Straumann [334, 335]; Vizgin [384, 385]^{Footnote 3} ; Bergia [11]; Goldstein and Ritter [146]; Straumann and OâRaifeartaigh [240]; Scholz [292], and Stachel [330]. The section on Einsteinâs unified field theories in Paisâ otherwise superb book presents the matter neither with the needed historical correctness nor with enough technical precision [241]. A recent contribution of van Dongen, focussing on Einsteinâs methodology, was also helpful [371]. As will be seen, with regard to interpretations and conclusions, my views are different in some instances. In Einstein biographies, the subject of âunified field theoriesâ â although keeping Einstein busy for the second half of his life â has been dealt with only in passing, e.g., in the book of Jordan [177], and in an unsatisfying way in excellent books by FĂ¶lsing [136] and by Hermann [159]. This situation is understandable; for to describe a genius stubbornly clinging to a set of ideas, sterile for physics in comparison with quantum mechanics, over a period of more than 30 years, is not very rewarding. For the short biographical notes, various editions of J. C. Poggendorffâs BiographischLiterarischem HandwĂ¶rterbuch and internet sources have been used (in particular [1]).
If not indicated otherwise, all nonEnglish quotations have been translated by the author; the original text of quotations is given in footnotes.
Introduction to part I
Past experience has shown that formerly unrelated parts of physics could be fused into one single conceptual formalism by a new theoretical perspective: electricity and magnetism, optics and electromagnetism, thermodynamics and statistical mechanics, inertial and gravitational forces. In the second half of the 20th century, the electromagnetic and weak nuclear forces have been bound together as an electroweak force; a powerful scheme was devised to also include the strong interaction (chromodynamics), and led to the standard model of elementary particle physics. Unification with the fourth fundamental interaction, gravitation, is in the focus of much present research in classical general relativity, supergravity, superstring, and supermembrane theory but has not yet met with success. These types of âunificationsâ have increased the explanatory power of present day physical theories and must be considered as highlights of physical research.
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 âa synthesis of the laws of mechanics and of electromagnetismâ ([16], p. 132). Einsteinâs 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 representation^{Footnote 4} In the second approach, predominantly the unification of the dynamics of physical fields is aimed at, i.e., a unification of the fundamental interactions. Maxwellâs 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 deductivehypothetical and an empiricalinductive method. As Dirac pointed out, however,
âThe successful development of science requires a proper balance between the method of building up from observations and the method of deducing by pure reasoning from speculative assumptions, [âŠ].â ([233], p. 1001)
In an unsuccessful hunt for progress with the deductivehypothetical 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, NordstrĂ¶m, and others^{Footnote 5}. 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 (KaluzaKlein), and much still might be learned in the future.
In the following I shall sketch, more or less chronologically, and by trailing Einsteinâs 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 âmesonicâ or ânuclearâ interaction. The physical fields considered in the framework of âunified field theoryâ including, after the advent of quantum (wave) mechanics, the wave function satisfying either SchrĂ¶dingerâs or Diracâs 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 Diracâs contributions, cf. [190]. Nowadays, it seems mandatory to approach unification in the framework of quantum field theory.
General relativityâs 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 stressenergy tensor T^{ik} (the sourceterm of his field equations G^{ik}=ÎșT^{ik}) 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 Einsteinâs equations and their generalisation, of the energymomentum tensor of matter by intrinsic geometrical structures, and, likewise, the removal of the electric current density vector as a nongeometrical source term in Maxwellâs equations.

The development of a unified field theory more geometrico for electromagnetism and gravitation, and in addition, later, of the âfield of the electronâ as a classical field of âde Brogliewavesâ without explicitly taking into account further matter sources^{Footnote 6}.
In a very Cartesian spirit, Tonnelat (Tonnelat 1955 [356], p. 5) gives a definition of a unified field theory as âa theory joining the gravitational and the electromagnetic field into one single hyperfield whose equations represent the conditions imposed on the geometrical structure of the universe.â No material source terms are taken into account^{Footnote 7}. 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 stressenergy tensor is in Einsteinâs theory of gravitation: They remain alien elements.
For the theories discussed, the representation of matter oscillated between the pointparticle 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 pointparticle is somewhat amiss. Nevertheless, geodesics of the Riemannian geometry underlying Einsteinâs theory of gravitation are identified with the worldlines of freely moving pointparticles. The field at the location of a pointparticle becomes unbounded, or âsingularâ, such that the derivation of equations of motion from the field equations is a nontrivial 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 âgeonâ and âgeometrodynamicsâ 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 1925â26, 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
âis in no way taking notice of the results of quantum calculation because he believes that by dealing with microscopic phenomena these will come out by themselves. Otherwise he would not support the theory.â ([91], p. 610)
However, in connection with one of his moves, i.e., the 5vector version of Kaluza^{Footnote 8}âs theory (cf. Sections 4.2, 6.3), which for him provided âa logical unity of the gravitational and the electromagnetic fieldsâ, he regretfully acknowledged:
âBut one hope did not get fulfilled. I thought that upon succeeding to find this law, it would form a useful theory of quanta and of matter. But, this is not the case. It seems that the problem of matter and quanta makes the construction fall apart.â^{Footnote 9} ([96], p. 442)
Thus, unfortunately, also the hopes of the eminent mathematician Schouten^{Footnote 10}, who knew some physics, were unfulfilled:
â[âŠ] collections of positive and negative electricity which we are finding in the positive nuclei of hydrogen and in the negative electrons. The older Maxwell theory does not explain these collections, but also by the newer endeavours it has not been possible to recognise these collections as immediate consequences of the fundamental differential equations studied. However, if such an explanation should be found, we may perhaps also hope that new light is shed on the [âŠ] mysterious quantum orbits.â^{Footnote 11} ([301], p. 39)
In this context, through all the years, Einstein vainly tried to derive, from the field equations of his successive unified field theories, the existence of elementary particles with opposite though otherwise equal electric charge but unequal mass. In correspondence with the state of empirical knowledge at the time (i.e., before the positron was found in 1932/33), but despite theoretical hints pointing into a different direction to be found in Diracâs papers, he always paired electron and proton ^{Footnote 12}.
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 particlelike quanta. Todayâs unified field theories appear in the form of gauge theories; matter is represented by operator valued spinhalf quantum fields (fermions) while the âforcesâ mediated by âexchange particlesâ are embodied in gauge fields, i.e., quantum fields of integer spin (bosons). The spacetime geometry used is rigidly fixed, and usually taken to be Minkowski space or, within string and membrane theory, some higherdimensional manifold also loosely called âspacetimeâ, 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 Lorentzinvariance, and by the statistical interpretation of the SchrĂ¶dinger wave function. Lanczos^{Footnote 13} , in 1929, was well aware of his being out of tune with those adherent to quantum mechanics:
âI therefore believe that between the âreactionary point of viewâ represented here, aiming at a complete fieldtheoretic description based on the usual spacetime structure and the probabilistic (statistical) point of view, a compromise [âŠ] no longer is possible.â^{Footnote 14} ([198], p. 486, footnote)
On the other hand, those working in quantum theory may have frowned upon the wealth of objects within unified field theories uncorrelated to a convincing physical interpretation and thus, in principle, unrelated to observation. In fact, until the 1930s, attempts still were made to âgeometrizeâ wave mechanics while, roughly at the same time, quantisation of the gravitational field had also been tried [284]. Einstein belonged to those who regarded the idea of unification as more fundamental than the idea of field quantisation [95]. His thinking is reflected very well in a remark made by Lanczos at the end of a paper in which he tried to combine Maxwellâs and Diracâs equations:
âIf the possibilities anticipated here prove to be viable, quantum mechanics would cease to be an independent discipline. It would melt into a deepened âtheory of matterâ which would have to be built up from regular solutions of nonlinear differential equations, â in an ultimate relationship it would dissolve in the âworld equationsâ of the Universe. Then, the dualism âmatterfieldâ would have been overcome as well as the dualism âcorpusclewaveâ.â^{Footnote 15} ([198], p. 493)
Lanczosâ 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 Maxwellâs 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^{Footnote 16} wrote a series of papers in the first of which he stated:
âIt is remarkable that not only no fundamental tensor [first fundamental form] or tensordensity, but also no connection, neither Riemannian nor projective, nor conformal, is needed for writing down the [Maxwell] equations. Matter is characterised by a bivectordensity [âŠ].â ([367], p. 422, and also [363, 364, 365, 366])
If one of the fields to be united asks for less âgeometryâ, why to mount all the effort needed for generalising Riemannian geometry?
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 â Einsteinâs equations in empty space;

no gravitational field â Maxwellâs equations;

âweakâ gravitational and electromagnetic fields â EinsteinMaxwell equations;

no gravitational field but a âstrongâ electromagnetic field â some sort of nonlinear electrodynamics.
A similar weakness occurred for the equations of motion; about the only limiting equation to be reproduced was Newtonâs equation augmented by the Lorentz force. Later, attempts were made to replace the relationship âgeodesics â freely falling point particlesâ by more general assumptions for charged or electrically neutral point particles â depending on the more general (nonRiemannian) connections introduced^{Footnote 17}. 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 gravitoelectromagnetic 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 ReichenbĂ€cher^{Footnote 18}, FĂ¶rster (alias Bach), Weyl^{Footnote 19}, Eddington^{Footnote 20}, and Einstein (see Section 3.1). In Section 4, the main ideas are developed. They include Weylâs generalization of Riemannian geometry by the addition of a linear form (see Section 4.1) and the reaction to this approach. To this, Kaluzaâs idea concerning a geometrization of the electromagnetic and gravitational fields within a fivedimensional 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). Kaluzaâs theory received a great deal of attention after O. Klein^{Footnote 21} intervention and extension of Kaluzaâs paper (see Section 6.3.2). Einsteinâs treatment of a special case of a metricaffine geometry, i.e., âdistant parallelismâ, set off an avalanche of research papers (see Section 6.4.4), the more so as, at the same time, the covariant formulation of Diracâs 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^{Footnote 22}, Veblen^{Footnote 23}), French (Cartan^{Footnote 24}), and the Dutch (Schouten, Struik^{Footnote 25}) 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.
The Possibilities of Generalizing General Relativity: A Brief Overview
As a rule, the point of departure for unified field theory was general relativity. The additional task then was to âgeometrizeâ the electromagnetic field. In this review, we will encounter essentially five different ways to include the electromagnetic field into a geometric setting:

by connecting an additional linear form to the metric through the concept of âgaugingâ (Weyl);

by introducing an additional space dimension (Kaluza);

by choosing an asymmetric Ricci tensor (Eddington);

by adding an antisymmetric tensor to the metric (Bach, Einstein);

by replacing the metric by a 4bein field (Einstein).
In order to bring some order into the wealth of these attempts towards âunified field theory,â I shall distinguish four main avenues extending general relativity, according to their mathematical direction: generalisation of

geometry,

dynamics (Lagrangians, field equations),

number field, and

dimension of space,
as well as their possible combinations. In the period considered, all four directions were followed as well as combinations between them like e.g., fivedimensional theories with quadratic curvature terms in the Lagrangian. Nevertheless, we will almost exclusively be dealing with the extension of geometry and of the number of space dimensions.
Geometry
It is very easy to get lost in the many constructive possibilities underlying the geometry of unified field theories. We briefly describe the mathematical objects occurring in an order that goes from the less structured to the more structured cases. In the following, only local differential geometry is taken into account^{Footnote 26}.
The space of physical events will be described by a real, smooth manifold M_{D} of dimension D coordinatised by local coordinates x^{i}, and provided with smooth vector fields X, Y, âŠ with components X^{i}, Y^{i}, âŠ and linear forms Ï, Îœ, âŠ, (Ï_{i}, Îœ_{i}) in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connections^{Footnote 27}. At each point, D linearly independent vectors (linear forms) form a linear space, the tangent space (cotangent space) of M_{D}. We will assume that the manifold M_{D} is space and timeorientable. On it, two independent fundamental structural objects will now be introduced.
Metrical structure
The first is a prescription for the definition of the distance ds between two infinitesimally close points on M_{D}, eventually corresponding to temporal and spatial distances in the external world. For ds, we need positivity, symmetry in the two points, and the validity of the triangle equation. We know that ds must be homogeneous of degree one in the coordinate differentials dx^{i} connecting the points. This condition is not very restrictive; it still includes Finsler geometry [281, 126, 224] to be briefly touched, below.
In the following, ds is linked to a nondegenerate bilinear form g(X, Y), called the first fundamental form; the corresponding quadratic form defines a tensor field, the metrical tensor, with D^{2} components g_{ij} such that
where the neighbouring points are labeled by x^{i} and x^{i}+dx^{i}, respectively^{Footnote 28}. Besides the norm of a vector \(\left X \right: = \sqrt {{g_{ij}}{X^i}{X^j}}\), the âangleâ between directions X, Y can be defined by help of the metric:
From this we note that an antisymmetric part of the metrical tensor does not influence distances and norms but angles.
With the metric tensor having full rank, its inverse g^{ik} is defined through^{Footnote 29}
We are used to g being a symmetric tensor field, i.e., with g_{ik}=g(_{ik}) and with only D(D+1)/2 components; in this case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is Â±(Dâ2)^{Footnote 30}. In the following this need not hold, so that the decomposition obtains^{Footnote 31}:
An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.
For an asymmetric metric, the inverse
is determined by the relations
and turns out to be [356]
where g, Ï, and Îł are the determinants of the corresponding tensors g_{ik}, Ï_{ik}, and Îł_{ik}. We also note that
where g := det g_{ik}, Ï := det Ï_{ik}, Îł := det Îł_{ik}. The results (6, 7, 8) were obtained already by ReichenbĂ€cher ([273], pp. 223â224)^{Footnote 32} and also by SchrĂ¶dinger [320]. Eddington also calculated Equation (8); in his expression the term âŒÏ_{ik}*Ï^{ik} is missing (cf. [59], p. 233).
The manifold is called spacetime if D=4 and the metric is symmetric and Lorentzian, i.e., symmetric and with signature sig g=Â±2. Nevertheless, sloppy contemporaneaous usage of the term âspacetimeâ includes arbitrary dimension, and sometimes is applied even to metrics with arbitrary signature.
In a manifold with Lorentzian metric, a nontrivial real conformal structure always exists; from the equation
results an equivalence class of metrics {Î»} with Î» being an arbitrary smooth function. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [67]. For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part Îł_{ik}=Îł(_{ik}) of the metric alone taken to be Lorentzian.
A special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by
A geometrical characterization of Minkowski space as an uncurved, flat space is given below. Let \({{\mathcal L}_X}\) be the Lie derivative with respect to the tangent vector X^{Footnote 33}; then \({{\cal L}_{{{_X}_p}{\eta _{ik}}}} = 0\) holds for the Lorentz group of generators X_{p}.
The metric tensor g may also be defined indirectly through D vector fields forming an orthonormal Dleg (bein) \(h_{\hat \iota }^k\). with
where the hatted indices (âbeinindicesâ) count the number of legs spanning the tangent space at each point (Ä”=1, 2, âŠ , D) and are moved with the Minkowski metric^{Footnote 34}. From the geometrical point of view, this can always be done (cf. theories with distant parallelism). By introducing 1forms \({\theta ^{\hat k}}: = h_l^{\hat k}d{x^l}\), Equation (11) may be brought into the form \(d{s^2} = {\theta ^{\hat \imath}}{\theta ^{\hat k}}{\eta _{\hat \imath\hat k}}\).
A new physical aspect will come in if the h ^{k}_{Ăź} are considered to be the basic geometric variables satisfying field equations, not the metric. Such tetradtheories (for the case D=4) are described well by the concept of fibre bundle. The fibre at each point of the manifold contains, in the case of an orthonormal Dbein (tetrad), all Dbeins (tetrads) related to each other by transformations of the group O(D), or the Lorentz group, and so on.
In Finsler geometry, the line element depends not only on the coordinates x^{i} of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points dx^{i}:
Again, g_{ij} is required to be homogeneous of rank 1.
Affine structure
The second structure to be introduced is a linear connection L with D3 components L_{ij}^{k}; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations^{Footnote 35}. The connection is a device introduced for establishing a comparison of vectors in different points of the manifold. By its help, a tensorial derivative â, called covariant derivative is constructed. For each vector field and each tangent vector it provides another unique vector field. On the components of vector fields X and linear forms Ï it is defined by
The expressions \({\mathop \nabla \limits^ + _k}{X^i}\) and \(\tfrac{{\partial {X^i}}}{{\partial {x^k}}}\) are abbreviated by \({X^i}_{\left\ k \right.}\) and X^{i}_{,k}, respectively, while for a scalar f covariant and partial derivative coincide: \({\nabla _i}f = {\tfrac{{\partial f}}{{\partial {x_i}}}} \equiv {\partial _i}f \equiv {f_{,i}}\).
We have adopted the notational convention used by Schouten [300, 310, 389]. Eisenhart and others [121, 234] change the order of indices of the components of the connection:
As long as the connection is symmetric, this does not make any difference as \({\mathop \nabla \limits^ + _k}{X^i}  {\mathop \nabla \limits^  _k}{X^i} = 2{L_{[kj]}}^i{X^j}\). For both kinds of derivatives we have:
Both derivatives are used in versions of unified field theory by Einstein and others^{Footnote 36}.
A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special role: With regard to it the connection transforms as a tensor (cf. Section 2.1.5).
For a vector density (cf. Section 2.1.5), the covariant derivative of XÌ contains one more term:
A smooth vector field Y is said to be parallely transported along a parametrised curve Î»(u) with tangent vector X if for its components \({Y^i}_{\left\ k \right.}{X^k}(u) = 0\) holds along the curve. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point^{Footnote 37}:
By a particular choice of the curve73x2019;s parameter, Ï=0 may be imposed.
A transformation mapping autoparallels to autoparallels is given by:
The equivalence class of autoparallels defined by Equation (18) defines a projective structure on M_{D} [404, 403].
The particular set of connections
with \({L_j}: = {L_{im}}^m\) is mapped into itself by the transformation (18) [348].
In Part II of this article, we shall find the set of transformations \({L_{ik}}^j \to {L_{ik}}^j + {\delta ^j}_i\tfrac{{\partial \omega }}{{\partial {x^k}}}\) playing a role in versions of Einsteinâs unified field theory.
From the connection L_{ij}^{k} further connections may be constructed by adding an arbitrary tensor field T to its symmetrised part^{Footnote 38}:
By special choice of T we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i.e.,
is called torsion; it is a tensor field. The trace of the torsion tensor \({S_i}: = {S_{il}}^l\) is called torsion vector; it connects to the two traces of the affine connection \({L_i}: = {L_{il}}^l;{{\tilde L}_j}: = {L_{lj}}^l\) as \({S_i} = \tfrac{1}{2}({L_i}  {\tilde L_i})\).
Different types of geometry
Affine geometry
Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric, i.e., with \({L_{ij}}^k = {\Gamma _{ij}}^k\). In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it. This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by^{Footnote 39}
respectively. In a geometry with symmetric affine connection both tensors coincide because of
In particular, in Riemannian geometry, both affine curvature tensors reduce to the one and only Riemann curvature tensor.
The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
For a vector density, the identity is given by
with the homothetic curvature V_{jk} to be defined below in Equation (31).
The curvature tensor (22) satisfies two algebraic identities:
where the curly bracket denotes cyclic permutation:
These identities can be found in Schoutenâs book of 1924 ([300], p. 88, 91) as well as the additional single integrability condition, called Bianchi identity:
A corresponding condition obtains for the curvature tensor KÌ from Equation (23).
From both affine curvature tensors we may form two different tensorial traces each. In the first case \({V_{kl}}: = {K^i}_{ikl} = {V_{[kl]}}\), and \({K_{jk}}: = {K^i}_{jki}\). V_{kl} is called homothetic curvature, while K_{jk} is the first of the two affine generalisations from \(\mathop {{\rm{ }}K}\limits^ +\) and \(\mathop {{\rm{ }}K}\limits^ \) of the Ricci tensor in Riemannian geometry. We get^{Footnote 40}
and the following identities hold:
where \({S_k}: = {S_{kl}}^l\). While V_{kl} is antisymmetric, K_{jk} has both tensorial symmetric and antisymmetric parts:
We use the notation \({A_{(i\left k \rightl)}}\) in order to exclude the index k from the symmetrisation bracket^{Footnote 41}.
In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor (cf., however, [356]).
For a symmetric affine connection, the preceding results reduce considerably due to \({S_{kl}}^m = 0\). From Equations (29,30,32) we obtain the identities:
i.e., only one independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor \({K_{[kl]}} =  {\partial _{[k}}{{\tilde L}_{l]}}\) holds. This equation will be important for the physical interpretation of affine geometry.
In affine geometry, the simplest way to define a fundamental tensor is to set g_{ij}:=Î±K_{(ij)}, or g_{ij}:=Î±KÌ _{(ij)}. It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by det (K_{ij})^{Footnote 42}.
As a final result in this section, we give the curvature tensor calculated from the connection \({{\bar L}_{ij}}^k = {\Gamma _{ij}}^k + {T_{ij}}^k\) (cf. Equation (20)), expressed by the curvature tensor of \({\Gamma _{ij}}^k\) and by the tensor \({T_{ij}}^k\):
where ^{(Î)}â is the covariant derivative formed with the connection \({\Gamma _{ij}}^k\) (cf. also [310], p. 141).
Mixed geometry
A manifold carrying both structural elements, i.e., metric and connection, is called a metricaffine space. If the first fundamental form is taken to be asymmetric, i.e., to contain an antisymmetric part \({g_{[ik]}}: = \tfrac{1}{2}({g_{ij}}  {g_{ji}})\), we speak of a mixed geometry. In principle, both metricaffine space and mixed geometry may always be reinterpreted as Riemannian geometry with additional geometric objects: the 2form field Ï(f) (symplectic form), the torsion S, and the nonmetricity Q (cf. Equation 41). It depends on the physical interpretation, i.e., the assumed relation between mathematical objects and physical observables, which geometry is the most suitable.
From the symmetric part of the first fundamental form h_{ij}=g_{(ij)}, a connection may be constructed, often called after LeviCivita^{Footnote 43} [204],
and from it the Riemannian curvature tensor defined as in Equation (22) with \({L_{ij}}^k = \{ \,_{ij}^k\}\) (cf. Section 2.1.3); { ^{k}_{ij} } is called the Christoffel symbol. Thus, in metricaffine and in mixed geometry, two different connections arise in a natural way. In the remaining part of this section we will deal with a symmetric fundamental form Îł_{ij} only, and denote it by g_{ij}.
With the help of the symmetric affine connection, we may define the tensor of nonmetricity \({Q_{ij}}^k\) by^{Footnote 44}
Then the following identity holds:
where the contorsion tensor \({K_{ij}}^k\), a linear combination of torsion \({S_{ij}}^k\), is defined by^{Footnote 45}
The inner product of two tangent vectors A^{i}, B^{k} is not conserved under parallel transport of the vectors along X^{l} if the nonmetricity tensor does not vanish:
A connection for which the nonmetricity tensor vanishes, i.e.,
holds, is called metriccompatible^{Footnote 46}
J. M. Thomas^{Footnote 47} introduced a combination of the terms appearing in \(\mathop \nabla \limits^ +\) and \(\mathop \nabla \limits^ \) to define a covariant derivative for the metric ([346], p. 188),
and extended it for tensors of arbitrary rank â„ 3.
Einstein later used as a constraint on the metrical tensor
a condition that cannot easily be interpreted geometrically [97]. We will have to deal with Equation (47) in Section 6.1 and, more intensively, in Part II of this review.
Connections that are not metriccompatible have been used in unified field theory right from the beginning. Thus, in Weylâs theory [397, 395] we have
In case of such a relationship, the geometry is called semimetrical [300, 310]. According to Equation (44), in Weylâs theory the inner product multiplies by a scalar factor under parallel transport:
This means that the light cone is preserved by parallel transport.
We may also abbreviate the last term in the identity (42) by introducing
Then, from Equation (39), the curvature tensor of a torsionless affine space is given by
where ({ ^{i}_{jk} })â is the covariant derivative formed with the Christoffel symbol.
RiemannCartan geometry is the subcase of a metricaffine geometry in which the metriccompatible connection contains torsion, i.e., an antisymmetric part \({L_{[ij]}}^k\); torsion is a tensor field to be linked to physical observables. A linear connection whose antisymmetric part \({S_{ij}}^k\) has the form
is called semisymmetric [300].
Riemannian geometry is the further subcase with vanishing torsion of a metricaffine geometry with metriccompatible connection. In this case, the connection is derived from the metric: \({\Gamma _{ij}}^k = \{ \,_{ij}^k\,\}\), where \(\{ \,_{ij}^k\,\}\) is the usual Christoffel symbol (40). The covariant derivative of A with respect to the LeviCivita connection \(\mathop \nabla \limits^{\{ \,_{ij}^k\,\} }\) is abbreviated by A_{;k}. The Riemann curvature tensor is denoted
An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:
This is an invariant characterisation irrespective of whether the Minkowski metric Î· is given in Cartesian coordinates as in Equation (10), or in an arbitrary coordinate system. We also have \({{\mathcal L}_X}{R_{ijkl}} = 0\) where \({\mathcal L}\) is the Liederivative (see below under âsymmetriesâ), and X stands for the generators of the Lorentz group.
In Riemanian geometry, the socalled geodesic equation,
determines the shortest and the straightest curve between two infinitesimally close points. However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished.
A conformal transformation of the metric,
with a smooth function Î» changes the components of the nonmetricity tensor,
as well as the LeviCivita connection,
with \({\sigma _i}: = {\lambda ^{  1}}{\partial _i}\lambda\). As a consequence, the Riemann curvature tensor \({R^i}_{jkl}\) is also changed; if, however, \({{R'}^i}_{jkl} = 0\) can be reached by a conformal transformation, then the corresponding spacetime is called conformally flat. In M_{D}, for D>3, the vanishing of the Weyl curvature tensor
is a necessary and sufficient condition for M_{D} to be conformally flat ([397], p. 404, [300], p. 170).
Even before Weyl, the question had been asked (and answered) as to what extent the conformal and the projective structures were determining the geometry: According to Kretschmann (and then to Weyl) they fix the metric up to a constant factor ([196]; see also [401], Appendix 1; for a modern approach, cf. [67]).
The geometry needed for the pre and nonrelativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form (NewtonCartan geometry, cf. [65, 66]). In the following we shall deal only with relativistic unified field theories.
Projective geometry
Projective geometry is a generalisation of Riemannian geometry in the following sense: Instead of tangent spaces with the light cone Î·_{ik}dx^{i}dx^{k}=0, where Î· is the Minkowski metric, in each event now a tangent space with a general, nondegenerate surface of second order Îł will be introduced. This leads to a tangential cone g_{ik}dx^{i}dx^{k}=0 in the origin (cf. Equation (9)), and to a hyperplane, the polar plane, formed by the contact points of the tangential cone and the surface Îł. In place of the D inhomogeneous coordinates x^{i} of M_{D}, D + 1 homogeneous coordinates \({X^\alpha }(\alpha = 0,1,2, \ldots ,D)\) are defined^{Footnote 48} such that they transform as homogeneous functions of first degree:
The connection to the inhomogeneous coordinates x^{i} is given by homogeneous functions of degree zero, e.g., by \({x^i} = \tfrac{{{x^i}}}{{{\phi _\alpha }{X^\alpha }}}\)^{Footnote 49}. Thus, the \({X^\alpha }\) themselves form the components of a tangent vector. Furthermore, the quadratic form \({g_{\alpha \beta }}{X^\alpha }{X^\beta } = \epsilon = \pm 1\) is adopted with \({g_{\alpha \beta }}\) being a homogeneous function of degree 2. A tensor field \(T_{\;{n_1}\;{n_2}\;{n_3} \ldots }^{{m_1}{m_2}{m_3} \ldots }\) (cf. Section 2.1.5) depending on the homogeneous coordinates \({X^\mu }\) with u contravariant (upper) and l covariant (lower) indices is required to be a homogeneous function of degree \(r: = u  l\).
If we define \({\gamma _\mu }^i: = \tfrac{{\partial {x^i}}}{{\partial {X^\mu }}}\), with \({\gamma _\mu }^i{X^\mu } = 0\), then \({\gamma _\mu }^i\) transforms like a tangent vector under point transformations of the x^{i}, and as a covariant vector under homogeneous transformations of the \({X^\alpha }\). The \({\gamma _\mu }^i\) may be used to relate covariant vectors a_{i} and \({A_\mu }\) by \({A_\mu } = {\gamma _\mu }^i{a_i}\). Thus, the metric tensor in the space of homogeneous coordinates \({g_{\alpha \beta }}\) and the metric tensor \({g_{ik}}\) of M_{D} are related by \({g_{ik}} = {\gamma _i}^\alpha {\gamma _k}^\beta {g_{\alpha \beta }}\) with \({\gamma _\mu }^i{\gamma _k}^\mu = \delta _k^i\). The inverse relationship is given by \({g_{\alpha \beta }} = {\gamma _\alpha }^i{\gamma _\beta }^k{g_{ik}} + \epsilon {X_\alpha }{X_\beta }\) with \({X_\alpha } = {g_{\alpha \beta }}{X^\beta }\). The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf. Section 2.1.2):
The covariant derivative of the quantity \({\gamma _k}^\mu\) interconnecting both spaces is given by
Cartanâs method
In this section, we briefly present Cartanâs oneform formalism in order to make understandable part of the literature. Cartan introduces oneforms \({\theta ^{\hat a}}(\hat a = 1, \ldots ,4)\) by \({\theta ^{\hat a}}: = h_l^{\hat a}d{x^l}\). The reciprocal basis in tangent space is given by \({e_{\hat \jmath}} = h_{\hat \jmath}^l\tfrac{\partial }{{\partial {x^l}}}\). Thus, \({\theta ^{\hat a}}({e_{\hat \jmath}}) = \delta _{\hat \jmath}^{\hat a}\). The metric is then given by \({\eta _{\hat \imath\hat k}}{\theta ^{\hat \imath}} \otimes {\theta ^{\hat k}}\). The covariant derivative of a tangent vector with beincomponents X^{kÌ} is defined via Cartanâs first structure equations,
where \({\omega ^{\hat \imath }}_{\hat k}\) is the connection1form, and \({\Theta ^{\hat \imath}}\) is the torsion2form, \({\Theta ^{\hat \imath }} =  {S_{\hat l\hat m}}^{\hat \imath}{\theta ^{\hat l}} \wedge {\theta ^{\hat m}}\). We have \({\omega _{\hat \imath\hat k}} =  {\omega _{\hat k\hat \imath}}\). The link to the components \({L_{[ij]}}^k\) of the affine connection is given by \({\omega ^{\hat \imath}}_{\hat k} = h_l^{\hat \imath}h_{\hat k}^m{L_{\hat rm}}^l{\theta ^{\hat r}}\)^{Footnote 50}. The covariant derivative of a tangent vector with beincomponents X^{kÌ} then is
By further external derivation ^{Footnote 51} on Î we arrive at the second structure relation of Cartan,
In Equation (65) the curvature2form \({\Omega ^{\hat k}}_{\hat l} = \tfrac{1}{2}{R^{\hat k}}_{\hat l\hat m\hat n}{\theta ^{\hat m}} \wedge {\theta ^{\hat n}}\) appears, which is given by
\({\Omega ^{\hat k}}_{\hat k}\) is the homothetic curvature.
Tensors, spinors, symmetries
Tensors
Up to here, no definitions of a tensor and a tensor field were given: A tensor T_{p}(M_{D}) of type (r, s) at a point p on the manifold M_{D} is a multilinear function on the Cartesian product of r cotangent and s tangent spaces in p. A tensor field is the assignment of a tensor to each point of M_{D}. Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:
where x^{kâČ}=x^{kâČ}(x^{i}) with smooth functions on the r.h.s. are taken from the set (âgroupâ) of coordinate transformations (diffeomorphisms). Strictly speaking, tensors are representations of the abstract group at a point on the manifold^{Footnote 52}.
A relative tensor T_{p}(M_{D}) of type (r, s) and of weight Ï at a point p on the manifold M_{D} transforms like
An example is given by the totally antisymmetric object â_{ijkl} with â_{ijkl}=Â±1, or â_{ijkl}=0 depending on whether (ijkl) is an even or odd permutation of (0123), or whether two indices are alike. Ï=1 for â_{ijkl}; in this case, the relative tensor is called tensor density. We can form a tensor from â_{ijkl} by introducing \({\eta _{ijkl}}: = \sqrt {  g} { \epsilon _{ijkl}}\), where g_{ik} is a Lorentzmetric. Note that \({\eta ^{ijkl}}: = \tfrac{1}{{\sqrt {  g} }}{ \epsilon ^{ijkl}}=\). The dual to a 2form (skewsymmetric tensor) then is defined by \(*{F^{ij}} = \tfrac{1}{2}{\eta ^{ijkl}}{F_{kl}}\).
In connection with conformal transformations \(g \to \lambda g\), the concept of the gaugeweight of a tensor is introduced. A tensor \(T_{\;\;\;\;...}^{...}\) is said to be of gauge weight q if it transforms by Equation (56) as
Objects that transform as in Equation (67) but with respect to a subgroup, e.g., the linear group, affine group G(D), orthonormal group O(D), or the Lorentz group \({\mathcal L}\), are tensors in a restricted sense; sometimes they are named affine or Cartesian tensors. All the subgroups mentioned are Liegroups, i.e., continuous groups with a finite number of parameters. In general relativity, both the âgroupâ of general coordinate transformations and the Lorentz group are present. The concept of tensors used in Special Relativity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinateindependent (âgenerally covariantâ) form, then the full tensor concept comes into play.
Spinors
Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the spacetime manifold. To see how spinor representations can be obtained, we must use the 21 homomorphism of the group SL(2,C) and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group^{Footnote 53}. Let AâSL(2,C); then A is a complex (2by2)matrix with det A=1. By picking the special Hermitian matrix
where 1 is the (2 by 2)unit matrix and Ï are the Pauli matrices satisfying
Then, by a transformation A from SL(2,C),
where A+ is the Hermitian conjugate matrix^{Footnote 54}. Moreover, det S=det SâČ which, according to Equation (70), expresses the invariance of the spacetime distance to the origin:
The link between the representation of a Lorentz transformation L_{ik} in spacetime and the unimodular matrix A mapping spin space (cf. below) is given by
Thus, the map is two to one: +A and A give the same L_{ik}.
Now, contravariant 2spinors ÎŸ^{A}(A=1, 2) are the elements of a complex linear space, spinor space, on which the matrices A are acting^{Footnote 55}. The spinor is called elementary if it transforms under a Lorentztransformation as
Likewise, contravariant dotted spinors Î¶^{ÈŠ} are those transforming with the complexconjugate matrix Ä: {\zeta ^{\dot B'}} = \pm {{\bar \mathbf{A}}^{\dot B}}_{\,\dot D}{\zeta ^{\dot C}}. Covariant and covariant dotted 2spinors correspondingly transform with the inverse matrices,
and
The space of 2spinors can be used as a representation space for the (proper, orthochronous) Lorentz group, with the 2spinors being the elements of the most simple representation D_{(1/2,0)}.
Higherorder spinors with dotted and undotted indices \(S_{\,\,\,\,C \ldots \dot D \ldots }^{A \ldots \dot B \ldots }\) transform correspondingly. For the raising and lowering of indices now a real, antisymmetric (2Ă2)matrix â with components \({ \epsilon ^{AB}} = \delta _1^A\delta _B^2  \delta _2^A\delta _B^1 = { \epsilon _{AB}}\) is needed, such that
Next to a spinor, bispinors of the form \({\zeta ^{AB}},\;{\xi ^{A\dot B}}\), etc. are the simplest quantities (spinors of 2nd order). A vector X^{k} can be represented by a bispinor X^{Aáž},
where \(\sigma _k^{A\dot B}\) (k=0,âŠ3) is a quantity linking the tangent space of spacetime and spinor space. If k numerates the matrices and A, áž designate rows and columns, then we can chose Ï ^{Aáž}_{0} to be the unit matrix while for the other three indices Ï ^{Aáž}_{j} are taken to be the Pauli matrices. Often the quantity \(s_k^{A\dot B} = \tfrac{1}{{\sqrt 2 }}\sigma _k^{A\dot B}\) is introduced. The reciprocal matrix \(s_{A\dot B}^k\) is defined by
whereas
In order to write down spinorial field equations, we need a spinorial derivative,
with \({\partial _{A\dot B}}{\partial ^{A\dot B}} = {\partial _k}{\partial ^k}\). The simplest spinorial equation is the Weyl equation:
The next simplest spinor equation for two spinors Ï^{áž}, Ï_{A} would be
where m is a mass. Equation (85) is the 2spinor version of Diracâs equation.
Dirac or 4spinors with 4 components Ï^{k}, k=1,âŠ, 4, may be constructed from 2spinors as a direct sum of contravariant undotted and covariant dotted spinors Ï and Ï;: For k=1, 2, we enter Ï^{1} and Ï^{2}; for k=3, 4, we enter Ï_{1Ì} and Ï_{2Ì} . In connection with Dirac spinors, instead of the Paulimatrices the Dirac Îłmatrices (4Ă4matrices) appear; they satisfy
The Dirac equation is in 4spinor formalism [53, 54]:
with the 4component Dirac spinor Ï. In the first version of Diracâs equation, Î± and ÎČmatrices were used, related to the Îłâs by
where the matrices ÎČ and Î±^{m} are given by \( \left( {\begin{array}{*{20}{c}} 0&{  {\sigma _i}}\\ {{\sigma _i}}&0 \end{array}} \right),\;\;\left( {\begin{array}{*{20}{c}} 0&{  1}\\ 1&0 \end{array}} \right) \).
The generallycovariant formulation of spinor equations necessitates the use of nbeins \(h_{\hat \imath}^k\), whose internal ârotationâ group, operating on the âhattedâ indices, is the Lorentz group. The group of coordinate transformations acts on the Latin indices. In Cartanâs oneform formalism (cf. Section 2.1.4), the covariant derivative of a 4spinor is defined by
where \({\sigma ^{\hat \imath\hat k}}: = \tfrac{1}{2}\left[ {{\gamma ^{\hat \imath}}{\gamma ^{\hat k}}} \right].\).
Equation (89) is a special case of the general formula for the covariant derivative of a tensorial form Ï, i.e., a vector in some vector space V, whose components are differential forms,
where \(\rho ({e_\alpha })\) is a particular representation of the corresponding Lie algebra in V with basis vectors \({e_\alpha }\). For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used^{Footnote 56}.
Symmetries
In Section 2.1.1 we briefly met the Lie derivative of a vector field \({{\mathcal L}_X}\) with respect to the tangent vector X defined by \({({L_X}Y)^k}: = {([X,Y])^k} = {X^i}{\partial _i}{Y^k}  {Y^i}{\partial _i}{X^k}\). With its help we may formulate the concept of isometries of a manifold, i.e., special mappings, also called âmotionsâ, locally generated by vector fields X satisfying
The generators X solving Equation (91) given some metric, form a Lie group G_{r}, the group of motions of M_{D}. If a group G_{r} is prescribed, e.g., the group of spatial rotations O(3), then from Equation (91) the functional form of the metric tensor having O(3) as a symmetry group follows.
A Riemannian space is called (locally) stationary if it admits a timelike Killing vector; it is called (locally) static if this Killing vector is hypersurface orthogonal. Thus if, in a special coordinate system, we take X^{i}=ÎŽ ^{i}_{0} then from Equation (91) we conclude that stationarity reduces to the condition â_{0}g_{ik}= 0. If we take X to be the tangent vector field to the congruence of curves x^{i}=x^{i}(u), i.e., if \({X^k} = \tfrac{{d{x^k}}}{{du}}\), then a necessary and sufficient condition for hypersurfaceorthogonality is \({\epsilon^{ijkl}}{X_j}{X_{[k,l]}} = 0\).
A generalisation of Killing vectors are conformal Killing vectors for which \({{\mathcal L}_X}\;{g_{ik}} = \Phi {g_{ik}}\) with an arbitrary smooth function ÎŠ holds. In purely affine spaces, another type of symmetry may be defined: \({{\mathcal L}_X}\;{\Gamma _{ik}}^l = 0\); they are called affine motions [425].
Dynamics
Within a particular geometry, usually various options for the dynamics of the fields (field equations, in particular as following from a Lagrangian) exist as well as different possibilities for the identification of physical observables with the mathematical objects of the formalism. Thus, in general relativity, the field equations are derived from the Lagrangian
where R(g_{ik}) is the Ricci scalar, g:=det g_{ik}, Î the cosmological constant, and L_{M} the matter Lagrangian depending on the metric, its first derivatives, and the matter variables. This Lagrangian leads to the wellknown field equations of general relativity,
with the energymomentum(stress) tensor of matter
and \(\kappa = \tfrac{{8\pi G}}{{{c^4}}}\), where G is Newtonâs gravitational constant. \({G^{ik}}: = {R^{ik}}  \tfrac{1}{2}R{g^{ik}}\) is called the Einstein tensor. In empty space, i.e., for T^{ik}= 0, Equation (92) reduces to
If only an electromagnetic field \({F_{ik}} = \frac{{\partial {A_k}}}{{\partial {x^i}}}  \frac{{\partial {A_i}}}{{\partial {x^k}}}\) derived from the 4vector potential A_{k} is present in the energymomentum tensor, then the Einstein.Maxwell equations follow:
The components of the metrical tensor are identified with gravitational potentials. Consequently, the components of the (LeviCivita) connection correspond to the gravitational âfield strengthâ, and the components of the curvature tensor to the gradients of the gravitational field. The equations of motion of material particles should follow, in principle, from Equation (92) through the relation
implied by it^{Footnote 57}. For point particles, due to the singularities appearing, in general this is a tricky task, up to now solved only approximately. However, the world lines for point particles falling freely in the gravitational field are, by definition, the geodesics of the Riemannian metric. This definition is consistent with the rigourous derivation of the geodesic equation for noninteracting dust particles in a fluid matter description. It is also consistent with all observations.
For most of the unified field theories to be discussed in the following, such identifications were made on internal, structural reasons, as no linkup to empirical data was possible. Due to the inherent wealth of constructive possibilities, unified field theory never would have come off the ground proper as a physical theory even if all the necessary formal requirements could have been satisfied. As an example, we take the identification of the electromagnetic field tensor with either the skew part of the metric, in a âmixed geometryâ with metric compatible connection, or the skew part of the Ricci tensor in metricaffine theory, to list only two possibilities. The latter choice obtains likewise in a purely affine theory in which the metric is a derived secondary concept. In this case, among the many possible choices for the metric, one may take it proportional to the variational derivative of the Lagrangian with respect to the symmetric part of the Ricci tensor. This does neither guarantee the proper signature of the metric nor its full rank. Several identifications for the electromagnetic 4potential and the electric current vector density have also been suggested (cf. below and [143]).
Number field
Complex fields may also be introduced on a real manifold. Such fields have also been used for the construction of unified field theories, although mostly after the period dealt with here (cf. Part II, in preparation). In particular, manifolds with a complex fundamental form were studied, e.g., with \({g_{ik}} = {s_{ik}} + i{a_{ik}}\), where \(i = \sqrt {  1}\) [97]. Also, geometries based on Hermitian forms were studied [313]. In later periods, hypercomplex numbers, quaternions, and octonions also were used as basic number fields for gravitational or unified theories (cf. Part II, forthcoming).
In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive, e.g., at complex manifolds and so on. In this part of the article we do not need to take into account this generalisation.
Dimension
Since the suggestions by NordstrĂ¶m and Kaluza [238, 181], manifolds with D>4 have been used for unified field theories. In most of the cases, the additional dimensions were taken to be spacelike; nevertheless, manifolds with more than one direction of time also have been studied.
Early Attempts at a Unified Field Theory
First steps in the development of unified field theories
Even before (or simultaneously with) the introduction and generalisation of the concept of parallel transport and covariant derivative by Hessenberg (1916/17) [160], LeviCivita (1917), [204], Schouten (1918) [294], Weyl (1918) [397], and KĂ¶nig (1919) [193], the introduction of an asymmetric metric was suggested by Rudolf FĂ¶rster^{Footnote 58} in 1917. In his letter to Einstein of 11 November 1917, he writes ([321], Doc. 398, p. 552):
âPerhaps, there exists a covariant 6vector by which the appearance of electricity is explained and which springs lightly from the \({g_{\mu \nu }}\), not forced into it as an alien element.â
^{Footnote 59} Einstein replied:
âThe aim of dealing with gravitation and electricity on the same footing by reducing both groups of phenomena to \({g_{\mu \nu }}\) has already caused me many disappointments. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence. But it is easier to suspect something than to discover it.â^{Footnote 60} (16 November 1917 [321], Vol. 8A, Doc. 400, p. 557)
In his next letter, FĂ¶rster gave results of his calculations with an asymmetric \({g_{\mu \nu }} = {s_{\mu \nu }} + {a_{\mu \nu }}\)introduced an asymmetric âthreeindexsymbolâ and a possible generalisation of the Riemannian curvature tensor as well as tentative Maxwellâs equations and interpretations for the 4potential \({A_\mu }\), and special solutions (28 December 1917) ([321], Volume 8A, Document 420, pp. 5817#x2013;587). Einsteinâs next letter of 17 January 1918 is skeptical:
âSince long, I also was busy by starting from a nonsymmetric \({g_{\mu \nu }}\); however, I lost hope to get behind the secret of unity (gravitation, electromagnetism) in this way. Various reasons instilled in me strong reservations: [âŠ] your other remarks are interesting in themselves and new to me.â^{Footnote 61} ([321], Volume 8B, Document 439, pp. 610â611)
Einsteinâs remarks concerning his previous efforts must be seen under the aspect of some attempts at formulating a unified field theory of matter by G. Mie [229, 230, 231]^{Footnote 62}, J. Ishiwara, and G. NordstrĂ¶m, and in view of the unified field theory of gravitation and electromagnetism proposed by David Hilbert.
âAccording to a general mathematical theorem, the electromagnetic equations (generalized Maxwell equations) appear as a consequence of the gravitational equations, such that gravitation and electrodynamics are not really different.â^{Footnote 63} (letter of Hilbert to Einstein of 13 November 1915 [162])
The result is contained in (Hilbert 1915, p. 397)^{Footnote 64}.
Einsteinâs answer to Hilbert on 15 November 1915 shows that he had also been busy along such lines:
âYour investigation is of great interest to me because I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism. The hints dropped by you on your postcards bring me to expect the greatest.â^{Footnote 65} [101]
Even before FĂ¶rster alias Bach corresponded with Einstein, a very early bird in the attempt at unifying gravitation and electromagnetism had published two papers in 1917, Reichenb#x00E4;cher [270, 269]. His paper amounts to a scalar theory of gravitation with field equation R=0 instead of Einsteinâs R_{ab}=0 outside the electrons. The electron is considered as an extended body in the sense of LorentzPoincarĂ©, and described by a metric joined continuously to the outside metric^{Footnote 66}:
ReichenbĂ€cher, at this point, seems to have had a limited understanding of general relativity: He thinks in terms of a variable velocity of light; he equates coordinate systems and reference systems, and apparently considers the transition from the Minkowskian to a nonflat metric as achieved by a coordinate rotation, a âDrehung gegen den Normalzustandâ (ârotation with respect to the normal stateâ) ([270], p. 137). According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:
âThe disturbance, which is generated by the electrons and which forces us to adopt a coordinate system different from the usual one, is interpreted as the electromagnetic sixvector, as is known.â^{Footnote 67} ([270], p. 136)
By his âcoordinate rotationâ, or, as he calls it in ([269], p. 174), âelectromagnetic rotationâ, he tries to geometrize the electromagnetic field. As Weylâs remark in RaumZeitMaterie ([398], p. 267, footnote 30) shows, he did not grasp ReichenbĂ€cherâs reasoning; I have not yet understood it either. Apparently, for ReichenbĂ€cher the metric deviation from Minkowski space is due solely to the electromagnetic field, whereas gravitation comes in by a single scalar potential connected to the velocity of light. He claims to obtain the same value for the perihelion shift of Mercury as Einstein ([269], p. 177). ReichenbĂ€cher was slow to fully accept general relativity; as late as in 1920 he had an exchange with Einstein on the foundations of general relativity [271, 71].
After ReichenbĂ€cher had submitted his paper to Annalen der Physik and seemingly referred to Einstein,
âPlanck was uncertain to which of Einsteinâs papers ReichenbĂ€cher appealed. He urged that ReichenbĂ€cher speak with Einstein and so dissolve their differences. The meeting was amicable. ReichenbĂ€cherâs paper appeared in 1917 as the first attempt at a unified field theory in the wake of Einsteinâs covariant field equations.â ([262], p. 208)
In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [126]. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [274].
Early disagreement about how to explain elementary particles by field theory
In his book on Einsteinâs relativity, Max Born, in 1920, had asked about the forces hindering âan electron or an atomâ to disintegrate.
âNow, these objects are tremendous concentrations of energy in the smallest place; therefore, they will house huge curvatures of space or, in other words, gravitational fields. The idea that they keep together the dispersing electrical charges lies close at hand.â^{Footnote 68} ([19], p. 235)
Thus, the idea of a program for building the extended constituents of matter from the fields the source of which they are, was very much alive around 1920. However, Pauliâs remark after Weylâs lecture in Bad Nauheim (86. Naturforscherversammlung, 19â25 September 1920) [245] showed that not everybody was a believer in it. He claimed that in bodies smaller than those carrying the elementary charge (electrons), an electric field could not be measured. There was no point of creating the âinteriorâ of such bodies with the help of an electric field. Pauli:
âNone of the present theories of the electron, also not Einsteinâs (Einstein 1919 [70]), up to now did achieve solving satisfactorily the problem of the electrical elementary quanta; it seems obvious to look for a deeper reason for this failure. I wish to see this reason in the fact that it is altogether not permitted to describe the electromagnetic field in the interior of an electron as a continuous space function. The electrical field is defined as the force on a charged test particle, and if no smaller test particles exist than the electron (vice versa the nucleus), the concept of electrical field at a certain point in the interior of the electron â with which all continuum theories are working â seems to be an empty fiction, because there are no arbitrarily small measures. Therefore, Iâd like to ask Mr. Einstein whether he approves of the opinion that a solution of the problem of matter may be expected only from a modification of our perception of space (perhaps also of time) and of electricity in the sense of atomism, or whether he thinks that the mentioned reservations are unconvincing and is of the opinion that the fundaments of continuum theory must be upheld.â^{Footnote 69}
Pauli referred to Einsteinâs paper about elementary particles and field theory in which he had exchanged his famous field equations for traceless equations with the electromagnetic field tensor as a source. Einsteinâs answer is tentative and evasive: We just donât know yet^{Footnote 70}.
âWith the progressing refinement of scientific concepts, the manner by which concepts are related to (physical) events becomes ever more complicated. If, in a certain stage of scientific investigation, it is seen that a concept can no longer be linked with a certain event, there is a choice to let the concept go, or to keep it; in the latter case, we are forced to replace the system of relations among concepts and events by a more complicated one. The same alternative obtains with respect to the concepts of timeand spacedistances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me.â^{Footnote 71}
In the same discussion Gustav Mie came back to FĂ¶rsterâs idea of an asymmetric metric but did not like it
â[âŠ] that an antisymmetric tensor was added to the symmetric tensor of the gravitational potential, which represented the sixvector of the electromagnetic field. But a more precise reasoning shows that in this way no reasonable world function is obtained.â^{Footnote 72}
It is to be noted that Weyl, at the end of 1920, already had given up on a possible field theory of matter:
âFinally I cut loose firmly from Mieâs theory and arrived at another position with regard to the problem of matter. To me, field physics no longer appears as the key to reality; in contrary, the field, the ether, for me simply is the totally powerless transmitter of causations, yet matter is a reality beyond the field and causes its states.â^{Footnote 73} (letter of Weyl to F. Klein on 28 December 1920, see [293], p. 83)
In the next year, Einstein had partially absorbed Pauliâs view but still thought it to be useful to apply field theory to the constituents of matter:
âThe physical interpretation of geometry (theory of the continuum) presented here, fails in its direct application to spaces of submolecular scale. Yet it retains part of its meaning also with regard to questions concerning the constitution of elementary particles. Because one may try to ascribe to these field concepts [âŠ] a physical meaning even if a description of the electrical elementary particles which constitute matter is to be made. Only success can decide whether such a procedure finds its justification [âŠ].â^{Footnote 74} [72]
During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:
âIn the program, Mr. Einstein expressed during his two talks given in November 1929 at the Institut Henri PoincarĂ©, he wished to search for the physical laws in solutions of his equations without singularities â with matter and the electromagnetic field thus being continuous. Let us move into the field chosen by him without too much surprise to see him apparently follow a road opposed to the one successfully walked by the contemporary physicists.â^{Footnote 75} ([36], p. 17 (1178))
The Main Ideas for Unification between about 1918 and 1923
After 1915, Einstein first was busy with extracting mathematical and physical consequences from general relativity (Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves). Although he kept thinking about how to find elementary particles in a field theory [70] and looked closer into Weylâs theory [72], at first he only reacted to the new ideas concerning unified field theory as advanced by others. The first such idea after FĂ¶rsterâs, of course, was Hermann Weylâs gauge approach to gravitation and electromagnetism, unacceptable to Einstein and to Pauli for physical reasons [246, 292].
Next came Kaluzaâs fivedimensional unification of gravitation and electromagnetism, and Eddingtonâs affine geometry.
Weylâs theory
The geometry
Weylâs fundamental idea for generalising Riemannian geometry was to note that, unlike for the comparison of vectors at different points of the manifold, for the comparison of scalars the existence of a connection is not required. Thus, while lengths of vectors at different points can be compared without a connection, directions cannot. This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:
âIf we make no further assumption, the points of a manifold remain totally isolated from each other with regard to metrical structure. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given.â ^{Footnote 76}
In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance.
âHowever, the possibility of such a comparison âat a distanceâ in no way can be admitted in a pure infinitesimal geometry.â ^{Footnote 77} ([397], p. 397)
In order to invent a purely âinfinitesimalâ geometry, Weyl introduced the 1dimensional, Abelian group of gauge transformations,
besides the diffeomorphism group (coordinate transformations). At a point, Equation (98) induces a local recalibration of lengths l while preserving angles, i.e., ÎŽl=Î»l. If the nonmetricity tensor is assumed to have the special form Q_{ijk}=Q_{k}g_{ij}, with an arbitrary vector field Q_{k}, then as we know from Equation (57), with regard to these gauge transformations
We see a striking resemblance with the electromagnetic gauge transformations for the vector potential in Maxwellâs theory. If, as Weyl does, the connection is assumed to be symmetric (i.e., with vanishing torsion), then from Equation (42) we get
Thus, unlike in Riemannian geometry, the connection is not fully determined by the metric but depends also on the arbitrary vector function Q_{i}, which Weyl wrote as a linear form dQ=Q^{i}dx^{i}. With regard to the gauge transformations (98), \({\Gamma _{ij}}^k\) remains invariant. From the 1form dQ, by exterior derivation a gaugeinvariant 2form \(F = {F_{ij}}d{x^i}\,\Lambda \;d{x^j}\) with \(F = {Q_{i,j}}  {Q_{j,i}}\) follows. It is named âStreckenkrĂŒmmungâ (âline curvatureâ) by Weyl, and, by identifying Q with the electromagnetic 4potential, he arrived at the electromagnetic field tensor F.
Let us now look at what happens to parallel transport of a length, e.g., the norm X of a tangent vector along a particular curve C with parameter u to a different (but infinitesimally neighbouring) point:
By a proper choice of the curveâs parameter, we may write (101) in the form \(d\left X \right =  {Q_k}{X^k}\left X \rightdu\) and integrate along C to obtain \(X = \int {\exp (  {Q_k}{X^k}du).} \). If X is taken to be tangent to C, i.e., \({X^k} = \tfrac{{d{x^k}}}{{{du}}}\), then
i.e., the length of a vector is not integrable; its value generally depends on the curve along which it is parallely transported. The same holds for the angle between two tangent vectors in a point (cf. Equation (44)). For a vanishing electromagnetic field, the 4potential becomes a gradient (âpure gaugeâ), such that \({Q_k}d{x^k} = \tfrac{{\partial \omega }}{{\partial {x^k}}}d{x^k} = d\omega\), and the integral becomes independent of the curve.
Thus, in Weylâs connection (100), both the gravitational and the electromagnetic fields, represented by the metrical field g and the vector field Q, are intertwined. Perhaps, having in mind Mieâs ideas of an electromagnetic world view and Hilbertâs approach to unification, in the first edition of his book, Weyl remained reserved:
âAgain physics, now the physics of fields, is on the way to reduce the whole of natural phenomena to one single law of nature, a goal to which physics already once seemed close when the mechanics of masspoints based on Newtonâs Principia did triumph. Yet, also today, the circumstances are such that our trees do not grow into the sky.â^{Footnote 78} ([396], p. 170; preface dated âEaster 1918â)
However, a little later, in his paper accepted on 8 June 1918, Weyl boldly claimed:
âI am bold enough to believe that the whole of physical phenomena may be derived from one single universal worldlaw of greatest mathematical simplicity.â^{Footnote 79} ([397], p. 385, footnote 4)
The adverse circumstances alluded to in the first quotation might be linked to the difficulties of finding a satisfactory Lagrangian from which the field equations of Weylâs theory can be derived. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gaugeweight within tensor calculus as in Section 2.1.5^{Footnote 80}. As the Lagrangian \({\mathcal L} = \sqrt {  g} L\) must have gaugeweight w=0, we are looking for a scalar L of gaugeweight 2. WeitzenbĂ¶ck^{Footnote 81} has shown that the only possibilities quadratic in the curvature tensor and the line curvature are given by the four expressions [391]
While the last invariant would lead to Maxwellâs equations, from the invariants quadratic in curvature, in general field equations of fourth order result.
Weyl did calculate the curvature tensor formed from his connection (100) but did not get the correct result^{Footnote 82}; it is given by Schouten ([310], p. 142) and follows from Equation (51):
If the metric field g and the 4potential Q_{i}âĄA_{i} are varied independently, from each of the curvaturedependent scalar invariants we do get contributions to Maxwellâs equations.
Perhaps Bach (alias FĂ¶rster) was also dissatisfied with Weylâs calculations: He went through the entire mathematics of Weylâs theory, curvature tensor, quadratic Lagrangian field equations and all; he even discussed exact solutions. His Lagrangian is given by \({\mathcal L} = \sqrt g (3{W_4}  6{W_3} + {W_2})\), where the invariants are defined by
with
where R_{pqik} is the Riemannian curvature tensor, \({f_{ik}} = {f_{i,k}}  {f_{k,i}}\), and f_{i} is the electromagnetic 4potential [4].
Physics
While Weylâs unification of electromagnetism and gravitation looked splendid from the mathematical point of view, its physical consequences were dire: In general relativity, the line element ds had been identified with space and time intervals measurable by real clocks and real measuring rods. Now, only the equivalence class {Î»g_{ik}Î» arbitrary} was supposed to have a physical meaning: It was as if clocks and rulers could be arbitrarily âregaugedâ in each event, whereas in Einsteinâs theory the same clocks and rulers had to be used everywhere. Einstein, being the first expert who could keep an eye on Weylâs theory, immediately objected, as we infer from his correspondence with Weyl.
In spring 1918, the first edition of Weylâs famous book on differential geometry, special and general relativity RaumZeitMaterie appeared, based on his course in ZĂŒrich during the summer term of 1917 [396]. Weyl had arranged that the page proofs be sent to Einstein. In communicating this on 1 March 1918, he also stated that
âAs I believe, during these days I succeeded in deriving electricity and gravitation from the same source. There is a fully determined action principle, which, in the case of vanishing electricity, leads to your gravitational equations while, without gravity, it coincides with Maxwellâs equations in first order. In the most general case, the equations will be of 4th order, though.â^{Footnote 83}
He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy ([321], Volume 8B, Document 472, pp. 663â664). At the end of March, Weyl visited Einstein in Berlin, and finally, on 5 April 1918, he mailed his note to him for the Berlin Academy. Einstein was impressed: In April 1918, he wrote four letters and two postcards to Weyl on his new unified field theory â with a tone varying between praise and criticism. His first response of 6 April 1918 on a postcard was enthusiastic:
âYour note has arrived. It is a stroke of genious of first rank. Nevertheless, up to now I was not able to do away with my objection concerning the scale.â^{Footnote 84} ([321], Volume 8B, Document 498, 710)
Einsteinâs âobjectionâ is formulated in his âAddendumâ (âNachtragâ) to Weylâs paper in the reports of the Academy, because Nernst had insisted on such a postscript. There, Einstein argued that if light rays would be the only available means for the determination of metrical relations near a point, then Weylâs gauge would make sense. However, as long as measurements are made with (infinitesimally small) rigid rulers and clocks, there is no indeterminacy in the metric (as Weyl would have it): Proper time can be measured. As a consequence follows: If in nature length and time would depend on the prehistory of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i.e., the frequencies would depend on the location of the emitter. He concluded with the words
âRegrettably, the basic hypothesis of the theory seems unacceptable to me, [of a theory] the depth and audacity of which must fill every reader with admiration.â^{Footnote 85} ([395], Addendum, p. 478)
Einsteinâs remark concerning the pathdependence of the frequencies of spectral lines stems from the pathdependency of the integral (102) given above. Only for a vanishing electromagnetic field does this objection not hold.
Weyl answered Einsteinâs comment to his paper in a âreply of the authorâ affixed to it. He doubted that it had been shown that a clock, if violently moved around, measures proper time â« ds. Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:
âThe most plausible assumption that can be made for a clock resting in a static field is this: that it measure the integral of the ds normed in this way [i.e., as in Einsteinâs theory]; the task remains, in my theory as well as in Einsteinâs, to derive this fact by a dynamics carried through explicitly.â^{Footnote 86} ([395], p. 479)
Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i.e., to give a theory of clocks and rulers within general relativity. Presumably, such a theory would have to include microphysics. In a letter to his former student Walter DĂ€llenbach, he wrote (after 15 June 1918):
â[Weyl] would say that clocks and rulers must appear as solutions; they do not occur in the foundation of the theory. But I find: If the ds, as measured by a clock (or a ruler), is something independent of prehistory, construction and the material, then this invariant as such must also play a fundamental role in theory. Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and welldefined chemical elements would not exist. [âŠ] In any case, I am as convinced as Weyl that gravitation and electricity must let themselves be bound together to one and the same; I only believe that the right union has not yet been found.â^{Footnote 87} ([321], Volume 8B, Document 565, 803)
Another famous theoretician who could not side with Weyl was H. A. Lorentz; in a paper on the measurement of lengths and time intervals in general relativity and its generalisations, he contradicted Weylâs statement that the worldlines of lightsignals would suffice to determine the gravitational potentials [211].
However, Weyl still believed in the physical value of his theory. As further âextraordinarily strong support for our hypothesis of the essence of electricityâ he considered the fact that he had obtained the conservation of electric charge from gaugeinvariance in the same way as he had linked with coordinateinvariance earlier, what at the time was considered to be âconservation of energy and momentumâ, where a nontensorial object stood in for the energymomentum density of the gravitational field ([398], pp. 252â253).
Moreover, Weyl had some doubts about the general validity of Einsteinâs theory which he derived from the discrepancy in value by 20 orders of magniture of the classical electron radius and the gravitational radius corresponding to the electronâs mass ([397], p. 476; [152]).
Reactions to Weylâs theory I: Einstein and Weyl
There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [321]. We subsume some of the relevant discussions. Even before Weylâs note was published by the Berlin Academy on 6 June 1918, many exchanges had taken place between him and Einstein.
On a postcard to Weyl on 8 April 1918, Einstein reaffirmed his admiration for Weylâs theory, but remained firm in denying its applicability to nature. Weyl had given an argument for dimension 4 of spacetime that Einstein liked: As the Lagrangian for the electromagnetic field F_{ik}F^{ik} is of gaugeweight 2 and \(\sqrt {  g}\) has gaugeweight D/2 in an M_{D}, the integrand in the Hamiltonian principle \(\sqrt {  g} {F_{ik}}{F^{ik}}\) can have weight zero only for D=4: âApart from the [lacking] agreement with reality it is in any case a grandiose intellectual performanceâ^{Footnote 88} ([321], Vol. 8B, Doc. 499, 711). Weyl did not give in:
âYour rejection of the theory for me is weighty; [âŠ] But my own brain still keeps believing in it. And as a mathematician I must by all means hold to [the fact] that my geometry is the true geometry âin the nearâ, that Riemann happened to come to the special case F_{ik}=0 is due only to historical reasons (its origin is the theory of surfaces), not to such that matter.â^{Footnote 89} ([321], Volume 8B, Document 544, 767)
After Weylâs next paper on âpure infinitesimal geometryâ had been submitted, Einstein put forward further arguments against Weylâs theory. The first was that Weylâs theory preserves the similarity of geometric figures under parallel transport, and that this would not be the most general situation (cf. Equation (49)). Einstein then suggested the affine group as the more general setting for a generalisation of Riemannian geometry ([321], Vol. 8B, Doc. 551, 777). He repeated this argument in a letter to his friend Michele Besso from his vacations at the Baltic Sea on 20 August 1918, in which he summed up his position with regard to Weylâs theory:
â[Weylâs] theoretical attempt does not fit to the fact that two originally congruent rigid bodies remain congruent independent of their respective histories. In particular, it is unimportant which value of the integral \(\int {\phi _\nu }d{x_\nu }\) is assigned to their world line. Otherwise, sodium atoms and electrons of all sizes would exist. But if the relative size of rigid bodies does not depend on past history, then a measurable distance between two (neighbouring) worldpoints exists. Then, Weylâs fundamental hypothesis is incorrect on the molecular level, anyway. As far as I can see, there is not a single physical reason for it being valid for the gravitational field. The gravitational field equations will be of fourth order, against which speaks all experience until now [âŠ].â^{Footnote 90} ([99], p. 133)
Einsteinâs remark concerning âaffine geometryâ is referring to the affine geometry in the sense it was introduced by Weyl in the 1st and 2nd edition of his book [396], i.e., through the affine group and not as a suggestion of an affine connexion.
From Einsteinâs viewpoint, in Weylâs theory the line element ds is no longer a measurable quantity â the electromagnetical 4potential never had been one. Writing from his vacations on 18 September 1918, Weyl presented a new argument in order to circumvent Einsteinâs objections. The quadratic form Rg_{ik}dx^{i}dx^{k} is an absolute invariant, i.e., also with regard to gauge transformations (gauge weight 0). If this expression would be taken as the measurable distance in place of ds, then
â[âŠ] by the prefixing of this factor, so to speak, the absolute norming of the unit of length is accomplished after allâ^{Footnote 91} ([321], Volume 8B, Document 619, 877â879)
Einstein was unimpressed:
âBut the expression Rg_{ik}dx^{i}dx^{k} for the measured length is not at all acceptable in my opinion because R is very dependent on the matter density. A very small change of the measuring path would strongly influence the integral of the square root of this quantity.â^{Footnote 92}
Einsteinâs argument is not very convincing: g_{ik} itself is influenced by matter through his field equations; it is only that now R is algebraically connected to the matter tensor. In view of the more general quadratic Lagrangian needed in Weylâs theory, the connection between R and the matter tensor again might become less direct. Einstein added:
âOf course I know that the state of the theory as I presented it is not satisfactory, not to speak of the fact that matter remains unexplained. The unconnected juxtaposition of the gravitational terms, the electromagnetic terms, and the Î»terms undeniably is a result of resignation.[âŠ] In the end, things must arrange themselves such that actiondensities need not be glued together additively.â^{Footnote 93} ([321], Volume 8B, Document 626, 893â894)
The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.
Reactions to Weylâs theory II: Schouten, Pauli, Eddington, and others
Sommerfeld seems to have been convinced by Weylâs theory, as his letter to Weyl on 3 June 1918 shows:
âWhat you say here is really marvelous. In the same way in which Mie glued to his consequential electrodynamics a gravitation which was not organically linked to it, Einstein glued to his consequential gravitation an electrodynamics (i.e., the usual electrodynamics) which had not much to do with it. You establish a real unity.â^{Footnote 94} [327]
Schouten, in his attempt in 1919 to replace the presentation of the geometrical objects used in general relativity in local coordinates by a âdirect analysisâ, also had noticed Weylâs theory. In his âaddendum concerning the newest theory of Weylâ, he came as far as to show that Weylâs connection is gauge invariant, and to point to the identification of the electromagnetic 4potential. Understandably, no comments about the physics are given ([295], pp. 89â91).
In the section on Weylâs theory in his article for the Encyclopedia of Mathematical Sciences, Pauli described the basic elements of the geometry, the loss of the lineelement ds as a physical variable, the convincing derivation of the conservation law for the electric charge, and the too many possibilities for a Lagrangian inherent in a homogeneous function of degree 1 of the invariants (103). As compared to his criticism with respect to Eddingtonâs and Einsteinâs later unified field theories, he is speaking softly, here. Of course, as he noted, no progress had been made with regard to the explanation of the constituents of matter; on the one hand because the differential equations were too complicated to be solved, on the other because the observed mass difference between the elementary particles with positive and negative electrical charge remained unexplained. In his general remarks about this problem at the very end of his article, Pauli points to a link of the asymmetry with timereflection symmetry (see [246], pp. 774â775; [244]). For Einstein, this criticism was not only directed against Weylâs theory
âbut also against every continuumtheory, also one which treats the electron as a singularity. Now as before I believe that one must look for such an overdetermination by differential equations that the solutions no longer have the character of a continuum. But how?â ([103], p. 43)
In a letter to Besso on 26 July 1920, Einstein repeated an argument against Weylâs theory which had been removed by Weyl â if only by a trick to be described below; Einstein thus said:
âOne must pass to tensors of fourth order rather than only to those of second order, which carries with it a vast indeterminacy, because, first, there exist many more equations to be taken into account, second, because the solutions contain more arbitrary constants.â^{Footnote 95} ([99], p. 153)
In his book âSpace, Time, and Gravitationâ, Eddington gave a nontechnical introduction into Weylâs âwelding together of electricity and gravitation into one geometryâ. The idea of gauging lengths independently at different events was the central theme. He pointed out that while the fourfold freedom in the choice of coordinates had led to the conservation laws for energy and momentum, âin the new geometry is a fifth arbitrariness, namely that of the selected gaugesystem. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge.â One natural gauge was formed by the âradius of curvature of the worldâ; âthe electron could not know how large it ought to be, unless it had something to measure itself againstâ ([57], pp. 174, 173, 177).
As Eddington distinguished natural geometry and actual space from world geometry and conceptual space serving for a graphical representation of relationships among physical observables, he presented Weylâs theory in his monograph âThe mathematical theory of relativityâ
âfrom the wrong end â as its author might consider; but I trust that my treatment has not unduly obscured the brilliance of what is unquestionably the greatest advance in the relativity theory after Einsteinâs work.â ([59], p. 198)
Of course, âwrong endâ meant that Eddington took Weylâs theory such
âthat his nonRiemannian geometry is not to be applied to actual spacetime; it refers to a graphical representation of that relationstructure which is the basis of all physics, and both electromagnetic and metrical variables appear in it as interrelated.â ([59], p. 197)
Again, Eddington liked Weylâs natural gauge encountered in Section 4.1.5, which made the curvature scalar a constant, i.e., K=4Î»; it became a consequence of Eddingtonâs own natural gauge in his affine theory, K_{ij}=Î»g_{ij} (cf. Section 4.3). For Eddington, Weylâs theory of gaugetransformation was a hybrid:
âHe admits the physical comparison of length by optical methods [âŠ]; but he does not recognise physical comparison of length by material transfer, and consequently he takes Î» to be a function fixed by arbitrary convention and not necessarily a constant.â ([59], pp. 220â221)
In the depth of his heart Weyl must have kept a fondness for his idea of âgaugingâ a field all during the decade between 1918 and 1928. As he had abandoned the idea of describing matter as a classical field theory since 1920, the linking of the electromagnetic field via the gauge idea could only be done through the matter variables. As soon as the new spinorial wave function (âmatter waveâ) in SchrĂ¶dingerâs and Diracâs equations emerged, he adapted his idea and linked the electromagnetic field to the gauging of the quantum mechanical wave function [407, 408]. In October 1950, in the preface for the first American printing of the English translation of the fourth edition of his book Space, Time, Matter from 1922, Weyl clearly expressed that he had given up only the particular idea of a link between the electromagnetic field and the local calibration of length:
âWhile it was not difficult to adapt also Maxwellâs equations of the electromagnetic field to this principle [of general relativity], it proved insufficient to reach the goal at which classical field physics is aiming: a unified field theory deriving all forces of nature from one common structure of the world and one uniquely determined law of action.[âŠ] My book describes an attempt to attain this goal by a new principle which I called gauge invariance. (Eichinvarianz). This attempt has failed.â ([410], p. V)
Reactions to Weylâs theory III: Further research
Pauli, still a student, and with his article for the Encyclopedia in front of him, pragmatically looked into the gravitational effects in the planetary system, which, as a consequence of Einsteinâs field equations, had helped Einstein to his fame. He showed that Weylâs theory had, for the static case, as a possible solution a constant Ricci scalar; thus it also admitted the Schwarzschild solution and could reproduce all desired effects [244, 243].
Weyl himself continued to develop the dynamics of his theory. In the third edition of his SpaceTimeMatter [398], at the Naturforscherversammlung in Bad Nauheim in 1920 [399], and in his paper on âthe foundations of the extended relativity theoryâ in 1921 [402], he returned to his new idea of gauging length by setting R=Î»= const. (cf. Section 4.1.3); he interpreted Î» to be the âradius of curvatureâ of the world. In 1919, Weylâs Lagrangian originally was \({\mathcal L} = \tfrac{1}{2}\sqrt g {K^2} + \beta {F_{ik}}{F^{ik}}\) together with the constraint K=2Î», with constant Î» ([398], p. 253). As an equivalent Lagrangian Weyl gave, up to a divergence^{Footnote 96}
with the 4potential Ï and the electromagnetic field F_{ik}. Due to his constraint, Weyl had navigated around another problem, i.e., the formulation of the Cauchy initial value problem for field equations of fourth order: Now he had arrived at second order field equations. In the paper in 1921, he changed his Lagrangian slightly into
with â a factor in Weylâs connection (100),
In both presentations, he considered as an advantage of his theory:
âMoreover, this theory leads to the cosmological term in a uniform and forceful manner, [a term] which in Einsteinâs theory was introduced ad hocâ^{Footnote 97} ([402], p. 474)
ReichenbĂ€cher seemingly was unhappy about Weylâs taking the curvature scalar to be a constant before the variation; in the discussion after Weylâs talk in 1920, he inquired whether one could not introduce Weylâs ânatural gaugeâ after the variation of the Lagrangian such that the field equations would show their gauge invariance first ([399], p. 651). Eddington criticised Weylâs choice of a Lagrangian as speculative:
âAt the most we can only regard the assumed form of action [âŠ] as a step towards some more natural combination of electromagnetic and gravitational variables.â ([59], p. 212)
The changes, which Weyl had introduced in the 4th edition of his book [401], and which, according to him, were of fundamental importance for the understanding of relativity theory, were discussed by him in a further paper [400]. In connection with the question of whether, in general relativity, a formulation might be possible such that âmatter whose characteristical traits are charge, mass, and motion generates the fieldâ, a question which was considered as unanswered by Weyl, he also mentioned a publication of ReichenbĂ€cher [272]. For Weyl, knowledge of the charge and mass of each particle, and of the extension of their âworldchannelsâ were insufficient to determine the field uniquely. Weylâs hint at a solution remains dark; nevertheless, for him it meant
âto reconciliate ReichenbĂ€cherâs idea: matter causes a âdeformationâ of the metrical field and Einsteinâs idea: inertia and gravitation are one.â ([400], p. 561, footnote)
Although Einstein could not accept Weylâs theory as a physical theory, he cherished âits courageous mathematical constructionâ and thought intensively about its conceptual foundation: This becomes clear from his paper âOn a complement at hand of the bases of general relativityâ of 1921 [73]. In it, he raised the question whether it would be possible to generate a geometry just from the conformal invariance of Equation (9) without use of the conception âdistanceâ, i.e., without using rulers and clocks. He then embarked on conformal invariants and tensors of gaugeweight 0, and gave the one formed from the square of Weylâs conformal curvature tensor (59), i.e. \({C^i}_{jkl}C_i^{jkl}\). His colleague in Vienna, Wirtinger^{Footnote 98}, had helped him in this^{Footnote 99}. Einsteinâs conclusion was that, by writing down a metric with gaugeweight 0, it was possible to form a theory depending only on the quotient of the metrical components. If J has gaugeweight 1, then Jg_{ik} is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation
would have to be solved.
Eisenhart wished to partially reinterpret Weylâs theory: In place of putting the vector potential equal to Weylâs gauge vector, he suggested to identify it with \(\tfrac{{  F_k^i{J^k}}}{\mu }\), where J^{i} is the electrical 4current vector (density) and ÎŒ the mass density. He referred to Weyl, Eddingtonâs book, and to Pauliâs article in the Encyclopedia of Mathematical Sciences [116].
Einsteinâs rejection of the physical value of Weylâs theory was seconded by Dienes^{Footnote 100}, if only with a not very helpful argument. He demanded that the connection remain metriccompatible from which, trivially, Weylâs gaugevector must vanish. Dienes applied the same argument to Eddingtonâs generalisation of Weylâs theory [51]. Other mathematicians took Weylâs theory at its face value and drew consequences; thus M. Juvet calculated Frenetâs formulas for an ânĂšdreâ in Weylâs geometry by generalising a result of Blaschke for Riemannian geometry [180]. More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. Johnâs College in Cambridge, M. H. A. Newman [237]. In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge. Newman applied his scheme to a variational principle with Lagrangian K^{2} and concluded:
âThe part independent of the âelectricalâ vector Ï_{i} is found to be \({K_{ij}}  \tfrac{1}{4}K{g_{ij}}\), a tensor which has been considered by Einstein from time to time in connection with the theory of gravitation.â ([237], p. 623)
After the Second World War, research following Weylâs classical geometrical approach with his original 1dimensional Abelian gaugegroup was resumed. The more important development, however, was the extension to nonAbelian gaugegroups and the combination with Kaluzaâs idea. We shall discuss these topics in Part II of this article. The shift in Weylâs interpretation of the role of the gauging from the link between gravitation and electromagnetism to a link between the quantum mechanical state function and electromagnetism is touched on in Section 7.
Kaluzaâs fivedimensional unification
What is now called KaluzaKlein theory in the physics community is a mixture of quite different contributions by both scientists^{Footnote 101}. Kaluzaâs idea of looking at four spatial and one time dimension originated in or before 1919; by then he had communicated it to Einstein:
âThe idea of achieving [a unified field theory] by means of a fivedimensional cylinder world never dawned on me. [â] At first glance I like your idea enormously.â (letter of Einstein to Kaluza of 21 April 1919)
This remark is surprising because NordstrĂ¶m had suggested a fivedimensional unification of his scalar gravitational theory with electromagnetism five years earlier [238], by embedding spacetime into a fivedimensional world in quite the same way as Kaluza did. In principle, Einstein could have known NordstrĂ¶mâs work. In the same year 1914, he and Fokker had given a covariant formulation of NordstrĂ¶mâs pure (scalar) theory of gravitation [104]. In a subsequent letter to Kaluza of 5 May 1919 Einstein still was impressed: âThe formal unity of your theory is startling.â However, on 29 May 1919, Einstein became somewhat reserved^{Footnote 102}:
âI respect greatly the beauty and boldness of your idea. But you understand that, in view of the existing factual concerns, I cannot take sides as planned originally.â^{Footnote 103}
Kaluzaâs paper was communicated by Einstein to the Academy, but for reasons unknown was published only in 1921 [181]. Kaluzaâs idea was to write down the Einstein field equations for empty space in a fivedimensional Riemannian manifold with metric \({g_{\alpha \beta }}\), i.e., \({R_{\alpha \beta }} = 0\), Î±, ÎČ=1,âŠ, 5, where \({R_{\alpha \beta }}\) is the Ricci tensor of M_{5}, and to look at small deviations Îł from Minkowski space: \({g_{\alpha \beta }} =  {\delta _{\alpha \beta }} + {\gamma _{\alpha \beta }}\).^{Footnote 104}. In order to obtain a theory in spacetime, he assumed the socalled âcylinder conditionâ
equivalent to the existence of a spacelike translational symmetry (Killing vector). Equation (109) is used for all âfunctions of stateâ (ZustandsgrĂ¶ssen), i.e., also for the matter variables. Kaluza did not normalize the Killing vector to a constant, i.e., he kept
Equation (110) is called the âsharpened cylinder conditionâ by some authors including Einstein. Of the 15 components of \({g_{\alpha \beta }}\), five had to get a new physical interpretation, i.e. \({g_{\alpha 5}}\) and g_{55}; the components g_{ik}, i, k=1, âŠ, 4, were to describe the gravitational field as before; Kaluza took g_{i5} proportional to the electromagnetic vector potential A_{i}. The component g_{55} turned out to be a (scalar) gravitational potential which, in the static case, satisfies the equation
with the constant matter density ÎŒ_{0}.
Kaluza also showed that the geodesics of the fivedimensional space reduce to the equations of motion for a charged point particle in spacetime, if a weakness assumption is made for the components of the 5velocity \({u^\alpha }:{u^1},{u^2},{u^3},{u^5} \ll 1\), u^{4}â1. The Lorentz force appears augmented by an additional term containing g_{55} of the order \({\left( {\tfrac{u}{c}} \right)^2}\) which thus may be neglected. From the fifth equation of motion Kaluza concluded that the fifth component of momentum p_{5}âŒe, with e being the particlesâ electric charge (up to a constant of proportionality). From the equations of motion, charge conservation also followed in Kaluzaâs linear approximation. Kaluza was well aware that his theory broke down if applied to elementary particles like electrons or protons, and speculated about an escape in which gravitation had to be considered as some âdifference effectâ, and the gravitational constant given âa statistical meaningâ. For him, any theory claiming universal validity was endangered by quantum theory, anyway.
From the cylinder condition, a grave objection toward Kaluzaâs approach results: Covariance with regard to the diffeomorphism group of M_{5} is destroyed. The remaining covariance group G_{5} is given by
The objects transforming properly under (112) are: the scalar \({g_{5'5'}} = {g_{55}}\), the vectorpotential \({g_{5'k'}} = {g_{5l}}\frac{{\partial {x^l}}}{{\partial {x^{k'}}}} + {g_{55}}\frac{{\partial {x^5}}}{{\partial {x^{k'}}}}\), and the projected metric
Klein identified the group; however, he did not comment on the fact that now further invariants are available for a Lagrangian, but started right away from the Ricci scalar of M_{5} [185]. The group G_{5} is isomorphic to the group H_{5} of transformations for five homogeneous coordinates \({X^{\mu '}} = {f^\mu }({X^\nu })\) with \({f^\nu }\) homogeneous functions of degree 1. Here, contact is made to the projective formulation of Kaluzaâs theory (cf. âprojective geometryâ in Sections 2.1.3 and 6.3.2).
While towards the end of May 1919 Einstein had not yet fully supported the publication of Kaluzaâs manuscript, on 14 October 1921 he thought differently:
âI am having second thoughts about having kept you from the publication of your idea on the unification of gravitation and electricity two years ago. I value your approach more than the one followed by H. Weyl. If you wish, I will present your paper to the Academy after all.â^{Footnote 105} (letter from Einstein to Kaluza reprinted in [49], p. 454)
It seems that at some point Einstein had set his calculational aide Grommer^{Footnote 106} to work on regular spherically symmetric solutions of Kaluzaâs theory. This led to a joint publication which was submitted just one month after Einstein had finally presented a rewritten manuscript of Kaluzaâs to the Berlin Academy [105]. The negative result of his own paper, i.e., that no nonsingular, statical, spherically symmetric exact solution exists, did not please Einstein. He also thought that Kaluzaâs assumption of general covariance in the fivedimensional manifold had no support from physics; he disliked the preference of the fifth coordinate due to Equation (109) which seemed to contradict the equivalence of all five coordinates used by Kaluza in the construction of the field equations [105]. In any case, apart from an encouraging letter to Kaluza in 1925 in which he called Kaluzaâs idea the only serious attempt at unified field theory besides the WeylEddington approach, Einstein kept silent on the fivedimensional theory until 1926.
Eddingtonâs affine theory
Eddingtonâs paper
The third main idea that emerged was Eddingtonâs suggestion to forego the metric as a fundamental concept and start right away with a (general) connection, which he then restricted to a symmetric one Î in order to avoid an âinfinitely crinkledâ world [58]. His motivation went beyond the unification of gravitation and electromagnetism:
âIn passing beyond Euclidean geometry, gravitation makes its appearance; in passing beyond Riemannian geometry, electromagnetic force appears; what remains to be gained by further generalisation? Clearly, the nonMaxwellian binding forces which hold together an electron. But the problem of the electron must be difficult, and I cannot say whether the present generalisation succeeds in providing the material for its solutionâ ([58], p. 104)
In the first, shorter, part of two, Eddington describes affine geometry; in the second he relates mathematical objects to physical variables. He distinguishes the affine geometry as the âgeometry of the worldstructureâ from Riemannian geometry as âthe natural geometry of the worldâ. He starts by calculating both the curvature and Ricci tensors from the symmetric connection according to Equation (39). The Ricci tensor K^{ij}(Î):=*G^{ij} is asymmetric^{Footnote 107},
with R_{kl}(Î) being the symmetric and F_{kl}(Î) the antisymmetric part. According to Equation (31) F_{kl} derives from a âvector potentialâ, i.e., \({F_{kl}} = {\partial _k}{\Gamma _l}  {\partial _l}{\Gamma _k}\) with \({\Gamma _{lr}}^r\), such that an immediate physical identification of F_{kl} with the electromagnetic field tensor is at hand. With half of Maxwellâs equations being satisfied automatically, the other half is used to define the electric charge current j^{k} by \({j^l}: = {F^{lk}}_{\left\ k \right.}\). By this, Eddington claims to guarantee charge conservation:
Now, by Equation (25),
For a symmetric connection thus, unlike in Riemannian geometry,
However, for a tensor density, due to Equation (16) we obtain
and thus for a torsionless connection (cf. Equation (38)) \({j^k}_{\left\ k \right.} = 0.\)
Eddington introduces the metrical tensor by the definition
âintroducing a universal constant Î», for convenience, in order to remain free to use the centimetre instead of the natural unit of lengthâ. This is called âEinsteinâs gaugeâ by Eddington; he is delighted that
âOur gaugingequation is therefore certainly true wherever light is propagated, i.e., everywhere inside the electron. Who shall say what is the ordinary gauge inside the electron?â ([58], p. 114)
While this remark certainly is true, there is no guarantee in Eddingtonâs approach that g_{kl} thus defined is a Lorentzian metric, i.e., that it could describe light propagation at all. Only connections leading to a Lorentz metric can be used if a physical interpretation is wanted. Note also, that the interpretation of R_{kl} as the metric implies that det R_{kl}â 0.
We must read Equation (118) as giving g_{kl}(Î) if the only basic variable in affine geometry, i.e., the connection \({\Gamma _{ij}}^k\), has been determined by help of some field equations. Thus, in general, g_{kl} is not metriccompatible; in order to make it such, we are led to the differential equations \({R_{ij\left\ k \right.}} = 0\) for \({\Gamma _{ij}}^k\), an equation not considered by Eddington. In the absence of an electromagnetic field, Equation (118) looks like Einsteinâs vacuum field equation with cosmological constant. In principle, now a fictitious âRiemannianâ connection (the Christoffel symbol) can be written down which, however, is a horribly complicated function of the affine connection â as the only fundamental geometrical quantity available. This is due to the expression for the inverse of the metric, a function cubic in R_{kl}. Eddingtonâs affine theory thus can also be seen as a biconnection theory. Note also that Eddington does not explicitly say how to obtain the contravariant form of the electromagnetic field F^{ij} from F_{ij}; we must assume that he thought of raising indices with the complicated inverse metric tensor.
In connection with cosmological considerations, Eddington cherished the Î»term in Equation (118):
âI would as soon think of reverting to Newtonian theory as of dropping the cosmic constant.â ([63], p. 35)
Now, Eddington was able to identify the energymomentum tensor T^{ik} of the electromagnetic field by decomposing the Ricci tensor K^{ij} formed from Equation (51) into a metric part R_{ik} and the rest. The energymomentum tensor T^{ik} of the electromagnetic field is then defined by Einsteinâs field equations with a fictitious cosmological constant \(\kappa {T^{ik}}: = {G^{ik}}  \tfrac{1}{2}{g^{ik}}(G  2\lambda )\).
Although Eddingtonâs interest did not rest on finding a proper set of field equations, he nevertheless discussed the Lagrangian \({\mathcal L} = \sqrt {  g} *{G^{ik}}*{G_{ik}}\), and showed that a variation with regard to g_{ik} did not lead to an acceptable field equation.
Eddingtonâs main goal in this paper was to include matter as an inherent geometrical structure:
âWhat we have sought is not the geometry of actual space and time, but the geometry of the worldstructure which is the common basis of space and time and things.â ([58], p. 121)
By âthingsâ he meant

(1)
the energymomentum tensor of matter, i.e., of the electromagnetic field,

(2)
the tensor of the electromagnetic field, and

(3)
the electric chargeandcurrent vector.
His aim was reached in the sense that all three quantities were fixed entirely by the connection; they could no longer be given from the outside. As to the question of the electron, it is seen as âa region of abnormal worldcurvatureâ, i.e., of abnormally large curvature.
While Pauli liked Eddingtonâs distinction between ânatural geometryâ and âworld geometryâ â with the latter being only âa graphical representationâ of reality â he was not sure at all whether âa point of view could be taken from which the gravitational and electromagnetical fields appear as unionâ. If so, then it must be a purely phenomenological one without any recourse to the nature of the charged elementary particles (cf. his letter to Eddington quoted below).
Lorentz did not like the large number of variables in Eddingtonâs theory; there were 4 components of the electromagnetic potential, 10 components of the metric and 40 components of the connection:
âIt may well be asked whether after all it would not be preferable simply to introduce the functions that are necessary for characterising the electromagnetic and gravitational fields, without encumbering the theory with so great a number of superfluous quantities.â ([211], p. 382)
Einsteinâs reaction and publications
Eddingtonâs publication early in 1921, generalising Einsteinâs and Weylâs theories started a new direction of research both in physics and mathematics. At first, Einstein seems to have been reserved (cf. his letters to Weyl in June and September 1921 quoted by Stachel in his article on Eddington and Einstein ([330], pp. 453â475; here p. 466)), but one and a half years later he became attracted by Eddingtonâs idea. To Bohr, Einstein wrote from Singapore on 11 January 1923:
âI believe I have finally understood the connection between electricity and gravitation. Eddington has come closer to the truth than Weyl.â ([139], p. 274)
He now tried to make Eddingtonâs theory work as a physical theory; Eddington had not given field equations:
âI must absolutely publish since Eddingtonâs idea must be thought through to the end.â (letter of Einstein to Weyl of 23 May 1923; cf. [241], p. 343)
And a few days later, he was still intrigued about this sort of unified field theory, in particular about its elusiveness: â[âŠ] Over it lingers the marble smile of inexorable nature, which has bestowed on us more longing than brains.â^{Footnote 108} (letter of Einstein to Weyl of 26 May 1923; cf.[241], p. 343) And indeed Einstein published fast, even while still on the steamer returning from Japan through Palestine and Spain: The paper of February 1923 in the reports of the Berlin Academy carries, as location of the sender, the ship âHaruna Maruâ of the Japanese Nippon Yushen Kaisha line^{Footnote 109} [77].
âIn past years, the wish to understand the gravitational and electromagnetic field as one in essence has dominated the endeavours of theoreticians. [âŠ] From a purely logical point of view only the connection should be used as a fundamental quantity, and the metric as a quantity derived thereof [âŠ] Eddington has done this.â^{Footnote 110} ([77], p. 32)
Like Eddington, Einstein used a symmetric connection and wrote down the equation^{Footnote 111}
where g_{kl}=g_{(kl)} and Ï_{kl}=Ï_{[kl]}, and Î» is a âlarge numberâ. By this, the metric was defined as the symmetric part of the Ricci tensor. Due to
one half of Maxwellâs equations is satisfied if Ï_{kl} is taken to be the electromagnetic field tensor. Let us note, however, that while \({{\Gamma _{kj}}^j}\) transforms inhomogeneously, its transformation law
is not exactly the same as that of the electric 4potential under gauge transformations.
For a Lagrangian, Einstein used \({\mathcal L} = 2\sqrt {  \det \;{K_{ij}}} ;\); he claims that for vanishing electromagnetic field the vacuum field equations of general relativity, with the cosmological term included, hold. Einstein varied with regard to g_{kl} and Ï_{kl}, not, as one might have expected, with regard to the connection \({\Gamma _{kj}}^j\). If \({\hat f^{kl}}: = \tfrac{{\delta {\mathcal L}}}{{\delta {\phi _{kl}}}}\), then the electric current density j^{l} is defined by \({{\hat \jmath}^l}: = \tfrac{{\partial {{\hat f}^{kl}}}}{{\partial {x^k}}}\). \({\hat f^{kl}}\) is interpreted as âthe contravariant tensor of the electromagnetic fieldâ.
The field equations are obtained from the Lagrangian by variation with regard to the connection \({\Gamma _{kj}}^l\) and are (Einstein worked in spacetime)
with the definition of the current density \(_{\hat \jmath }\,^k\) given before, and \({\hat s^{kl}} = \tfrac{{\delta {\cal L}}}{{\delta {g_{kl}}}}\). Besides \({{\hat s}^{kl}}\), Einstein also uses s^{kl} introduced by\
From Equation (121) the connection can be obtained. If \(_{\hat \jmath}\,^l = \sqrt {  \det \;{s_{kl}}} {j^l}\), and j_{k}=s_{kl}j^{l}, then the affine connection may formally be expressed by
This equation is an identity if a solution of the field equations (121) is inserted. From Equation (123),
If no electromagnetic field is present, \({\hat s^{kl}}\) reduces to \({\hat s^{kl}} = {g^{kl}}\sqrt {  \det \;{g_{kl}}}\); the definition of themetric g_{ij} in Equation (119) is reinterpreted by Einstein as giving his vacuum field equation with cosmological constant Î»^{2}. In order that this makes sense, the identifications in Equation (119) are always to be made after the variation of the Lagrangian is performed.
For nonvanishing electromagnetic field, due to Equation (124) the Equation (120) now becomes
which means that for vanishing current density no electromagnetic field is possible. Einstein concluded:
âBut the extraordinary smallness of 1/Î»^{2} implies that finite Ï_{kl} are possible only for tiny, almost vanishing current density. Except for singular positions, the current density is practically vanishing.â^{Footnote 112}
Einstein went on to show that Maxwellâs vacuum equations are holding in first order approximation. Up to the same order, \({\hat f^{kl}} \simeq {\hat \phi _{kl}}\). In general however, \({{\hat \phi }_{kl}} \ne {s_{km}}{s_{ln}}{{\hat f}^{nm}}\). Also, the geometrical theory presented here is energetically closed, i.e., the current density \({\hat \jmath l}\) cannot be given arbitrarily as in the usual Maxwell theory with external sources.
Einstein was not sure whether âelectrical elementary elementsâ, i.e., nonsingular electrons, are possible in this theory; they might be. He found it remarkable â[âŠ] that, according to this theory, positive and negative electricity cannot differ just in signâ^{Footnote 113} ([77], p. 38). His final conclusion was:
âthat EDDINGTONâS general idea in context with the Hamiltonian principle leads to a theory almost free of ambiguities; it does justice to our present knowledge about gravitation and electricity and unifies both kinds of fields in a truly accomplished manner.â^{Footnote 114} ([77], p. 38)
Until the end of May 1923, two further publications followed in which Einstein elaborated on the theory. In the second paper, he exchanged the Lagrangian \({\mathcal L} = 2\sqrt {  \det \;{K_{ij}}}\) for a new one, i.e., for \({\mathcal L} =  2\sqrt {  \det \;{K_{ij}}} + \hat R  \tfrac{1}{6}{{\hat s}^{lm}}{i_l}{i_m}\), where \(_{\hat \imath }\,^k = {i^k}\sqrt {\det \;{s_{kl}}}\). \({\mathcal L}\) is to be varied with respect to Ć^{kl} and fÌ^{kl}. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skewsymmetric part, respectively, of
Although the theory offered, for every solution with positive charge, also a solution with negative charge, the masses in the two cases were the same. However, the only known particle with positive charge at the time (what is now called the proton) had a mass greatly different from the particle with negative charge, the electron. Einstein noted:
âTherefore, the theory may not account for the difference in mass of positive and negative electrons.â^{Footnote 115} ([74], p. 77)
In the third paper [76], apart from changing notations^{Footnote 116}, Einstein set Î»=1. He also dropped the assumption (119) and replaced it by allowing his Lagrangian (Hamiltonian) Ä€ to be a function of the two independent variables,
The logic of the subsequent derivations in his paper is quite involved. The first step consisted in the definition of tensor densities
In the second step, the variations ÎŽÎł_{kl} and ÎŽÏ_{kl} were expressed by ÎŽÎ ^{l}_{ik} via (127) and inserted into ÎŽÄ€=0. The ensuing equation could be solved for Î ^{l}_{ik} and led to Equation (123). In the third step, the Lagrangian Ä€* is taken as a functional of the variables introduced in the first step, i.e., of Ä^{kl}, fÌ^{kl} such that in place of Equation (128) the relations
hold. Einstein then took âthe expression most natural visavis our present knowledgeâ, i.e., \(\hat H * = 2\alpha \sqrt {  g}  \tfrac{\beta }{2}{f_{ik}}{{\hat f}^{ik}}\). By using both Equation (127) and Equation (129), Einstein obtained the Einstein. Maxwell equations augmented by a term 1/6i_{k}i_{l} on the side of the energymomentum tensor of the electromagnetic field and Equation (125) with a changed l.h.s. now reading ÎČf_{ik}.
After a field rescaling, he then took a third expression to become his Lagrangian
where Î± and ÎČ are arbitrary constants, and Îș is the gravitational constant. i_{l} is defined to be proportional to the electromagnetic 4potential f_{k}, i.e., 1/ÎČi_{l}=f_{k}, and f_{ij} corresponds to Ï_{ij}, \(\sqrt {  g} {f^{ij}}\) to fÌ^{ij}. After the field equations had been obtained by this longwinded procedure, it became obvious that they could also be derived from HÌ=H. taken as an âeffectiveâ Lagrangian varied with respect to g_{ik} and f_{ik}. In Einsteinâs words: âR is the Riemannian curvature scalar formed from g_{ij}â^{Footnote 117}. In the third paper as well, Einsteinâs desire to create a unified field theory satisfying all his criteria still was not fulfilled: His equations, again, did not give a singularityfree electron. In a paper on Hilbertâs vision of a unified science, Sauer and Majer recently have found out from lectures of Hilbert given in Hamburg and ZĂŒrich in 1923, that Hilbert considered Einsteinâs work in affine theory a return to his own results of 1915 by â[âŠ] a colossal detour via LeviCivita, Weyl, Schouten, Eddington [âŠ]â [215]. It seems that, in this evaluation, Hilbert was influenced by Einsteinâs proportionality between the 4potential and the electrical current which Hilbert had assumed as early as in 1915 [161]^{Footnote 118}.
Comments by Einsteinâs colleagues
While, in the meantime, mathematicians had taken over the conceptual development of affine theory, some other physicists, including the perpetual piĂšce de resistance Pauli, kept a negative attitude:
â[âŠ] I now do not at all believe that the problem of elementary particles can be solved by any theory applying the concept of continuously varying field strengths which satisfy certain differential equations to regions in the interior of elementary particles. [âŠ] The quantities \(\Gamma _{\nu \alpha }^\mu\) cannot be measured directly, but must be obtained from the directly measured quantities by complicated calculational operations. Nobody can determine empirically an affine connection for vectors at neighbouring points if he has not obtained the line element before. Therefore, unlike you and Einstein, I deem the mathematicianâs discovery of the possibility to found a geometry on an affine connection without a metric as meaningless for physics, in the first place.â^{Footnote 119} (Pauli to Eddington on 20 September 1923; [251], pp. 115â119)
Also Weyl, in the 5th edition of RaumZeitMaterie ([398], Appendix 4), in discussing âworldgeometric extensions of Einsteinâs theoryâ, found Eddingtonâs theory not convincing. He criticised a theory that keeps only the connection as a fundamental building block for its lack of a guarantee that it would also house the conformal structure (light cone structure). This is needed for special relativity to be incorporated in some sense, and thus must be an independent fundamental input [405].
Likewise, Eddington himself did not appreciate much Einsteinâs followership. In Note 14, Â§ 100 appended to the second edition of his book, he laid out Einsteinâs theory but not without first having warned the reader:
âThe theory is intensely formal as indeed all such actiontheories must be, and I cannot avoid the suspicion that the mathematical elegance is obtained by a short cut which does not lead along the direct route of real physical progress. From a recent conversation with Einstein I learn that he is of much the same opinion.â ([64], pp. 257â261)
In fact, when Eddingtonâs book was translated into German in 1925 [60], Einstein wrote an appendix to it in which he repeated, with minor changes, the results of his last paper on the affine theory. His outlook on the state of the theory now was rather bleak:
âFor me, the final result of this consideration regrettably consists in the impression that the deepening of the geometrical foundations by WeylEddington is unable to bring progress for our physical understanding; hopefully, future developments will show that this pessimistic opinion has been unjustified.â^{Footnote 120} ([60], p. 371)
An echo of this can be found in Einsteinâs letter to Besso of 5 June 1925:
âI am firmly convinced that the entire chain of thought WeylEddingtonSchouten does not lead to something useful in physics, and I now have found another, physically better founded approach. To me, the quantumproblem seems to require something like a special scalar, for the introduction of which I have found a plausible way.â^{Footnote 121} ([99], p. 204)
This remark shows that Einstein must have taken some notice of Schoutenâs work in affine geometry. What the âspecial scalarâ was, remains an open question.
Overdetermination of partial differential equations and elementary particles
Einstein spent much time in thinking about the âquantum problemâ, as he confessed to Born:
âI do not believe that the theory will be able to dispense with the continuum. But I fail to succeed in giving my pet idea a tangible form: to understand the quantumstructure through an overdetermination by differential equations.â^{Footnote 122} ([103], pp. 48â49)
In a paper from December 1923, Einstein not only stated clearly the necessary conditions for a unified field theory to be acceptable to him, but also expressed his hope that this technique of âoverdeterminationâ of systems of differential equations could solve the âquantum problemâ.
âAccording to the theories known until now the initial state of a system may be chosen freely; the differential equations then give the evolution in time. From our knowledge about quantum states, in particular as it developed in the wake of Bohrâs theory during the past decade, this characteristic feature of theory does not correspond to reality. The initial state of an electron moving around a hydrogen nucleus cannot be chosen freely; its choice must correspond to the quantum conditions. In general: not only the evolution in time but also the initial state obey laws.â^{Footnote 123} ([75], pp. 360â361)
He then ventured the hope that a system of overdetermined differential equations is able to determine
âalso the mechanical behaviour of singular points (electrons) in such a way that the initial states of the field and of the singular points are subjected to constraints as well. [âŠ] If it is possible at all to solve the quantum problem by differential equations, we may hope to reach the goal in this direction.â
We note here Einsteinâs emphasis on the very special problem of the quantum nature of elementary particles like the electron, as compared to the general problem of embedding matter fields into a geometrical setting.
One of the crucial tests for an acceptable unified field theory for him now was:
âThe system of differential equations to be found, and which overdetermines the field, in any case must admit this static, spherically symmetric solution which describes, respectively, the positive and negative electron according to the equations given above [i.e the EinsteinMaxwell equations].â^{Footnote 124}
This attitude can also be found in a letter to M. Besso from 5 January 1924:
âThe idea I am wrestling with concerns the understanding of the quantum facts; it is: overdetermination of the laws by more field equations than field variables. In such a way, the unambiguity of the initial conditions ought to be understood without leaving field theory. [âŠ] The equations of motion of material points (electrons) will be given up totally; their motion ought to be codetermined by the field laws.â^{Footnote 125} ([99], p. 197)
In his answer, Besso asked for more information concerning the quantum aspect of the concept of âoverdeterminationâ, because:
âOn the one hand, this seems to be connected only formally with a field theory; on the other, it has not yet dawned on me how in this manner something corresponding to the discrete quantum orbits may be reached.â^{Footnote 126} ([99], p. 199)
Differential Geometryâs High Tide
In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [337]:

(1)
The generalisation of parallel transport in the sense of LeviCivita and Weyl. Schouten is the leading figure in this approach [300].

(2)
The âgeometry of pathsâ considering the lines of constant direction for a connection â with the proponents Veblen, Eisenhart [122, 114, 115, 373], J. M. Thomas [348], and T. Y. Thomas^{Footnote 127}[349, 347]. Here, only symmetric connections can appear.

(3)
The idea of mapping a manifold at one point to a manifold at a neighbouring point is central (affine, conformal, projective mappings). The names of KĂ¶nig [192] and Cartan [29, 302] are connected with this program.
In his assessment, Eisenhart [121] adds to this all the geometries whose metric is
âbased upon an integral whose integrand is homogeneous of the first degree in the differentials. Developments of this theory have been made by Finsler, Berwald, Synge, and J. H. Taylor. In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics. Affine properties of these spaces are obtained from a natural generalisation of the definition of LeviCivita for Riemannian spaces.â ([121], p. V)
In fact, already in May 1921 Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [297, 296]. In the first he wrote:
âMotivated by relativity theory, differential geometry received a totally novel, simple and satisfying foundation; I just refer to G. Hessenbergâs âVectorial foundationâŠâ, Math. Ann. 78, 1917, S. 187â217 and H. Weyl, RaumZeitMaterie, 2. Section, Leipzig 1918 (3. Aufl. Berlin 1920) as well as âReine Infinitesimalgeometrieâ etc.^{Footnote 128}. [âŠ] In the present investigation all 18 different linear connections are listed and determined in an invariant manner. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field [âŠ].â^{Footnote 129} ([297], p. 57)
The fields referred to are the torsion tensor S ^{k}_{ij} , the tensor of nonmetricity Q ^{k}_{ij} , the metric g_{ij}, and the tensor C ^{k}_{ij} which, in unified field theory, was rarely used. It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1form not by the connection L ^{k}_{ij} in Equation (13), but by
with \(\prime L \ne L\). In fact
In the first paper, Schouten had considered only the special case C ^{k}_{ij} =C_{i} ÎŽ ^{k}_{j} ^{Footnote 130}.
Furthermore, on p. 57 of [297] we read:
âThe general connection for n=4 at least theoretically opens the door for an extension of Weylâs theory. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression.â^{Footnote 131}
Through footnote 5 on the same page we learn the pedagogical reason why Schouten did not use the âdirectâ method [294, 336] in his presentation, but rather a coordinate dependent formalism^{Footnote 132}:
âAs the results of the present investigation might be of interest for a wider circle of mathematicians, and also for a number of physicists [âŠ].â^{Footnote 133}
At the end of the first paper we can find a section âEventual importance of the present investigation for physicsâ (p. 79â81) and the confirmation that during the proofreading Schouten received Eddingtonâs paper ([58], accepted 19 February 1921). Thus, while Einstein and Weyl influenced Eddington, Schouten apparently did his research without knowing of Eddingtonâs idea. Einstein, perhaps, got to know Schoutenâs work only later through the German translation of Eddingtonâs book where it is mentioned ([60], p. 319), and to which he wrote an addendum, or, more directly, through Schoutenâs book on the Ricci calculus, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, in the same famous yellow series of Springer Verlag [300]. On the other hand, Einsteinâs papers following Eddingtonâs [77, 74] inspired Schouten to publish on a theory with vector torsion that tried to remedy a problem Einstein had noted in his papers, i.e., that no electromagnetic field could be present in regions of vanishing electric current density. According to Schouten
â[âŠ] we see that the electromagnetic field only depends on the curl of the electric current vector, so that the difficulty arises that the electromagnetic field cannot exist in a place with vanishing current density. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical.â ([300], p. 850)
Schouten criticised Einsteinâs argument for using a symmetric connection^{Footnote 134} as unfounded (cf. Equation (15)). He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:
âWe will not consider the most general case, but the semisymmetric case in which the alternating part of the parameters has the form:
$$ 1/2({{\Gamma '}_\mu }^\nu _{\,\,\,\,\,\lambda }  {{\Gamma '}_\lambda }^\nu _{\,\,\,\,\,\mu }) = 1/2({S_\lambda }\;\delta _\mu ^\nu  {S_\mu }\;\delta _\lambda ^\nu ), $$in which S_{Î»} is a general covariant vector.â ([298], p. 851)
The affine connection ÎâČ can then be decomposed as follows:
Hence, besides the covariant derivative ââČ following from use of \({\Gamma '_{jk}}^l\), in his calculations Schouten also introduced a covariant derivative â* formed with \({\Lambda _{jk}}^l\). Schoutenâs point of departure for the field equations is Einsteinâs first Lagrangian \({\mathcal L} = \sqrt {\det \;{K_{ij}}}\) and, consequently, his field equations were the same as Einsteinâs apart from additional terms in vector torsion. Also, Schoutenâs definition of some of the observables is different; For example, the electromagnetic field tensor unlike in Equation (125) is now
where \({i^k}: = \nabla _l^*{f^{kl}}  {P_l}{f^{kl}}\), and \({P_k}: =  \tfrac{\partial }{{\partial {x^k}}}(\log \sqrt {  \det \;{K_{ij}}} ) + {\Lambda _{lk}}^l\). On the same topic, Schouten wrote a paper with Friedman in Leningrad [142]. A similar, but less detailed, classification of connections than Schoutenâs has also been given by Cartan. He relied on the curvature, torsion and homothetic curvature 2forms ([32], Section III; cf. also Section 2.1.4). In 1925, Eyraud^{Footnote 135} came back to Schoutenâs paper [298] and proved that his connection can be mapped projectively and conformally on a Riemannian space [124, 123].
Other mathematicians were also stimulated by Einsteinâs use of differential geometry in his general relativity and, particularly, by the idea of unified field theory. Examples are Eisenhart and Veblen, both in Princeton, who developed the âgeometry of pathsâ^{Footnote 136} under the influence of papers by Weyl, Eddington, and Einstein [122, 117, 383]. In Eisenhartâs paper, we may read that
âEinstein has said (in Meaning of Relativity) that âa theory of relativity in which the gravitational field and the electromagnetic field enter as an essential unityâ is desirable and recently has proposed such a theory.â ([117], pp. 367â368)
and
âHis geometry also is included in the one now proposed and it may be that the latter, because of its greater generality and adaptability will serve better as the basis for the mathematical formulation of the results of physical experiments.â ([117], p. 369)
The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad. Seven years after Schoutenâs classification of connections, FrĂ©edericksz of Leningrad â known better for his contributions to the physics of liquid crystals â put forward a classification of his own by using both the connection and the curvature tensor [138].
The Pursuit of Unified Field Theory by Einstein and His Collaborators
Affine and mixed geometry
Already in July 1925 Einstein had laid aside his doubts concerning âthe deepening of the geometric foundationsâ. He modified Eddingtonâs approach to the extent that he now took both a nonsymmetric connection and a nonsymmetric metric, i.e., dealt with a mixed geometry (metricaffine theory):
â[âŠ] Also, my opinion about my paper which appeared in these reports [i.e., Sitzungsberichte of the Prussian Academy, Nr. 17, p. 137, 1923], and which was based on Eddingtonâs fundamental idea, is such that it does not present the true solution of the problem. After an uninterrupted search during the past two years I now believe to have found the true solution.â^{Footnote 137} ([78], p. 414)
As in general relativity, he started from the Lagrangian \({\mathcal L} = {\hat g^{ik}}{R_{ik}}\), but now with Ä^{ik} and the connection \({\Gamma _{kj}}^l\) being varied separately as independent variables. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:
i.e., 64+16 equations for the same number of variables. Ï_{k} is an arbitrary covariant vector. The asymmetric g_{ik} is related to Ä^{lm} by
The three equations (134) and
were the result of the variation. In order to be able to interpret the symmetric part of g_{ik} as metrical tensor and its anti(skew)symmetric part as the electromagnetic field tensor, Einstein put Ï_{k}=0, i.e., overdetermined his system of partial differential equations. However, he cautioned:
âHowever, for later investigations (e.g., the problem of the electron) it is to be kept in mind that the HAMILTONian principle does not provide an argument for putting Ï_{k} equal to zero.â^{Footnote 138}
In comparing Equation (134) with Ï_{k}=0 and Equation (47), we note that the expression does not seem to correspond to a covariant derivative due to the + sign where a â sign is required. But this must be due to either a calculational error, or to a printerâs typo because in the paper of J. M. Thomas following Einsteinâs by six months and showing that Einsteinâs ânew equations can be obtained by direct generalisation of the equations of the gravitational field previously given by him. The process of generalisation consists in abandoning assumptions of symmetry and in adopting a definition of covariant differentiation which is not the usual one, but which reduces to the usual one in case the connection is symmetric.â ([346], p. 187) J. M. Thomas wrote Einsteinâs Equation (134) in the form
with Î© being the skewsymmetric part of the asymmetric connection \({H_{ij}}^k = \Gamma _{(ij)}^{\;\;\;k} + \Omega _{[ij]}^{\;\;\;\;k}\), and g_{ij} being the symmetric part of the asymmetric metric \({h_{ij}} = {g_{(ij)}} + {\omega _{[ij]}}\). The two covariant derivatives introduced by J. M. Thomas are \({g_{ij,k}} = \tfrac{{\partial {g_{ij}}}}{{\partial {x^k}}}  {g_{rj}}{\Gamma _{ik}}^r  {g_{ir}}{\Gamma _{jk}}^r\) and \({h_{ij/k}} = \tfrac{{\partial {h_{ij}}}}{{\partial {x^k}}}  {h_{rj}}{H_{ik}}^r  {h_{ir}}{H_{jk}}^r\). J. M. Thomas then could reformulate Equation (137) in the form
and derive the result
(see [346], p. 189).
After having shown that his new theory contains the vacuum field equations of general relativity for vanishing electromagnetic field, Einstein then proved that, in a firstorder approximation, Maxwellâs field equations result cum grano salis: Instead of \({F_{ik,l}} + {F_{li,k}} + {F_{kl,i}} = 0\) he only obtained \(\Sigma \tfrac{\partial }{{\partial {x^l}}}({F_{ik,l}} + {F_{li,k}} + {F_{kl,i}}) = 0\).
This was commented on in a paper by Eisenhart who showed âmore particularly what kind of linear connection Einstein has employedâ and who obtained âin tensor form the equations which in this theory should replace Maxwellâs equations.â He then pointed to some difficulty in Einsteinâs theory: When identification of the components of the antisymmetric part Ï_{ij} of the metric \({a_{ij}} = {g_{ij}} + {\phi _{ij}}\) with the electromagnetic field is made in first order,
âthey are not the components of the curl of a vector as in the classical theory, unless an additional condition is added.â ([120], p. 129)
Toward the end of the paper Einstein discussed timereversal; according to him, by it the sign of the magnetic field is changed, while the sign of the electric field vector is left unchanged^{Footnote 139}. As he wanted to obtain chargesymmetric solutions from his equations, Einstein now proposed to change the roles of the magnetic fields and the electric fields in the electromagnetic field tensor. In fact, the substitutions \(\mathbf{\tilde E} \to \mathbf{\tilde B}\) and \({\mathbf{\tilde B}} \to  {\mathbf{\tilde E}}\) leave invariant Maxwellâs vacuum field equations (duality transformations)^{Footnote 140}. Already Pauli had pointed to timereflection symmetry in relation with the problem of having elementary particles with charge Â±e and unequal mass ([246], p. 774).
At first, Einstein seems to have been proud about his new version of unified field theory; he wrote to Besso on 28 July 1925 that he would have liked to present him âorally, the egg laid recently, but now I do it in writingâ, and then explained the independence of metric and connection in his mixed geometry. He went on to say:
âIf the assumption of symmetry^{Footnote 141} is dropped, the laws of gravitation and Maxwellâs field laws for empty space are obtained in first approximation; the antisymmetric part of Ä^{ik} is the electromagnetic field. This is surely a magnificent possibility which likely corresponds to reality. The question now is whether this field theory is consistent with the existence of quanta and atoms. In the macroscopic realm, I do not doubt its correctness.â^{Footnote 142} ([99], p. 209) We have noted before that a similar suggestion within a theory with a geometry built from an asymmetric metric had been made, in 1917, by Bach alias FĂ¶rster.
Yet, in the end, also this novel approach did not convince Einstein. Soon after the publication discussed, he found his argument concerning charge symmetric solutions not to be helpful. The link between the occurrence of solutions with both signs of the charge with timesymmetry of the field equations induced him to doubt, if only for a moment, whether the endeavour of unifying electricity and gravitation made sense at all:
âTo me, the insight seems to be important that an explanation of the dissimilarity of the two electricities is possible only if time is given a preferred direction, and if this is taken into account in the definition of the decisive physical quantities. In this, electrodynamics is basically different from gravitation; therefore, the endeavour to melt electrodynamics with the law of gravitation into one unity, to me no longer seems to be justified.â^{Footnote 143} [79]
In a paper dealing with the field equations
which had been discussed earlier by Einstein [70], and to which he came back now after Rainich^{Footnote 144}âs insightful paper into the algebraic properties of both the curvature tensor and the electromagnetic field tensor ([263, 264, 265, 266]), Einstein indicated that he had lost hope in the extension of Eddingtonâs affine theory:
âThat the equations (140) have received only little attention is due to two circumstances. First, the attempts of all of us were directed to arrive, along the path taken by Weyl and Eddington or a similar one, at a theory melting into a formal unity the gravitational and electromagnetic fields; but by lasting failure I now have laboured to convince myself that truth cannot be approached along this path.â^{Footnote 145} (Einsteinâs italics; [80], p. 100)
The new field equation was picked up by R. N. Sen of Kalkutta who calculated âthe energy of an electric particleâ according to it [323].
In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas 1925 in words similar to those in his letter in June:
âRegrettably, I had to throw away my work in the spirit of Eddington. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by WeylEddington. The equations
$$ {R_{ik}}  \frac{1}{4}R\;{g_{ik}} =  \kappa {T_{ik}}\;\;\;\;\;{\rm{electromagnetic}} $$I take as the best we have nowadays. They are 9 equations for the 14 variables g_{ik} and Îł_{ik} New calculations seem to show that these equations yield the motion of the electrons. But it appears doubtful whether there is room in them for the quanta.â ^{Footnote 146} ([99], p. 216)
According to the commenting note by Tonnelat, the 14 variables are given by the 10 components of the symmetric part g_{(ik)}^{Footnote 147} of the metric and the 4 components of the electromagnetic vector potential âthe rotation of which are formed by the Îł_{[ik]}â^{Footnote 148}.
But even âthe best we have nowadaysâ did not satisfy Einstein; half a year later, he expressed his opinion in a letter to Besso:
âAlso, the equation put forward by myself^{Footnote 149},
$$ {R_{ik}} = {g_{ik}}{f_{lm}}{f^{lm}}  \frac{1}{2}{f_l}{f_{km}}{g^{lm}} $$gives me little satisfaction. It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items (as the l.h.s. and the r.h.s. of an equation) which from a logicalmathematical point of view have nothing to do with each other.â ^{Footnote 150} ([99], p. 230)
Further work on (metric) affine and mixed geometry
Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school. Thus J. M. Thomas, after having given a review of Weylâs, Einsteinâs, and Schoutenâs approaches, said about his own work:
âI show in the present paper that his [Einsteinâs] new equations can be obtained by a direct generalisation of the equations of the gravitational field previously given by him [g_{ij;k}=0; R_{ij}=0]. [âŠ] In the final section I show that the adoption of the ordinary definition of covariant differentiation leads to a geometry which includes as a special case that proposed by Weyl as a basis for the electric theory; further that the asymmetric connection for this special case is of the type adopted by Schouten for the geometry at the basis of his electric theory.â ([346], p. 187)
We met J. M. Thomasâ paper before in Section 6.1.
During the period considered here, a few physicists followed the path of Eddington and Einstein. One who had absorbed Eddingtonâs and Einsteinâs theories a bit later was Infeld^{Footnote 151} of Warsaw^{Footnote 152}. In January 1928, he followed Einstein by using an asymmetric metric the symmetric part Îł_{ik} of which stood for the gravitational potential, the skewsymmetric part Ï_{ik} for the electromagnetic field. However, he set the nonmetricity tensor (of the symmetric part Îł of the metric) \({Q_{ij}}^k = 0\), and assumed for the skewsymmetric part Ï,
with an arbitrary tensor J_{ijl}. The electric current vector then is defined by \({J^i} = {J^{il}}_l\) where the indices, as I assume, are moved with Îł_{ik}. In a weakfield approximation for the metric, Infeldâs connection turned out to be \({L_{ik}}^l = \{ \,_{ik}^l\,\} + \tfrac{1}{2}({\phi _{i,}}^l_k + {\phi _{k,}}^l_i + {\delta ^{ls}}{\phi _{ik,s}})\). For field equations Infeld postulated the (generalised) Einstein field equations in empty space, K_{ij}=0. He showed that, in first approximation, he got what is wanted, i.e., Einsteinâs and Maxwellâs equations [166].
Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as
where Î± is âan extremely small numerical factorâ. By neglecting terms âŒÎ±^{2} he could gain both Einsteinâs field equation in empty space (94) and Maxwellâs equation, if the electric current vector is identified with \({\alpha ^{  1}}({L_{il}}^l  {L_{li}}^l)\). Thus, he is back at vector torsion treated before by Schouten [298].
The Japanese physicist Hattori embarked on a metricaffine geometry derived purely from an asymmetric metrical tensor \({h_{ik}} = {g_{(ik)}} + {f_{[ik]}}\). He defined an affine connection
where \({g^{il}}{g_{lk}} = {\delta ^i}_k\), and the Christoffel symbol is formed from g. The electromagnetic field was not identified with f_{ik} by Hattori, but with the skewsymmetric part of the (generalised) Ricci tensor formed from \({L_{ik}}^j\). By introducing the tensor \({f_{ijk}}: = \tfrac{{\partial {f_{ij}}}}{{\partial {x^k}}} + \tfrac{{\partial {f_{jk}}}}{{\partial {x^i}}} + \tfrac{{\partial {f_{ki}}}}{{\partial {x^j}}}\), he could write the (generalised) Ricci tensor as
where the covariant derivative â is formed with the LeviCivita connection of g_{ij}. The electromagnetic field tensor F_{ik} now is introduced through a tensor potential by \({F_{ik}}: = {\nabla _l}{f_{ik}}^l\) and leads to half of âMaxwellâsâ equations. In the sequel, Hattori started from a Lagrangian \( {\mathcal L} = ({g^{ik}} + {\alpha ^2}{F^{ik}}){K_{ik}} \) with the constant Î±^{2} and varied, alternatively, with respect to g_{ij} and f_{ij}. He could write the field equations in the form of Einsteinâs, with the energymomentum tensor of the electromagnetic field F_{ik} and a âmatterâ tensor M^{ik} on the r.h.s., M^{ik} being a complicated, purely geometrical quantity depending on K_{ik}, K, f_{ikl}, and F_{ikl}. F_{ikl} is formed from F_{ik} as f_{ikl} from f_{ik}. From the variation with regard to f_{ik}, in addition to Maxwellâs equation, a further field equation resulted, which could be brought into the form
i.e., f_{ikl}âŒF_{ikl} Hattoriâs conclusion was:
âThe preceding equation shows that electrical charge and electrical current are distributed wherever an electromagnetic field exists.â^{Footnote 153}
Thus, the same problem obtained as in Einsteinâs theory: A field without electric current or charge density could not exist [155]^{Footnote 154}.
Infeld quickly reacted to Hattoriâs paper by noting that Hattoriâs voluminous calculations could be simplified by use of Schoutenâs Equation (39) of Section 2.1.2. As in Hattoriâs theory two connections are used, Infeld criticised that Hattori had not explained what his fundamental geometry should be: Riemannian or nonRiemannian? He then gave another example for a theory allowing the identification of the electromagnetic field tensor with the antisymmetric part of the Ricci tensor: He displayed again the wellknown connection with vector torsion used by Schouten [298] without referring to Schoutenâs paper [165]. He also claimed that Hattoriâs Equation (145) is the same as the one that had been deduced from Eddingtonâs theory by Einstein in the Appendix to the German translation of Eddingtonâs book ([60], p. 367). All in all, Infeldâs critique tended to deny that Hattoriâs theory was more general than Einsteinâs, and to point out
âthat the problem of generalising the theory of relativity cannot be solved along a purely formal way. At first, one does not see how a choice can be made among the various nonRiemannian geometries providing us with the gravitational and Maxwellâs equations. The proper world geometry which ought to lead to a unified theory of gravitation and electricity can only be found by an investigation of its physical content.â^{Footnote 155} ([165], p. 811)
Infeld could as well have applied this admonishment to his own unified field theory discussed above. Perhaps, he became irritated by comparing his expression for the connection (142) with Hattoriâs (145).
In June 1931, von Laue submitted a paper of the Genuese mathematical physicist Paolo Straneo to the Berlin Academy [331]. In it Straneo took note of Einsteinâs teleparallel geometry, but decided to take another route within mixed geometry; he started with a symmetric metric and the asymmetric connection
with both nonvanishing curvature tensor \({K^i}_{jkl} = {R^i}_{jkl} + 2\delta _j^i(\tfrac{{\partial {\psi _l}}}{{\partial {x^k}}}  \tfrac{{\partial {\psi _k}}}{{\partial {x^l}}})\) and torsion \({S_{ik}}^j = 2{\delta ^j}_{[i}{\psi _{k]}}\). Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [298, 142]) without referring to him. The field equations Straneo wrote down, i. e.
where T_{ik} is the symmetric and Îš_{ik} the antisymmetric part of the l.h.s., do not fulfill Einsteinâs conception of unification: Straneo kept the energymomentum tensor of matter as an extraneous object (including the electromagnetic field) as well as the electric current vector. The antisymmetric part of (147) just is \({\Psi _{ik}} = (\tfrac{{\partial {\psi _l}}}{{\partial {x^k}}}  \tfrac{{\partial {\psi _k}}}{{\partial {x^l}}})\); thus Îš_{ik} is identified with the electromagnetic field tensor, and the electric current vector J^{i} defined by \({\Psi ^{il}}_l = {J^i}\). Straneo wrote further papers on the subject [332, 333].
By a remark of Straneo, that autoparallels and geodesics have to be distinguished in an affine geometry, the Indian mathematician Kosambi^{Footnote 156} felt motivated to approach affine geometry from the system of curves solving áș^{i}+Î±^{i}(x, x, t) with an arbitrary parameter t. He then defined two covariant âvectorderivationsâ along an arbitrary curve and arrived at an (asymmetric) affine connection. By this, he claimed to have made superfluous the fivevectors of Einstein and Mayer^{Footnote 157} [107]. This must be read in the sense that he could obtain the Einstein.Mayer equations from his formalism without introducing a connecting quantity leading from the space of 5vectors to spacetime [195].
Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning. Due to this complication â for example even a condition of metric compatibility would not have the physical meaning of the conservation of the norm of an angle between vectors under parallel transport, and the further difficulty that much of the formalism was very clumsy to manipulate; essential work along this line was done only much later in the 10940s and 1950s (Einstein, Einstein and Strauss, SchrĂ¶dinger, Lichnerowicz, Hlavaty, Tonnelat, and many others). In this work a generalisation of the equation for metric compatibility, i.e., Equation (47), will play a central role. The continuation of this research line will be presented in Part II of this article.
Kaluzaâs idea taken up again
Kaluza: Act I
Einstein became interested in Kaluzaâs theory again due to O. Kleinâs paper concerning a relation between âquantum theory and relativity in five dimensionsâ (see Klein 1926 [185], received by the journal on 28 April 1926). Einstein wrote to his friend and colleague Paul Ehrenfest on 23 August 1926: âSubject Kaluza, Schroedinger, general relativityâ, and, again on 3 September 1926: âKleinâs paper is beautiful and impressive, but I find Kaluzaâs principle too unnatural.â However, less than half a year later he had completely reversed his opinion:
âIt appears that the union of gravitation and Maxwellâs theory is achieved in a completely satisfactory way by the fivedimensional theory (KaluzaKleinFock).â (Einstein to H. A. Lorentz, 16 February 1927)
On the next day (17 February 1927), and ten days later Einstein was to give papers of his own in front of the Prussian Academy in which he pointed out the gaugegroup, wrote down the geodesic equation, and derived exactly the Einstein.Maxwell equations â not just in first order as Kaluza had done [81, 82]. He came too late: Klein had already shown the same before [185]. Einstein himself acknowledged indirectly that his two notes in the report of the Berlin Academy did not contain any new material. In his second communication, he added a postscript:
âMr. Mandel brings to my attention that the results reported by me here are not new. The entire content can be found in the paper by O. Klein.â^{Footnote 158}
He then referred to the papers of Klein [185, 186] and to âFochs Arbeitâ which is a paper by Fock^{Footnote 159} 1926 [130], submitted three months later than Kleinâs paper. That Klein had published another important clarifying note in Nature, in which he closed the fifth dimension, seems to have escaped Einstein^{Footnote 160} [184]. Unlike in his paper with Grommer, but as in Kleinâs, Einstein, in his notes, applied the âsharpened cylinder conditionâ, i.e., dropped the scalar field. Thus, the three of them had no chance to find out that Kaluza had made a mistake: For g_{55}â const., even in first approximation the new field will appear in the fourdimensional Einstein.Maxwell equations ([145], p. 5).
Mandel^{Footnote 161} of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. Kleinâs results [216]. In a footnoote, Mandel stated that he had learned of Kaluzaâs (whom he spelled âKalusaâ) paper only through Kleinâs article. He started by embedding spacetime as a hypersurface x^{5}=const. into M_{5}, and derived the field equations in spacetime by assuming that the fivedimensional curvature tensor vanishes; by this procedure he obtained also a matterenergy tensor âclosely linked to the second fundamental form of this hypersurfaceâ. From the geodesics in M_{5} he derived the equations of motion of a charged point particle. One of the two additional terms appearing besides the Lorentz force could be removed by a weakness assumption; as to the second, Mandel opinioned
âthat the experimental discovery of the second term appears difficult, yet perhaps not entirely impossible.â ([216], p. 145)
As to Fockâs paper, it is remarkable because it contains, in nuce, the coupling of the SchrĂ¶dinger wave function Ï and the electromagnetic potential by the gauge transformation Ï=Ï0 e^{2Ïip/h}, where h is Planckâs constant and p âa new parameter with the unit of the quantum of actionâ [130]. In Fockâs words:
âThe importance of the additional coordinate parameter p seems to lie in the fact that it causes the invariance of the equations [i.e., the relativistic wave equations] with respect to addition of an arbitrary gradient to the 4potential.â^{Footnote 162} ([130], p. 228)
Fock derived the general relativistic wave equation and the equations of motion of a charged point particle; the latter is identified with the null geodesics of M_{5}. Neither Mandel nor Fock used the âsharpened cylinder conditionâ (110).
A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated fivedimensional wave equation
and by neglecting the gravitational field, he arrived at the fourdimensional SchrĂ¶dinger equation after insertion of the quantum mechanical differential operators \( \tfrac{{ih}}{{2\pi }}\tfrac{\partial }{{\partial {x^i}}}\). It was Kleinâs papers and the magical lure of a link between classical field theory and quantum theory that raised interest in Kaluzaâs idea â seven years after Kaluza had sent his manuscript to Einstein. Klein acknowledged Mandelâs contribution in his second paper received on 22 October 1927, where he also gave further references on work done in the meantime, but remained silent about Einsteinâs papers [189]. Likewise, Einstein did not comment on Kleinâs new idea of âdimensional reductionâ as it is now called and which justifies Kleinâs name in the âKaluzaKleinâ theories of our time. By this, the reduction of fivedimensional equations (as e.g., the fivedimensional wave equation) to fourdimensional equations by Fourier decomposition with respect to the new 5th spacelike coordinate x^{5}, taken as periodic with period L, is understood:
with an integer n. Klein had only the lowest term in the series. The 5th dimension is assumed to be a circle, topologically, and thus gets a finite linear scale: This is at the base of what now is called âcompactificationâ. By adding to this picture the idea of de Broglie waves, Klein brought in Planckâs constant and determined the linear scale of x^{5} to be unmeasurably small (âŒ10^{30}). From this, the possibility of âforgettingâ the fifth dimension arose which up to now has not been observed.
In his papers, Einstein took over Kleinâs condition g_{55}=1, which removed the additional scalar field admitted by the theory. It was ReichenbĂ€cher who apparently first tried to perform the projection into spacetime of the most general fivedimensional metric, and without using the cylinder condition (109):
âNow, a rather laborious calculation of the fivedimensional curvature quantities in terms of a fourdimensional submanifold contained in it has shown to me also in the general case (g_{55}â const., dependence of the components of the f u n d a m e n t a l [tensor] of x^{5} is admitted) that the c h a r a c t e r i s t i c properties of the field equations are then conserved as well, i.e., they keep the form
only the T^{ik} contain further terms besides the electromagnetic energy tensor S^{ik}, and the quantities collected in s^{i} do not vanish. [âŠ] The appearance of the new terms on the right hand sides could even be welcomed in the sense that now the field equations are obtained not only for a field point free of matter and charge^{Footnote 163}.â ^{Footnote 164} ([276], p. 426) Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail [102].
It is likely that ReichenbĂ€cher had been led to this excursion into fivedimensional space, an idea which he had rejected before as unphysical, because his attempt to build a unified field theory in spacetime through the ansatz for the metric \({\gamma _{ik}} = {g_{ik}}  { \epsilon ^2}{\phi _i}{\phi _k}\) with Ï_{k} the electromagnetic 4potential, had failed. Beyond incredibly complicated field equations nothing much had been gained [275]. ReichenbĂ€cherâs ansatz is well founded: As we have seen in Section 4.2, due to the violation of covariance in M_{5}, Îł_{ik} transforms as a tensor under the reduced covariance group.
Even L. de Broglie became interested in Kaluzaâs âbold but very beautiful theoryâ and rederived Kleinâs results his way [46], but not without getting into a squabble with Klein, who felt misunderstood [188, 47]. He also suggested that one should not accept the cylinder condition, a suggestion looked into by Darrieus who introduced an electrical 5potential and 5current, and deduced Maxwellâs equations from the fivedimensional homogeneous wave equation and the fivedimensional equation of continuity [43].
In 1929 Mandel tried to âaxiomatiseâ the fivedimensional theory: His two axioms were the cylinder condition (109) and its sharpening, Equation (110). He then weakened the second assumption by assuming that âan objective meaning does not rest in the g_{ik} proper, but only in their quotientsâ, an idea he ascribed to O. Klein and Einstein. He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [220].
Kleinâs lure lasted for some years. In 1930, N. R. Sen claimed to have investigated the âKeplerproblem for the fivedimensional wave equation of Kleinâ. What he did was to calculate the energy levels of the hydrogen atom (as a one particlesystem) with the general relativistic wave equation in spacetime (148) with a^{ik}=Îł^{ik}, where \({\gamma _{ik}} = {g_{ik}} + {\alpha ^2}{\gamma _{55}}\) is the metric on spacetime following from the 5metric \({\gamma _{\alpha \beta }}\) by dx^{5}=0. For g_{ik} he took the ReissnerNordstrĂ¶m solution and did not obtain a discrete spectrum [324]. He continued his approach by trying to solve SchrĂ¶dingerâs wave equation [325]
Presently, the different contributions of Kaluza and O. Klein are lumped together by most physicists into what is called âKaluzaKlein theoryâ. An early criticism of this unhistorical attitude has been voiced in [210].
Kaluza: Act II
Four years later, Einstein returned to Kaluzaâs idea. Perhaps, he had since absorbed Mandelâs ideas which included a projection formalism from the fivedimensional space to spacetime [216, 217, 218, 219].
In a paper with his assistant Mayer, Einstein now presented Kaluzaâs approach in the form of an implicit projective fourdimensional theory, although he did not mention the word âprojectiveâ [107]:
âPsychologically, the theory presented here connects to Kaluzaâs wellknown theory; however, it avoids extending the physical continuum to one of five dimensions.â^{Footnote 165}
In the eyes of Einstein, by avoiding the artificial cylinder condition (109), the new method removed a serious objection to Kaluzaâs theory.
Another motivation is also put forward: The linearity of Maxwellâs equations âmay not correspond to realityâ; thus, for strong electromagnetic fields, Einstein expected deviations from Maxwellâs equations. After a listing of all the shortcomings of Kaluzaâs theory, the new approach is introduced: At every event a fivedimensional vector space V_{5} is affixed to spacetime V_{4}, and âmixedâ tensors \({\gamma _\imath }^{\;k}\) are defined linking the tangent space of spacetime V_{4} with a V_{5} such that
where g_{lm} is the metric tensor of V_{4}, and \({g_{\iota \kappa }}\) a nonsingular, symmetric tensor on V_{5} with Îč, Îș=1, âŠ , 5, and k, l=1, âŠ , 4^{Footnote 166}. Indices are raised and lowered with the metrics of V_{5} or V_{4}, respectively. There exists a âpreferred direction of V_{5}â defined by \({\gamma _\iota }^{\;k}{A^\iota } = 0\), and which is the normal to a âpreferred planeâ \({\gamma _\iota }^{\;k}{\omega _k} = 0\)^{Footnote 167}. A consequence then is
A covariant derivative for fivevectors in V_{4} is defined with a âthreeindexsymbolâ \({\Gamma ^\iota }_{\pi l}\) with two indices in V_{5}, and one in V_{4} standing in for the connection coefficients:
The covariant derivative of 4vectors is defined as usual,
where { ^{i}_{jl} } is calculated from the metric of V_{4} as given in Equation (149). Both covariant derivatives are abbreviated by the same symbol A_{;k}. The covariant derivative of tensors with both indices referring to V_{5} and those referring to V_{4}, is formed correspondingly. In this context, Einstein and Mayer mention an extension of absolute differential calculus by âWAERDEN and BARTOLOTTIâ without giving any reference to their respective papers. They may have had in mind van der Waerdenâs [368] and Bortolotti^{Footnote 168}âs [24] papers. The autoparallels of V_{5} lead to the exact equations of motion of a charged particle, not the geodesics of V_{4}.
Einstein and Mayer made three basic assumptions:
where \({A^\iota }\) is the preferred direction and F_{kl} an arbitrary 2form, later to be interpreted as the electromagnetic field tensor. From them \({A_{\sigma ;l}} = {\gamma _\sigma }^k{F_{lk}}\) follows. They also noted that a symmetric tensor F_{kl} could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V_{4} into V_{5}.
Einstein and Mayer introduced what they called âFĂŒnferkrĂŒmmungâ (5curvature) via the threeindex symbol given above by
It is related to the Riemannian curvature \({R^r}_{mlk}\) of V_{4} by
and
From (154), by transvection with \({\gamma ^{\tau k}}\), the 5curvature itself appears:
By contraction, \({P_{\iota k}}: = {\gamma _\tau }^r{P^\tau }_{\iota rk}\) and \(P: = {\gamma ^{\iota k}}{P_{\iota k}}\). Two new quantities are introduced:

(1)
\({U_{\iota k}}: = {P_{\iota k}}  \tfrac{1}{4}(P + R)\), where R is the Ricci scalar of the Riemannian curvature tensor of V_{4}, and

(2)
the tensor \({N_{klm}}: = {F_{\{ kl;m\} }}\)^{Footnote 169}.
It turns out that \(P = R  {F_{kp}}{F^{kp}}\).
The field equations put forward in the paper by Einstein and Mayer now are
and turn out to be exactly the EinsteinMaxwell vacuum field equations. Thus, by another formalism, Einstein and Mayer rederived what Klein had obtained in his first paper on Kaluzaâs theory [185].
The authorsâ conclusion is:
âFrom the theory presented here, the equations for the gravitational and the electromagnetic fields follow effortlessly by a unifying method; however, up to now, [the theory] does not bring any understanding for the way corpuscles are built, nor for the facts comprised by quantum theory.â^{Footnote 170} ([107], p. 19)
After this paper Einstein wrote to Ehrenfest in a letter of 17 September 1931 that this theory âin my opinion definitively solves the problem in the macroscopic domainâ ([241], p. 333). Also, in a lecture given on 14 October 1931 in the Physics Institute of the University of Wien, he still was proud of the 5vector approach. In talking about the failed endeavours to reconcile classical field theory and quantum theory (âa cemetery of buried hopesâ) he is reported to have said:
âSince 1928 I also tried to find a bridge, yet left that road again. However, following an idea half of which came from myself and half from my collaborator, Prof. Dr. Mayer, a startlingly simple construction became successful. [âŠ] According to my and Mayerâs opinion, the fifth dimension will not show up. [âŠ] according to which relationships between a hypothetical fivedimensional space and the fourdimensional can be obtained. In this way, we succeeded to recognise the gravitational and electromagnetic fields as a logical unity.â^{Footnote 171} [96]
In his letter to Besso of 30 October 1931, Einstein seemed intrigued by the mathematics used in his paper with Mayer, but not enthusiastic about the physical content of this projective formulation of Kaluzaâs unitary field theory:
âThe only result of our investigation is the unification of gravitation and electricity, whereby the equations for the latter are just Maxwellâs equations for empty space. Hence, no physical progress is made, [if at all] at most only in the sense that one can see that Maxwellâs equations are not just first approximations but appear on as good a rational foundation as the gravitational equations of empty space. Electrical and massdensity are nonexistent; here, splendour ends; perhaps this already belongs to the quantum problem, which up to now is unattainable from the point of view of field [theory] (in the same way as relativity is from the point of view of quantum mechanics). The witty point is the introduction of 5vectors \({a^\sigma }\) in fourdimensional space, which are bound to space by a linear mechanism. Let a^{s} be the 4vector belonging to \({a^\sigma }\); then such a relation \({a^s} = \gamma _\sigma ^s{a^\sigma }\) obtains. In the theory equations are meaningful which hold independently of the special relationship generated by \(\gamma _\sigma ^s\). Infinitesimal transport of \({a^\sigma }\) in fourdimensional space is defined, likewise the corresponding 5curvature from which spring the field equations.â^{Footnote 172} ([99], pp. 274â25)
In his report for the MacyFoundation, which appeared in Science on the very same day in October 1931, Einstein had to be more optimistic:
âThis theory does not yet contain the conclusions of the quantum theory. It furnishes, however, clues to a natural development, from which we may anticipate further developments in this direction. In any event, the results thus far obtained represent a definite advance in knowledge of the structure of physical space.â ([94], p. 439)
Unfortunately, as in the case of his previous papers on Kaluzaâs theory, Einstein came in only second: Veblen had already worked on projective geometry and projective connections for a couple of years [374, 376, 375]. One year prior to Einsteinâs and Mayerâs publication, with his student Hoffmann^{Footnote 173}, he had suggested an application to physics equivalent to the KaluzaKlein theory [381, 163]. However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:
âBut these authors choose a formulation that, due to an unnecessary specialisation of the coordinate system, prefers the fifth coordinate relative to the remaining [coordinates] in much the same way as this had happened in KaluzaKlein theory by means of the cylinder condition [âŠ].â^{Footnote 174} ([249], p. 307)
By using the idea that an affine (n+1)space can be represented by a projective nspace [413], Veblen and Hoffmann avoided the five dimensions of Kaluza: There is a onetoone correspondence between the points of spacetime and a certain congruence of curves in a fivedimensional space for which the fifth coordinate is the curvesâ parameter, while the coordinates of spacetime are fixed. The fivedimensional space is just a mathematical device to represent the events (points) of spacetime by these curves. Geometrically, the theory of Veblen and Hoffmann is more transparent and also more general than Einstein and Mayerâs: It can house the additional scalar field inherent in Kaluzaâs original approach. Thus, Veblen and Hoffmann also gained the Klein.Gordon equation in curved space, i.e., an equation with the Ricci scalar R appearing besides its mass term. Interestingly, the curvature term reads as 5/27R ([381], p. 821). In his note, Hoffmann generalised the formalism such as to include Diracâs equations (without gravitation), although some technical difficulties remained. Nevertheless, Hoffman remained optimistic:
âThere is thus a possibility that the complete system will constitute an improved unification within the relativity theory of the gravitational, electromagnetic and quantum aspects of the field.â ([163], p. 89)
In his book, Veblen emphasised
â[âŠ] that our theory starts from a physical and geometrical point of view totally different from KALUZAâs. In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly fourdimensional.â^{Footnote 175} [379]
Shortly after Einsteinâs and Mayerâs paper had appeared, Schouten and van Dantzig also proved that the 5vector formalism of this paper can be brought into a projective form [314].
In a second note, Einstein and Mayer extended the 5vectorformalism to include Maxwellâs equations with a nonvanishing current density [109]. Of the three basic assumptions of the previous paper, the second had to be given up. The expression in the middle of Equation (153) is replaced by
where, again, F_{kl}=F_{lk}, and the new V_{rlk}=V_{rkl} are arbitrary tensors. The field equations were set up according to the method of the first paper; now the 5curvature scalar was \(P = R  {F_{kp}}{F^{kp}}  {V_{rqp}}{V^{rpq}}\). It also turned out that \({V^{rpq}} = { \epsilon ^{lrpq}}{\phi _l}\) with \({\phi _l} = \tfrac{{\partial \phi }}{{\partial {x^l}}}\), i.e., that the introduction of V_{rpq} brought only one additional variable. The electric current density became âŒV^{V}_{prq}F^{rq}.
In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:
âWe note that Mr. Cartan, in a general and very illuminating investigation, has analysed more deeply the property of systems of differential equations that has been termed by us âcompatibilityâ in this paper and in previous papers.â^{Footnote 176} [37]
At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on projective geometry [361, 362, 360]. He used homogeneous coordinates \({X^\alpha }\), with Î±= 1, âŠ, 5, and the invariant \({g_{\alpha \beta }}{X^\alpha }{X^\beta }\), and introduced projectors and covariant differentiation (cf. Section 2.1.3). Together with him, Schouten wrote a series of papers on projective geometry as the basis of a unified field theory [316, 317, 315, 318]^{Footnote 177}, which, according to Pauli, combine
âall advantages of the formulations of KaluzaKlein and EinsteinMayer while avoiding all their disadvantages.â ([249], p. 307)
Both the EinsteinMayer theory and Veblen and Hoffmannâs approach turned out to be subcases of the more general scheme of Schouten and van Dantzig intending
âto give a unification of general relativity not only with Maxwellâs electromagnetic theory but also with SchrĂ¶dingerâs and Diracâs theory of material waves.â ([318], p. 271)
In this paper ([318], p. 311, Figure 2), we find an early graphical representation of the parametrised set of all possible theories of a kind^{Footnote 178}. The formalism of Schouten and van Dantzig allows for taking the additional dimension to be timelike; in their physical applications the metric of spacetime is taken as a Lorentz metric; torsion is also included in their geometry.
Pauli, with his student J. Solomon^{Footnote 179}, generalised Klein, and Einstein and Mayer by allowing for an arbitrary signature in an investigation concerning âthe form that take Diracâs equations in the unitary theory of Einstein and Mayerâ^{Footnote 180} [253]. In a note added after proofreading, the authors showed that they had noted Schouten and Dantzigâs papers [316, 317]. The authors pointed out that
â[âŠ] even in the absence of gravitation we must pay attention to a difference between Diracâs equation in the theory of Einstein and Mayer, and Diracâs equation as it is written out, usually.â^{Footnote 181} ([253], p. 458)
The second order wave equation iterated from their form of Diracâs equation, besides the spin term contained a curvature term 1/4R, with the numerical factor different from Veblenâs and Hoffmannâs. In a sequel to this publication, Pauli and Solomon corrected an error:
âWe examine from a general point of view the theory of spinors in a fivedimensional space. Then we discuss the form of the energymomentum tensor and of the current vector in the theory of EinsteinMayer.[âŠ] Unfortunately, it turned out that the considerations of Â§in the first part are marred by a calculational errorâŠ This has made it necessary to introduce a new expression for the energymomentum tensor and [âŠ] likewise for the current vector [âŠ].â^{Footnote 182} ([254], p. 582)
In the California Institute of Technology, Einsteinâs and Mayerâs new mathematical technique found an attentive reader as well; A. D. Michal and his coauthor generalised the EinsteinMayer 5vectorformalism:
âThe geometry considered by Einstein and Mayer in their âUnified field theoryâ leads to the consideration of an ndimensional Riemannian space V_{n} with a metric tensor g_{ij}, to each point of which is associated an mdimensional linear vector space V_{m}, (m>n), for which vector spaces a general linear connection is defined. For the general case (mânâ 1) we find that the calculation of the mân âexceptional directionsâ is not unique, and that an additional postulate on the linear connection is necessary. Several of the new theorems give new results even for n=4, m=5, the EinsteinMayer case.â [228]
Michal had come from Cartan and Schoutenâs papers on group manifolds and the distant parallelisms defined on them [227]. H. P. Robertson found a new way of applying distant parallelism: He studied groups of motion admitted by such spaces, e.g., by Einsteinâs and Mayerâs spherically symmetric exact solution [282] (cf. Section 6.4.3).
Cartan wrote a paper on the EinsteinMayer theory as well ([39], an article published only posthumously) in which he showed that this could be interpreted as a fivedimensional flat geometry with torsion, in which spacetime is embedded as a totally geodesic subspace.
Distant parallelism
The next geometry Einstein took as a fundament for unified field theory was a geometry with Riemannian metric, vanishing curvature, and nonvanishing torsion, named âabsolute parallelismâ, âdistant parallelismâ, âteleparallelism, or âFernparallelismusâ. The contributions from the LeviCivita connection and from contorsion^{Footnote 183} in the curvature tensor cancel. In place of the metric, tetrads are introduced as the basic variables. As in Euclidean space, in the new geometry these 4beins can be parallely translated to retain the same fixed directions everywhere. Thus, again, a degree of absoluteness is reintroduced into geometry in contrast to Weylâs first attempt at unification which tried to soften the ârigidityâ of Riemannian geometry.
The geometric concept of âfields of parallel vectorsâ had been introduced on the level of advanced textbooks by Eisenhart as early as 1925â1927 [119, 121] without use of the concept of a metric. In particular, the vanishing of the (affine) curvature tensor was given as a necessary and sufficient condition for the existence of D linearly independent fields of parallel vectors in a Ddimensional affine space ([121], p. 19).
Cartan and Einstein
As concerns the geometry of âFernparallelismâ, it is a special case of a space with Euclidean connection introduced by Cartan in 1922/23 [31, 30, 32]. Pais let Einstein âinventâ and âdiscoverâ distant parallelism, and he states that Einstein âdid not know that Cartan was already aware of this geometryâ ([241], pp. 344â345). However, when Einstein published his contributions in June 1928 [84, 83], Cartan had to remind him that a paper of his introducing the concept of torsion had
âappeared at the moment at which you gave your talks at the CollĂšge de France. I even remember having tried, at Hadamardâs place, to give you the most simple example of a Riemannian space with Fernparallelismus by taking a sphere and by treating as parallels two vectors forming the same angle with the meridians going through their two origins: the corresponding geodesics are the rhumb lines.â^{Footnote 184} (letter of Cartan to Einstein on 8 May 1929; cf. [50], p. 4)
This remark refers to Einsteinâs visit in Paris in March/April 1922. Einstein had believed to have found the idea of distant parallelism by himself. In this regard, Pais may be correct. Every researcher knows how an idea, heard or read someplace, can subconsciously work for years and then surface all of a sudden as his or her own new idea without the slightest remembrance as to where it came from. It seems that this happened also to Einstein. It is quite understandable that he did not remember what had happened six years earlier; perhaps, he had not even fully followed then what Cartan wanted to explain to him. In any case, Einsteinâs motivation came from the wish to generalise Riemannian geometry such that the electromagnetic field could be geometrized:
âTherefore, the endeavour of the theoreticians is directed toward finding natural generalisations of, or supplements to, Riemannian geometry in the hope of reaching a logical building in which all physical field concepts are unified by one single viewpoint.â^{Footnote 185} ([84], p. 217)
In an investigation concerning spaces with simply transitive continuous groups, Eisenhart already in 1925 had found the connection for a manifold with distant parallelism given 3 years later by Einstein [118]. He also had taken up Cartanâs idea and, in 1926, produced a joint paper with Cartan on âRiemannian geometries admitting absolute parallelismâ [40], and Cartan also had written about absolute parallelism in Riemannian spaces [33]. Einstein, of course, could not have been expected to react to these and other purely mathematical papers by Cartan and Schouten, focussed on group manifolds as spaces with torsion and vanishing curvature ([41, 34], pp. 50â54). No physical application had been envisaged by these two mathematicians.
Nevertheless, this story of distant parallelism raises the question of whether Einstein kept up on mathematical developments himself, or whether, at the least, he demanded of his assistants to read the mathematical literature. Against his familiarity with mathematical papers speaks the fact that he did not use the name âtorsionâ in his publications to be described in the following section. In the area of unified field theory including spinor theory, Einstein just loved to do the mathematics himself, irrespective of whether others had done it before â and done so even better (cf. Section 7.3).
Anyhow, in his response (Einstein to Cartan on 10 May 1929, [50], p. 10), Einstein admitted Cartanâs priority and referred also to Eisenhartâs book of 1927 and to WeitzenbĂ¶ckâs paper [393]. He excused himself byWeitzenbĂ¶ckâs likewise omittance of Cartanâs papers among his 14 references. In his answer, Cartan found it curious that WeitzenbĂ¶ck was silent because
â[âŠ] he indicates in his bibliography a note by Bortolotti in which he several times refers to my papers.â^{Footnote 186} (Cartan to Einstein on 15 May 1929; [50], p. 14)
The embarrassing situation was solved by Einsteinâs suggestion that he had submitted a comprehensive paper on the subject to Zeitschrift fĂŒr Physik, and he invited Cartan to add his description of the historical record in another paper (Einstein to Cartan on 10 May 1929). After Cartan had sent his historical review to Einstein on 24 May 1929, the latter answered three months later:
âI am now writing up the work for the Mathematische Annalen and should like to add yours [âŠ]. The publication should appear in the Mathematische Annalen because, at present, only the mathematical implications are explored and not their applications to physics.â^{Footnote 187} (letter of Einstein to Cartan on 25 August 1929 [50, 35, 89])
In his article, Cartan made it very clear that it was not WeitzenbĂ¶ck who had introduced the concept of distant parallelism, as valuable as his results were after the concept had become known. Also, he took Einsteinâs treatment of Fernparallelism as a special case of his more general considerations. Interestingly, he permitted himself to interpet the physical meaning of geometrical structures^{Footnote 188}:
âLet us say simply that mechanical phenomena are of a purely affine nature whereas electromagnetic phenomena are essentially metric; therefore it is rather natural to try to represent the electromagnetic potential by a not purely affine vector.â^{Footnote 189} ([35], p. 703)
Einstein explained:
âIn particular, I learned from Mr. WeitzenbĂ¶ck and Mr. Cartan that the treatment of continua of the species which is of import here, is not really new.[âŠ] In any case, what is most important in the paper, and new in any case, is the discovery of the simplest field laws that can be imposed on a Riemannian manifold with Fernparallelismus.â^{Footnote 190} ([89], p. 685)
For Einstein, the attraction of his theory consisted
âFor me, the great attraction of the theory presented here lies in its unity and in the allowed highly overdetermined field variables. I also could show that the field equations, in first approximation, lead to equations that correspond to the NewtonPoisson theory of gravitation and to Maxwellâs theory. Nevertheless, I still am far from being able to claim that the derived equations have a physical meaning. The reason is that I could not derive the equations of motion for the corpuscles.â^{Footnote 191} ([89], p. 697)
The split, in first approximation, of the tetrad field h_{ab} according to \({h_{ab}} = {\eta _{ab}} + {\bar h_{ab}}\) lead to homogeneous wave equations and divergence relations for both the symmetric and the antisymmetric part identified as metric and electromagnetic field tensors, respectively.
How the word spread
Einstein in 1929 really seemed to have believed that he was on a good track because, in this and the following year, he published at least 9 articles on distant parallelism and unified field theory before switching off his interest. The press did its best to spread the word: On 2 February 1929, in its column News and Views, the respected British science journal Nature reported:
âFor some time it has been rumoured that Prof. Einstein has been about to publish the results of a protracted investigation into the possibility of generalising the theory of relativity so as to include the phenomena of electromagnetism. It is now announced that he has submitted to the Prussian Academy of Sciences a short paper in which the laws of gravitation and of electromagnetism are expressed in a single statement.â
Nature then went on to quote from an interview of Einstein of 26 January 1929 in a newspaper, the Daily Chronicle. According to the newspaper, among other statements Einstein made, in his wonderful language, was the following:
âNow, but only now, we know that the force which moves electrons in their ellipses about the nuclei of atoms is the same force which moves our earth in its annual course about the sun, and it is the same force which brings to us the rays of light and heat which make life possible upon this planet.â [2]
Whether Einstein used this as a metaphorical language or, whether he at this time still believed that the system ânucleus and electronsâ is dominated by the electromagnetic force, remains open.
The paper announced by Nature is Einsteinâs âZur einheitlichen Feldtheorieâ, published by the Prussian Academy on 30 January 1929 [88]. A thousand copies of this paper had been sold within 3 days, so the presiding secretary of the Academy ordered the printing of a second thousand. Normally, only a hundred copies were printed ([183], Dokument Nr. 49, p. 136). On 4 February 1929, The Times (of London) published the translation of an article by Einstein, âwritten as an explanation of his thesis for readers who do not possess an expert knowledge of mathematicsâ. This article then became reprinted in March by the British astronomy journal The Observatory [86]. In it, Einstein first gave a historical sketch leading up to the introduction of relativity theory, and then described the method that guided him to the new theory of distant parallelism. In fact, the only formulas appearing are the line elements for twodimensional Riemannian and Euclidean space. At the end, by one figure, Einstein tried to convey to the reader what consequence a Euclidean geometry with torsion would have â without using that name. His closing sentences are^{Footnote 192}:
âWhich are the simplest and most natural conditions to which a continuum of this kind can be subjected? The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electromagnetism.â ([86], p. 118)
A few months later in that year, again in Nature, the mathematician H. T. H. Piaggio gave an exposition for the general reader of âEinsteinâs and other Unitary Field Theoriesâ. He was a bit more explicit than Einstein in his article for the educated general reader. However, he was careful to end it with a warning:
âOf course the ultimate test of the theory must be by experiment. It may succeed in predicting some interaction between gravitation and electromagnetism which can be confirmed by observation. On the other hand, it may be only a âgraphâ and so outside the ken of the ordinary physicist.â ([258], p. 879)
The use of the concept âgraphâ had its origin in Eddingtonâs interpretation of his and other peoplesâ unified field theories to be only graphs of the world; the true geometry remained the Riemannian geometry underlying Einsteinâs general relativity.
Even the FrenchBelgian writer and poet Maurice Maeterlinck had heard of Einsteinâs latest achievement in the area of unified field theory. In his poetic presentation of the universe âLa grande fĂ©erieâ^{Footnote 193} we find his remark:
âEinstein, in his last publications comments to which are still to appear, again brings us mathematical formulae which are applicable to both gravitation and electricity, as if these two forces seemingly governing the universe were identical and subject to the same law. If this were true it would be impossible to calculate the consequences.â^{Footnote 194} ([214], p. 68)
Einsteinâs research papers
We are dealing here with Einsteinâs, and Einstein and Mayerâs joint papers on distant parallelism in the reports of the Berlin Academy and Mathematische Annalen, which were taken as the starting point by other researchers following suit with further calculations. Indeed, there was a lot of work to do, only in part because Einstein, from one paper to the next, had changed his field equations^{Footnote 195}.
In his first note [84], dynamics was absent; Einstein made geometrical considerations his main theme: Introduction of a local ânbeinfieldâ \(h_{\hat \imath}^k\) at every point of a differentiable manifold and the related object \({h_{k\hat \imath}}\) defined as the collection of the ânormed subdeterminants of the \(h_{\hat \imath}^k\)â^{Footnote 196} such that \(h_{i\hat l}h_{\hat l}^k = \delta _i^k\). As we have seen before, the components of the metric tensor are defined by
where summation over \(\hat \jmath = 1, \ldots ,n\) is assumed^{Footnote 197}.
âFernparallelismâ now means that if the components referred to the local nbein of a vector \({A^{\hat k}} = h_l^{\hat k}{A^l}\) at a point p, and of a vector \({B^{\hat k}}\) at a different point q are the same, i.e., \({A^{\hat k}} = {B^{\hat k}}\), then the vectors are to be considered as âparallelâ. There is an underlying symmetry, called ârotational invarianceâ by Einstein: joint rotations of each nbein by the same angle. All relations with a physical meaning must be ârotationally invariantâ. Of course, in spacetime with a Lorentz metric, the 4beintransformations do form the proper Lorentz group.
If parallel transport of a tangent vector A is defined as usual by \(d{A^k} =  {\Delta _{lm}}{A^l}d{x^m}\), then the connection components turn out to be
An immediate consequence is that the covariant derivative of each beinvector vanishes,
by use of Equation (161). Also, the metric is covariantly constant
Neither fact is mentioned in Einsteinâs note. Also, no reference is given to Eisenhartâs paper of 1925 [118], in which the connection (161) had been given (Equation (3.5) on p. 248 of [118]), as noted above, its metriccompatibility shown, and the vanishing of the curvature tensor concluded.
The (Riemannian) curvature tensor calculated from Equation (161) turns out to vanish. As Einstein noted, by g_{ij} from Equation (160) also the usual Riemannian connection \({\Gamma _{lm}}^k(g)\) may be formed. Moreover, \({Y_{lm}}^k: = {\Gamma _{lm}}^k(g)  {\Delta _{lm}}^k\) is a tensor that could be used for building invariants. In principle, distant parallelism is a particular biconnection theory. The connection \({\Gamma _{lm}}^k(g)\) does not play a role in the following (cf., however, de Donder^{Footnote 198}âs paper [48]).
From Equation (161), obviously the torsion tensor \({S_{lm}}^k = \tfrac{1}{2}({\Delta _{lm}}^k  {\Delta _{ml}}^k) \ne 0\) follows (cf. Equation (21)). Einstein denoted it by \({\Lambda _{lm}}^k\) and, in comparison with the curvature tensor, considered it as the âformally simplestâ tensor of the theory for building invariants by help of the linear form \({\Lambda _{lj}}^jd{x^l}\) and of the scalars \({g^{ij}}{\Lambda _{im}}^l{\Lambda _{jl}}^m\) and \({g_{ij}}\;{g^{lr}}{g^{ms}}{\Lambda _{lm}}^i{\Lambda _{rs}}^{j.}\). He indicated how a Lagrangian could be built and the 16 field equations for the field variables h_{lj} obtained.
At the end of the note Einstein compared his new approach to Weylâs and Riemannâs:

WEYL: Comparison at a distance neither of lengths nor of directions;

RIEMANN: Comparison at a distance of lengths but not of directions;

Present theory: Comparison at a distance of both lengths and directions.
In his second note [83], Einstein departed from the Lagrangian \({\mathcal L} = h\;{g^{ij}}{\Lambda _{im}}^l{\Lambda _{jl}}^m\), i.e., a scalar density corresponding to the first scalar invariant of his previous note^{Footnote 199}. He introduced \({\phi _k}: = {\Lambda _{kl}}^l\), and took the case Ï=0 to describe a âpurely gravitational fieldâ. However, as he added in a footnote, pure gravitation could have been characterised by \(\tfrac{{\partial {\phi _i}}}{{\partial {x^k}}}  \tfrac{{\partial {\phi _k}}}{{\partial {x^i}}}\) as well. In his first paper on distant parallelism, Einstein did not use the names âelectrical potentialâ or âelectrical fieldâ. He then showed that in a firstorder approximation starting from \({h_{i\hat \jmath}} = {\delta _{i\hat \jmath}} + {j_{i\hat \jmath}} + \ldots\), both the Einstein vacuum field equations and Maxwellâs equations are surfacing. To do so he replaced \({h_{i\hat \jmath }}\) by \({g_{ij}} = {\delta _{ij}} + 2{k_{(ij)}}\) and introduced \({\phi _k}: = \tfrac{1}{2}\left( {\tfrac{{\partial {k_{kj}}}}{{\partial {x^j}}}  \tfrac{{\partial {k_{jj}}}}{{\partial {x^k}}}} \right)\). Einstein concluded that
âThe separation of the gravitational and the electromagnetic field appears artificial in this theory. [âŠ] Furthermore, it is remarkable that, according to this theory, the electromagnetic field does not enter the field equations quadratically.â^{Footnote 200} ([83], p. 6)
In a postscript, Einstein noted that he could have obtained similar results by using the second scalar invariant of his previous note, and that there was a certain indeterminacy as to the choice of the Lagrangian.
This shows clearly that the ambiguity in the choice of a Lagrangian had bothered Einstein. Thus, in his third note, he looked for a more reassuring way of deriving field equations [88]. He left aside the Hamiltonian principle and started from identities for the torsion tensor, following from the vanishing of the curvature tensor^{Footnote 201}. He thus arrived at the identity given by Equation (29), i.e., (Einsteinâs equation (3), p. 5; his convention is \({\Lambda _{\hat k\hat l}}\,^{\hat \imath}: = 2{S_{\hat k\hat l}}\,^{\hat \imath}\))
By defining \({\phi _{\hat k}}: = {\Lambda _{\hat k\hat l}}^{\;\hat l} = 2\;{S_{\hat k\hat l}}^{\;\hat l}\), and contracting equation (164), Einstein obtained another identity^{Footnote 202}:
where the covariant divergence refers to the connection components \({\Delta _{\hat m\hat l}}^{\hat k}\), and the tensor density \({\hat V_{\hat k\hat l}}^{\hat \jmath}\) is given by
For the proof, he used the formula for the covariant vector density given in Equation (16), which, for the divergence, reduces to \({\mathop \nabla \limits^ + _i}{X^i} = \tfrac{{\partial {{\hat X}^i}}}{{\partial {x^i}}}  2{S_j}{\hat X^j}\).
The second identity used by Einstein follows with the help of Equation (27) for vanishing curvature (Einsteinâs equation (5), p. 5):
As we have seen in Section 2, if he had read it, Einstein could have taken these identities from Schoutenâs book of 1924 [300].
By replacing Ă^{jk} by VÌ^{kÌlÌj} and using Equation (165), the final form of the second identity now is
Einstein first wrote down a preliminary set of field equations from which, in first approximation, both the gravitational vacuum field equations (in the limit â=0, cf. below) and Maxwellâs equations follow:
Here,
replaces \({\hat V_{kl}}^j\) such that the necessary number of equations is obtained. With this first approximation as a hint, Einstein, after some manipulations, postulated the 20 exact field equations:
among which 8 identities hold.
Einstein seems to have sensed that the average reader might be able to follow his path to the postulated field equations only with difficulty. Therefore, in a postscript, he tried to clear up his motivation:
âThe field equations suggested in this paper may be characterised with regard to other such possible ones in the following way. By staying close to the identity (167), it has been accomplished that not only 16, but 20 independent equations can be imposed on the 16 quantities h ^{kÌ}_{i} By âindependentâ we understand that none of these equations can be derived from the remaining ones, even if there exist 8 identical (differential) relations among them.â^{Footnote 203} ([88], p. 8)
He still was not entirely sure that the theory was physically acceptable:
âA deeper investigation of the consequences of the field equations (170) will have to show whether the Riemannian metric, together with distant parallelism, really gives an adequate representation of the physical qualities of space.â^{Footnote 204}
In his second paper of 1929, the fourth in the series in the Berlin Academy, Einstein returned to the Hamiltonian principle because his collaborators Lanczos and MĂŒntz^{Footnote 205} had doubted the validity of the field equations of his previous publication [88] on grounds of their unproven compatibility. In the meantime, however, he had found a Lagrangian such that the compatibilityproblem disappeared. He restricted the many constructive possibilities for \({\mathcal L} = {\mathcal L}(h_i^{\hat k},{\partial _l}h_i^{\hat k})\) by asking for a Lagrangian containing torsion at most quadratically. His Lagrangian is a particular linear combination of the three possible scalar densities, as follows:

(1)
\(\hat H = {\tfrac{1}{2}}{{\hat \jmath }_1} + {\tfrac{1}{4}}{{\hat \jmath }_2}  {{\hat \jmath }_3},\)

(2)
\(\hat H* = {\tfrac{1}{2}}{\hat \jmath _1}  {\tfrac{1}{4}}{\hat \jmath _2},\)

(3)
\(\hat H** = {{\hat \jmath }_3},\)
with \({{\hat \jmath }_1}: = h\;{S_{kl}}^m{S^k}{_m^l}\), \({{\hat \jmath }_2}: = h\;{S_{kl}}^m{S^{kl}}_m\), and \({{\hat \jmath }_3}: = h\;{S_j}{S^j}\). If â_{1}, â_{2} are small parameters, then his final Lagrangian is \({\mathcal L} = \hat H + { \epsilon _1}\hat H* + { \epsilon _2}\hat H**\). In order to prove that Maxwellâs equations follow from his Lagrangian, Einstein had to perform the limit \(\sigma : = \tfrac{{{ \epsilon _1}}}{{{ \epsilon _2}}} \to 0\) in an expression termed \(\hat G*{\;^{\;ik}}\), which he assumed to depend homogeneously and quadratically on a linear combination of torsion^{Footnote 206}.
In a Festschrift for his former teacher and colleague in ZĂŒrich, A. Stodola, Einstein summed up what he had reached^{Footnote 207}. He exchanged the definition of the invariants named \({{\hat \jmath}_2}\), and \({{\hat \jmath}_3}\), and stated that a choice of A=B, C=0 in the Lagrangian \(\hat \jmath = A{{\hat \jmath }_1} + B{{\hat \jmath }_2} + C{{\hat \jmath }_3}\) would give field equations
â[âŠ] that coincide in first approximation with the known laws for the gravitational and electromagnetic field [âŠ]â^{Footnote 208}
with the proviso that the specialisation of the constants A, B, C must be made only after the variation of the Lagrangian, not before. Also, together with MĂŒntz, he had shown that for an uncharged mass point the Schwarzschild solution again obtained [87].
Einsteinâs next publication was the note preceding Cartanâs paper in Mathematische Annalen [89]. He presented it as an introduction suited for anyone who knew general relativity. It is here that he first mentioned Equations (162) and (163). Most importantly, he gave a new set of field equations not derived from a variational principle; they are^{Footnote 209}.
where S^{ikl}=g^{im}g^{kn}S_{mn}^{l}. There exist 4 identities among the 16+6 field equations
As Cartan remarked, Equation (172) expresses conservation of torsion under parallel transport:
âIn fact, in the new theory of Mr. Einstein, it is natural to call a universe homogeneous if the torsion vectors that are associated to two parallel surface elements are parallel themselves; this means that parallel transport conserves torsion.â^{Footnote 210} ([35], p. 703)
From Equation (173) with the help of Equation (171), (172), Einstein wrote down two more identities. One of them he had obtained from Cartan:
âBut I am very grateful to you for the identity
$$ {G^{ik}}_{\left\ i \right.}  {S_{lm}}^{\;k}{G^{lm}} = 0, $$which, astonishingly, had escaped me. [âŠ] In a new presentation in the Sitzungsberichten, I used this identity while taking the liberty of pointing to you as its source.â ^{Footnote 211} (letter of Einstein to Cartan from 18 December 1929, Document X of [50], p. 72)
In order to show that his field equations were compatible he counted the number of equations, identities, and field quantities (in ndimensional space) to find, in the end, n^{2}+n equations for the same number of variables. To do so, he had to introduce an additional variable Ï via \({F_k} = {\phi _k}  \tfrac{{\partial \log \psi }}{{\partial {x^k}}}\). Here, F_{k} is introduced by \({F_{ik}} = {\partial _k}{F_i}  {\partial _i}{F_k} = {\partial _k}{\phi _i}  {\partial _i}{\phi _k}\). Einstein then showed that \({\partial _l}{F_{ik}} + {\partial _k}{F_{li}} + {\partial _i}{F_{kl}} = 0\).
The changes in his approach Einstein continuously made, must have been hard on those who tried to follow him in their scientific work. One of them, Zaycoff^{Footnote 212}, tried to make the best out of them:
In November 1929, Einstein gave two lectures at the Institute Henri PoincarĂ© in Paris which had been opened one year earlier in order to strengthen theoretical physics in France ([14], pp. 263â272). They were published in 1930 as the first article in the new journal of this institute [92]. On 23 pages he clearly and leisurely outlined his theory of distant parallelism and the progress he had made. As to references given, first Cartanâs name is mentioned in the text:
âIt is not for the first time that such spaces are envisaged. From a purely mathematical point of view they were studied previously. M. Cartan was so amiable as to write a note for the Mathematische Annalen exposing the various phases in the formal development of these concepts.â^{Footnote 214} ([92], p. 4)
Note that Einstein does not say that it was Cartan who first âenvisagedâ these spaces before. Later in the paper, he comes closer to the point:
âThis type of space had been envisaged before me by mathematicians, notably by WEITZENBĂCK, EISENHART et CARTAN [âŠ].â^{Footnote 215} [92]
Again, he held back in his support of Cartanâs priority claim.
Some of the material in the paper overlaps with results from other publications [85, 90, 93]. The counting of independent variables, field equations, and identities is repeated from Einsteinâs paper in Mathematische Annalen [89]. For n=4, there were 20 field equations (G^{ik}=0, F^{l}=0) for 16+1 variables \(h_{\hat \imath}^k\) and Ï, four of which were arbitrary (coordinate choice). Hence 7 identities should exist, four of which Einstein had found previously. He now presented a derivation of the remaining three identities by a calculation of two pagesâ length. The field equations are the same as in [89]; the proof of their compatibility takes up, in a slightly modified form, the one communicated by Einstein to Cartan in a letter of 18 December 1929 ([92], p. 20). It is reproduced also in [90].
Interestingly, right after Einsteinâs article in the instituteâs journal, a paper of C. G. Darwin, âOn the wave theory of matterâ, is printed, and, in the same first volume, a report of Max Born on âSome problems in Quantum Mechanics.â Thus, French readers were kept uptodate on progress made by both parties â whether they worked on classical field theory or quantum theory [45, 21].
A. Proca, who had attended Einsteinâs lectures, gave an exposition of them in a journal of his native Romania. He was quite enthusiastic about Einsteinâs new theory:
âA great step forward has been made in the pursuit of this total synthesis of phenomena which is, right or wrong, the ideal of physicists. [âŠ] the splendid effort brought about by Einstein permits us to hope that the last theoretical difficulties will be vanquished, and that we soon will compare the consequences of the theory with [our] experience, the great stepping stone of all creations of the mind.â^{Footnote 216} [260, 261]
Einsteinâs next paper in the Berlin Academy, in which he reverts to his original notation h ^{kÌ}_{l} , consisted of a brief critical summary of the formalism used in his previous papers, and the announcement of a serious mistake in his first note in 1930, which made invalid the derivation of the field equations for the electromagnetic field ([90], p. 18). The mistake was the assumption on the kind of dependence on torsion of the quantity Ä*^{ik}, which was mentioned above. Also, Einstein now found it better âto keep the concept of divergence, defined by contraction of the extension of a tensorâ and not use the covariant derivative \(\nabla { * _l}\) introduced by him in his third paper in the Berlin Academy [88].
Then Einstein presented the same field equations as in his paper in Annalen der Mathematik, which he demanded to be

(1)
covariant,

(2)
of second order, and

(3)
linear in the second derivatives of the field variable h ^{k}_{i} .
while these demands had been sufficient to uniquely lead to the gravitational field equations (with cosmological constant) of general relativity, in the teleparallelism theory a great deal of ambiguity remained. Sixteen field equations were needed which, due to covariance, induced four identities.
âTherefore equations must be postulated among which identical relations are holding. The higher the number of equations (and consequently also the number of identities among them), the more precise and stronger than mere determinism is the content; accordingly, the theory is the more valuable, if it is also consistent with the empirical facts.â^{Footnote 217} ([90], p. 21)
He then gave a proof of the compatibility of his field equations:
âThe proof of the compatibility, as given in my paper in the Mathematische Annalen, has been somewhat simplified due to a communication which I owe to a letter of Mr. CARTAN (cf. Â§3, [16]).â^{Footnote 218}
The reader had to make out for himself what Cartanâs contribution really was.
In linear approximation, i.e., for \({h_{ik}} = {\delta _{ik}} + {\bar h_{ik}}\), Einstein obtained dâAlembertâs equation for both the symmetric and the antisymmetric part of hÌ_{ik}, identified with the gravitational and the electromagnetic field, respectively.
Einsteinâs next note of one and a half pages contained a mathematical result within teleparallelism theory: From any tensor with an antisymmetric pair of indices a vector with vanishing divergence can be derived [93].
In order to test the field equations by exhibiting an exact solution, a simple case would be to take a spherically symmetric, asymptotically (Minkowskian) 4bein. This is what Einstein and Mayer did, except with the additional assumption of spacereflection symmetry [106]. Then the 4bein contains three arbitrary functions of one parameter s:
where Î±, \(\hat \imath = 1,2,3\). As an exact solution of the field equations (171, 172), Einstein and Mayer obtained \(\lambda = \tau = e{s^{  3}}{(1  {e^2}/{s^4})^{  \tfrac{1}{4}}})\) and \(u = 1 + m\;\int {ds\;{s^{  2}}(1  {e^2}/{s^4})} {\;^{  \tfrac{1}{4}}}\). The constants e and m were interpreted as electric charge and âponderomotive mass,â respectively. A further exact solution for uncharged point particles was also derived; it is static and corresponds to âtwo or more unconnected electrically neutral masses which can stay at rest at arbitrary distancesâ. Einstein and Mayer do not take this physically unacceptable situation as an argument against the theory, because the equations of motion for such singularities could not be derived from the field equations as in general relativity. Again, the continuing wish to describe elementary particles by singularityfree exact solutions is stressed.
Possibly, W. F. G. Swan of the Bartol Research Foundation in Swarthmore had this paper in mind when he, in April 1930, in a brief description of Einsteinâs latest publications, told the readers of Science:
âIt now appears that Einstein has succeeded in working out the consequences of his general law of gravity and electromagnetism for two special cases just as Newton succeeded in working out the consequences of his law for several special cases. [âŠ] It is hoped that the present solutions obtained by Einstein, or if not these, then others which may later evolve, will suggest some experiments by which the theory may be tested.â ([339], p. 391)
Two days before the paper by Einstein and Mayer became published by the Berlin Academy, Einstein wrote to his friend Solovine:
âMy field theory is progressing well. Cartan has already worked with it. I myself work with a mathematician (S. Mayer^{Footnote 219} from Vienna), a marvelous chap [âŠ].â^{Footnote 220} ([98], p. 56)
The mentioning of Cartan resulted from the intensive correspondence of both scientists between December 1929 and February 1930: About a dozen letters were exchanged which, sometimes, contained long calculations [50] (cf. Section 6.4.6). In an address given at the University of Nottingham, England, on 6 June 1930, Einstein also must have commented on the exact solutions found and on his program concerning the elementary particles. A report of this address stated about Einsteinâs program:
âThe problem is nearly solved; and to the first approximations he gets laws of gravitation and electromagnetics. He does not, however, regard this as sufficient, though those laws may come out. He still wants to have the motions of ordinary particles to come out quite naturally. [The program] has been solved for what he calls the âquasistatical motionsâ, but he also wants to derive elements of matter (electrons and protons) out of the metric structure of space.â ([91], p. 610)
With his âassistantâ Walther Mayer, Einstein then embarked on a very technical, systematic study of compatible field equations for distant parallelism [108]. In addition to the assumptions (1), (2), (3) for allowable field equations given above, further restrictions were made:

(4)
the field equations must contain the first derivatives of the field variable h ^{kÌ}_{i} only quadratically;

(5)
the identities for the left hand sides G^{ik} of the field equations must be linear in G^{ik} and contain only their first derivatives;

(6)
torsion must occur only linearly in G^{ik}.
For the field equations, the following ansatz was made:
where R^{ik} is a collection of terms quadratic in torsion S, and p, q, a_{1}, a_{2}, a_{3} are constants. They must be determined in such a way that the âdivergenceidentityâ
is satisfied. Here, 8 new constants A, B, c_{r} with r=1, âŠ, 6 to be fixed in the process also appear. After inserting Equation (175) into Equation (176), Einstein and Mayer reduced the problem to the determination of 10 constants by 20 algebraic equations by a lengthy calculation. In the end, four different types of compatible field equations for the teleparallelism theory remained:
âTwo of these are (nontrivial) generalisations of the original gravitational field equations, one of them being known already as a consequence of the Hamiltonian principle. The remaining two types are denoted in the paper by [âŠ].â^{Footnote 221}
With no further restraining principles at hand, this ambiguity in the choice of field equations must have convinced Einstein that the theory of distant parallelism could no longer be upheld as a good candidate for the unified field theory he was looking for, irrespective of the possible physical content^{Footnote 222}. Once again, he dropped the subject and moved on to the next. While aboard a ship back to Europe from the United States, Einstein, on 21 March 1932, wrote to Cartan:
â[âŠ] In any case, I have now completely given up the method of distant parallelism. It seems that this structure has nothing to do with the true character of space [âŠ].â ([50], p. 209)
What Cartan might have felt, after investing the forty odd pages of his calculations printed in Debeverâs book, is unknown. However, the correspondence on the subject came to an end in May 1930 with a last letter by Cartan. ^{Footnote 223}.
Reactions I: Mostly critical
About half a year after Einsteinâs two papers on distant parallelism of 1928 had appeared, Reichenbach^{Footnote 224}, who always tended to defend Einstein against criticism, classified the new theory [268] according to the lines set out in his book [267] as âhaving already its precisely fixed logical position in the edifice of WeylEddington geometryâ ([267], p. 683). He mentioned as a possible generalization an idea of Einsteinâs, in which the operation of parallel transport might be taken as integrable not with regard to length but with regard to direction: âa generalisation which already has been conceived by Einstein as I learned from himâ ([267], p. 687)^{Footnote 225}.
As concerns parallelism at a distance, Reichenbach was not enthusiastic about Einsteinâs new approach:
â[âŠ] it is the aim of Einsteinâs new theory to find such an entanglement between gravitation and electricity that it splits into the separate equations of the existing theory only in first approximation; in higher approximation, however, a mutual influence of both fields is brought in, which, possibly, leads to an understanding of questions unanswered up to now as [is the case] for the quantum riddle. But this aim seems to be in reach only if a direct physical interpretation of the operation of transport, even of the immediate field quantities, is given up. From the geometrical point of view, such a path [of approach] must seem very unsatisfactory; its justifications will only be reached if the mentioned link does encompass more physical facts than have been brought into it for building it up.â^{Footnote 226} ([267], p. 689)
A first reaction from a competing colleague came from Eddington, who, on 23 February 1929, gave a cautious but distinct review of Einsteinâs first three publications on distant parallelism [84, 83, 88] in Nature. After having explained the theory and having pointed out the differences to his own affine unified field theory of 1921, he confessed:
âFor my own part I cannot readily give up the affine picture, where gravitational and electric quantities supplement one another as belonging respectively to the symmetrical and antisymmetrical features of world measurement; it is difficult to imagine a neater kind of dovetailing. Perhaps one who believes that Weylâs theory and its affine generalisation afford considerable enlightenment, may be excused for doubting whether the new theory offers sufficient inducement to make an exchange.â [62]
Weyl was the next unhappy colleague; in connection with the redefinition of his gauge idea he remarked (in April/May 1929):
â[âŠ] my approach is radically different, because I reject distant parallelism and keep to Einsteinâs general relativity. [âŠ] Various reasons hold me back from believing in parallelism at a distance. First, my mathematical intuition a priori resists to accept such an artificial geometry; I have difficulties to understand the might who has frozen into rigid togetherness the local frames in different events in their twisted positions. Two weighty physical arguments join in [âŠ] only by this loosening [of the relationship between the local frames] the existing gaugeinvariance becomes intelligible. Second, the possibility to rotate the frames independently, in the different events, [âŠ] is equivalent to the s y m m e t r y o f t h e e n e r g y  m o m e n t u m t e n s o r, or to the validity of the conservation law for angular momentum.â^{Footnote 227} ([407], pp. 330â332.)
As usual, Pauli was less than enthusiastic; he expressed his discontent in a letter to Hermann Weyl of 26 August 1929:
âFirst let me emphasize that side of the matter about which I fully agree with you: Your approach for incorporating gravitation into Diracâs theory of the spinning electron [âŠ] I am as adverse with regard to Fernparallelismus as you are [âŠ] (And here I must do justice to your work in physics. When you made your theory with gâČ_{ik}=Î»g_{ik} this was pure mathematics and unphysical; Einstein rightly criticised and scolded you. Now the hour of revenge has come for you, now Einstein has made the blunder of distant parallelism which is nothing but mathematics unrelated to physics, now you may scold [him].)â^{Footnote 228} ([251], pp. 518â519)
Another confession of Pauliâs went to Paul Ehrenfest:
âBy the way, I now no longer believe in one syllable of teleparallelism; Einstein seems to have been abandoned by the dear Lord.â^{Footnote 229} (Pauli to Ehrenfest 29 September 1929; [251], p. 524)
Pauliâs remark shows the importance of ideology in this field: As long as no empirical basis exists, beliefs, hopes, expectations, and rationally guided guesses abound. Pauliâs letter to Weyl from 1 July 1929 used nonstandard language (in terms of science):
âI share completely your skeptical position with regard to Einsteinâs 4bein geometry. During the Easter holidays I have visited Einstein in Berlin and found his opinion on modern quantum theory reactionary.â^{Footnote 230} ([251], p. 506)
while the wealth of empirical data supporting Heisenbergâs and SchrĂ¶dingerâs quantum theory would have justified the use of a word like âuninformedâ or even ânot up to dateâ for the description of Einsteinâs position, use of âreactionaryâ meant a definite devaluation.
Einstein had sent a further exposition of his new theory to the Mathematische Annalen in August 1928. When he received its proof sheets from Einstein, Pauli had no reservations to criticise him directly and bluntly:
âI thank you so much for letting be sent to me your new paper from the Mathematische Annalen [89], which gives such a comfortable and beautiful review of the mathematical properties of a continuum with Riemannian metric and distant parallelism [âŠ]. Unlike what I told you in spring, from the point of view of quantum theory, now an argument in favour of distant parallelism can no longer be put forward [âŠ]. It just remains [âŠ] to congratulate you (or should I rather say condole you?) that you have passed over to the mathematicians. Also, I am not so naive as to believe that you would change your opinion because of whatever criticism. But I would bet with you that, at the latest after one year, you will have given up the entire distant parallelism in the same way as you have given up the affine theory earlier. And, I do not wish to provoke you to contradict me by continuing this letter, because I do not want to delay the approach of this natural end of the theory of distant parallelism.â^{Footnote 231} (letter to Einstein of 19 December 1929; [251], 526â527)
Einstein answered on 24 December 1929:
âYour letter is quite amusing, but your statement seems rather superficial to me. Only someone who is certain of seeing through the unity of natural forces in the right way ought to write in this way. Before the mathematical consequences have not been thought through properly, is not at all justified to make a negative judgement. [âŠ] That the system of equations established by myself forms a consequential relationship with the space structure taken, you would probably accept by a deeper study â more so because, in the meantime, the proof of the compatibility of the equations could be simplified.â^{Footnote 232} ([251], p. 582)
Before he had written to Einstein, Pauli, with lesser reservations, complained visavis Jordan:
âEinstein is said to have poured out, at the Berlin colloquium, horrible nonsense about new parallelism at a distance. The mere fact that his equations are not in the least similar to Maxwellâs theory is employed by him as an argument that they are somehow related to quantum theory. With such rubbish he may impress only American journalists, not even American physicists, not to speak of European physicists.â^{Footnote 233} (letter of 30 November 1929, [251], p. 525)
Of course, Pauliâs spells of rudeness are well known; in this particular case they might have been induced by Einsteinâs unfounded hopes for eventually replacing the SchrĂ¶dingerHeisenbergDirac quantum mechanics by one of his unified field theories.
The question of the compatibility of the field equations played a very important role because Einstein hoped to gain, eventually, the quantum laws from the extra equations (cf. his extended correspondence on the subject with Cartan ([50] and Section 6.4.6).
That Pauli had been right (except for the time span envisaged by him) was expressly admitted by Einstein when he had given up his unified field theory based on distant parallelism in 1931 (see letter of Einstein to Pauli on 22 January 1932; cf. [241], p. 347).
Bornâs voice was the lonely approving one (Born to Einstein on 23 September 1929)^{Footnote 234}:
âYour report on progress in the theory of Fernparallelism did interest me very much, particularly because the new field equations are of unique simplicity. Until now, I had been uncomfortable with the fact that, aside from the tremendously simple and transparent geometry, the field theory did look so very involvedâ^{Footnote 235} ([154], p. 307)
Born, however, was not yet a player in unified field theory, and it turned out that Einsteinâs theory of distant parallelism became as involved as the previous ones.
Einsteinâs collaborator Lanczos even wrote a review article about distant parallelism with the title âThe new field theory of Einsteinâ [201]. In it, Lanczos cautiously offers some criticism after having made enough bows before Einstein:
âTo be critical with regard to the creation of a man who has long since obtained a place in eternity does not suit us and is far from us. Not as a criticism but only as an impression do we point out why the new field theory does not house the same degree of conviction, nor the amount of inner consistency and suggestive necessity in which the former theory excelled.[âŠ] The metric is a sufficient basis for the construction of geometry, and perhaps the idea of complementing RIEMANNian geometry by distant parallelism would not occur if there were the wish to implant something new into RIEMANNian geometry in order to geometrically interpret electromagnetism.â^{Footnote 236} ([201], p. 126)
When Pauli reviewed this review, he started with the scathing remark
âIt is indeed a courageous deed of the editors to accept an essay on a new field theory of Einstein for the âResults in the Exact Sciencesâ [literal translation of the journalâs title]. His neverending gift for invention, his persistent energy in the pursuit of a fixed aim in recent years surprise us with, on the average, one such theory per year. Psychologically interesting is that the author normally considers his actual theory for a while as the âdefinite solutionâ. Hence, [âŠ] one could cry out: âEinsteinâs new field theory is dead. Long live Einsteinâs new field theory!ââ^{Footnote 237} ([248], p. 186)
For the remainder, Pauli engaged in a discussion with the philosophical background of Lanczos and criticised his support for Mieâs theory of matter of 1913 according to which
âthe atomism of electricity and matter, fully separated from the existence of the quantum of action, is to be reduced to the properties of (singularityfree) eigensolutions of stilltobefound nonlinear differential equations for the field variables.â^{Footnote 238}
Thus, Pauli lightly pushed aside as untenable one of Einsteinâs repeated motivations and hopedfor tests for his unified field theories.
Lanczos, being dissatisfied with Einsteinâs distant parallelism, then tried to explain âelectromagnetism as a natural property of Riemannian geometryâ by starting from the Lagrangian quadratic in the components of the Ricci tensor: \({\mathcal L} = {R_{ik}}{R_{lm}}{g^{il}}{g^{km}} + C{({R_{ik}}{g^{ik}})^2}\) with an arbitrary constant C. He varied g_{ik} and R_{ik} independently [202]. (For Lanczos see J. Stachelâs essay âLanczosâ early contributions to relativity and his relation to Einsteinâ in [330], pp. 499â518.)
Reactions II: Further research on distant parallelism
The first reactions to Einsteinâs papers came quickly. On 29 October 1928, de Donder suggested a generalisation by using two metric tensors, a spacetime metric g_{ik}, and a beinmetric g ^{â}_{ik} , connected to the 4bein components \({h_{l\hat \jmath }}\) by
In place of Einsteinâs connection (161), defined through the 4bein only, he took:
where the dotsymbol denotes covariant derivation by help of the LeviCivita connection derived from g ^{â}_{ik} . If the Minkowski metric is used as a bein metric g^{â}, then the dot derivative reduces to partial derivation, and Einsteinâs original connection is obtained [48].
Another application of Einsteinâs new theory came from Eugen Wigner in Berlin whose paper showing that the tetrads in distantparallelismtheory permitted a generally covariant formulation of âDiracs equation for the spinning electronâ, was received by Zeitschrift fĂŒr Physik on 29 December 1928 [419]. He did point out that âup to now, grave difficulties stood in the way of a general relativistic generalisation of Diracâs theoryâ and referred to a paper of Tetrode [344]. Tetrode, about a week after Einsteinâs first paper on distant parallelism had appeared on 14 June 1928, had given just such a generally relativistic formulation of Diracâs equation through coordinate dependent Gammamatrices^{Footnote 239}; he also wrote down a (symmetric) energymomentum tensor for the Dirac field and the conservation laws. However, he had kept the metric g_{ik} introduced into the formalism by
to be conformally flat. For the matrixvalued 4vector Îł^{i} he prescribed the condition of vanishing divergence. Wigner did not fully accept Tetrodeâs derivations because there, implicitly and erroneously, it had been assumed that the twodimensional representation of the Lorentz group (2spinors) could be extended to a representation of the affine group. Wigner stated that such difficulties would disappear if Einsteinâs teleparallelism theory were used. Nevertheless, nowhere did he claim that the Dirac equation could only be formulated covariantly with the help of Einsteinâs new theory.
Zaycoff of the Physics Institute of the University in Sofia also followed Einsteinâs work closely. Half a year after Einsteinâs first two notes on distant parallelism had appeared [84, 83], i.e., shortly before Christmas 1928, Zaycoff sent off his first paper on the subject, whose arrival in Berlin was acknowledged only after the holidays on 13 January 1929 [429]. In it he described the mathematical formalism of distant parallelism theory, gave the identity (42), and calculated the new curvature scalar in terms of the Ricci scalar and of torsion. He then took a more general Lagrangian than Einstein and obtained the variational derivatives in linear and, in a simple example, also in second approximation. In his presentation, he used both the teleparallel and the LeviCivita connections. His second and third papers came quickly after Einsteinâs third note of January 1929 [88], and thus had to take into account that Einstein had dropped derivation of the field equations from a variational principle. In his second paper, Zaycoff followed Einsteinâs method and gave a somewhat simpler derivation of the field equations. An exact, complicated wave equation followed:
where \({X_k}: = \tfrac{1}{2}{D_k}({V^{lmn}}{S_{lnm}}) + {S_{lkm}}{D_r}{S^{lrm}}  \tfrac{1}{2}{S_k}{V^{lmn}}{S_{lnm}} + {V^{lmn}}{S_{mn}}^r{S_{lkr}} + {S_{rkm}}{S^r}{S^m}\) with torsion \({S_{mn}}^r\) and the torsion vector S_{k}, and the covariant derivative \({D_l}: = {\nabla _l}  {S_l}\), â_{l}being the teleparallel connection (161). In linear approximation, the Einstein vacuum and the vacuum Maxwell equations are obtained, supplemented by the homogeneous wave equation for a vector field [431]. In his third note, Zaycoff criticised Einstein âfor not having shown, in his most recent publication, whether his constraints on the world metric be permissible.â He then derived additional exact compatibility conditions for Einsteinâs field equations to hold; according to him, their effect would show up only in second approximation [430]. In his fourth publication Zaycoff came back to Einsteinâs Hamiltonian principle and rederived for himself Einsteinâs results. He also defended Einstein against critical remarks by Eddington [62] and Schouten [304], although Schouten, in his paper, had mentioned neither Einstein nor his teleparallelism theory, but only gave a geometrical interpretation of the torsion vector in a geometry with semisymmetric connection. Zaycoff praised Einsteinâs teleparallelism theory in words reminding me of the creation of the world as described in Genesis:
âWe may say that A. Einstein built a plane world which is no longer waste like the Euclidean spacetimeworld of H. Minkowski, but, on the contrary, contains in it all that we usually call physical reality.â^{Footnote 240} ([428], p. 724)
A conference on theoretical physics at the Ukrainian PhysicalTechnical Institute in Charkow in May 1929, brought together many German and Russian physicists. Unified field theory, quantum mechanics, and the new quantum field theory were all discussed. Einsteinâs former calculational assistant Grommer, now on his own in Minsk, in a brief contribution stressed Einsteinâs path for getting an overdetermined system of differential equations: Vary with regard to the 16 beinquantities but consider only the 10 metrical components as relevant. He claimed that Einstein had used only the antisymmetric part of the tensor \({P_{lm}}^k: = \Gamma {(g)_{lm}}^k  {\Delta _{lm}}^k\), where both \(\Gamma {(g)_{lm}}^k\) and \({\Delta _{lm}}^k\) were mentioned above (in Einsteinâs first note) although Einstein never used \(\Gamma {(g)_{lm}}^k\). According to Grommer the antisymmetry of P is needed, because its contraction leads to the electromagnetic 4potential and because the symmetric part can be expressed by the antisymmetric part and the metrical tensor. He also played the true voice of his (former) master by repeating Einsteinâs program of deriving the equations of motion from the overdetermined system:
âIf the law of motion of elementary particles could be derived from the overdetermined field equation, one could imagine that this law of motion permit only discrete orbits, in the sense of quantum theory.â^{Footnote 241} ([153], p. 646)
LeviCivita also had sent a paper on distant parallelism to Einstein, who had it appear in the reports of the Berlin Academy [207]. LeviCivita introduced a set of four congruences of curves that intersect each other at right angles, called their tangents \(\lambda _{\hat \imath}^k\) and used Equation (160) in the form:
He also employed the Ricci rotation coefficients defined by \({\gamma _{\hat \imath \hat k\hat l}} = {\nabla _\sigma }h_{\hat \imath}^\beta {h_{\hat k\beta }}h_{\hat l}^\sigma\), where the hatted indices are âbeinâindices; the Greek letters denote coordinates. They obey
The electromagnetic field tensor F_{ik} was entered via
LeviCivita chose as his field equations the EinsteinMaxwell equations projected on a rigidly fixed âworldlatticeâ of 4beins. He used the time until the printing was done to give a short preview of his paper in Nature [206]. About a month before LeviCivitaâs paper was issued by the Berlin Academy, Fock and Ivanenko [135] had had the same idea and compared Einsteinâs notation and the one used by LeviCivita in his monograph on the absolute differential calculus [205]:
âEinsteinâs new gravitational theory is intimately linked to the known theory of the orthogonal congruences of curves due to Ricci. In order to ease a comparison between both theories, we may bring together here the notations of R i c c i and L e v i  C i v i t a [âŠ] with those of Einstein.â^{Footnote 242}
A little after the publication of LeviCivitaâs papers, Heinrich Mandel embarked on an application of Kaluzaâs fivedimensional approach to Einsteinâs theory of distant parallelism [218]. Einstein had sent him the corrected proof sheets of his fourth paper [85]. The basic idea was to consider the points of M_{4} as equivalent to the ensemble of congruences with tangent vector X ^{i}_{5} in M_{5} (with cylindricity condition) werden.â^{Footnote 243}. The spacetime interval is defined as the distance of two lines of the congruence on \({M_5}:d{\tau ^2} = ({\gamma _{il}}  {X_{5i}}\;{X_{5l}})(\delta _k^l  {X_5}^l{X_{5k}})d{x^i}d{x^k}\). Mandel did not identify the torsion vector with the electromagnetic 4potential, but introduced the covariant derivative \({\Delta _k}{A^i}: = \tfrac{{\partial {A^i}}}{{\partial {x^k}}} + \{ _{kj}^i\} {A^j} + \tfrac{e}{c}{X_{5k}}{\mathcal M}_l^i{A^l}\), where the tensor \({\mathcal M}\) is skewsymmetric. We may look at this paper also as a forerunner of some sort to the Einstein.Mayer 5vector formalism (cf. Section 6.3.2).
Before Einstein dropped the subject of distant parallelism, many more papers were written by a bakerâs dozen of physicists. Some were more interested in the geometrical foundations, in exact solutions to the field equations, or in the variational principle.
One of those hunting for exact solutions was G. C. McVittie who referred to Einsteinâs paper [88]:
â[âŠ] we test whether the new equations proposed by Einstein are satisfied. It is shown that the new equations are satisfied to the first order but not exactly.â
He then goes on to find a rigourous solution and obtains the metric \(d{s^2} = {e^{a{x_1}}}dx_4^2  {e^{  2a{x_1}}}dx_1^2  {e^{  a{x_1}}}dx_2^2  {e^{  a{x_1}}}dx_3^2\) and the 4potential \({\phi _4} = \tfrac{1}{{2\sqrt \pi }}{e^{\tfrac{1}{2}a{x_1}}}\) [225]. He also wrote a paper on exact axially symmetric solutions of Einsteinâs teleparallelism theory [226].
Tamm and Leontowich treated the field equations given in Einsteinâs fourth paper on distant parallelism [85]. They found that these field equations did not have a spherically symmetric solution corresponding to a charged point particle at rest^{Footnote 244}. The corresponding solution for the uncharged particle was the same as in general relativity, i.e., Schwarzschildâs solution. Tamm and Leontowitch therefore guessed that a charged point particle at rest would lead to an axiallysymmetric solution and pointed to the spin for support of this hypothesis [342, 342].
Wiener^{Footnote 245} and Vallarta^{Footnote 246} were after particular exact solutions of Einsteinâs field equations in the teleparallelism theory. By referring to Einsteinâs first two papers concerning distant parallelism, they set out to show that the
â[âŠ] electromagnetic field is incompatible in the new Einstein theory with the assumption of static spherical symmetry and symmetry of the past and the future. [âŠ] the new Einstein theory lacks at present all experimental confirmation.â
In footnote 4, they added:
âSince writing this paper the authors have learned from Dr. H. MĂŒntz that the new Einstein field equations of the 1929 paper do not yield the vanishing of the gravitational field in the case of spherical symmetry and time symmetry. In this case he has been able to obtain results checking the observed perihelion of mercuryâ ([416], p. 356)
MĂŒntz is mentioned in [88, 85].
In his paper âOn unified field theoryâ of January 1929, Einstein acknowledges work of a Mr. MĂŒntz:
âI am pleased to dutifully thank Mr. Dr. H. MĂŒntz for the laborious exact calculation of the centrallysymmetric problem based on the Hamiltonian principle; by the results of this investigation I was led to the discovery of the road following here.â^{Footnote 247}
Again, two months later in his next paper, âUnified field theory and Hamiltonian principleâ, Einstein remarks:
âMr. Lanczos and MĂŒntz have raised doubt about the compatibility of the field equations obtained in the previous paper [âŠ].â
and, by deriving field equations from a Lagrangian shows that the objection can be overcome. In his paper in July 1929, the physicist Zaycoff had some details:
âSolutions of the field equations on the basis of the original formulation of unified field theory to first approximation for the spherically symmetric case were already obtained by MĂŒntz.â
In the same paper, he states: âI did not see the papers of Lanczos and MĂŒntz.â Even before this, in the same year, in a footnote to the paper of Wiener and Vallarta, we read:
âSince writing this paper the authors have learned from Dr. H. MĂŒntz that the new Einstein field equations of the 1929 paper do not yield the vanishing of the gravitational field in the case of spherical symmetry and time symmetry. In this case he has been able to obtain results checking the observed perihelion of mercury.â
The latter remark refers to a constant query Pauli had about what would happen, within unified field theory, to the gravitational effects in the planetary system, described so well by general relativity^{Footnote 248}.
Unfortunately, as noted by Meyer Salkover of the Mathematics Department in Cincinatti, the calculations by Wiener and Vallarta were erroneous; if corrected, one finds the Schwarzschild metric is indeed a solution of Einsteinâs field equations. In the second of his two brief notes, Salkover succeeded in gaining the most general, spherically symmetric solution [288, 287]. This is admitted by the authors in their second paper, in which they present a new calculation.
âIn a previous paper the authors of the present note have treated the case of a spherically symmetrical statical field, and stated the conclusions: first, that under Einsteinâs definition of the electromagnetic potential an electromagnetic field is incompatible with the assumption of static spherical symmetry and symmetry of the past and future; second, that if one uses the Hamiltonian suggested in Einsteinâs second 1928 paper, the electromagnetic potential vanishes and the gravitational field also vanishes.â
And they hasten to reassure the reader:
âNone of the conclusions of the previous paper are vitiated by this investigation, although some of the final formulas are supplemented by an additional term.â ([417], p. 802)
Vallarta also wrote a paper by himself ([358], p. 784) whose abstract reads:
âIn recent papers Wiener and the author have determined the tensors \(^s{h_\lambda }\) of Einsteinâs unified theory of electricity and gravitation under the assumption of static spherical symmetry and of symmetry of past and future. It was there shown that the field equations suggested in Einsteinâs second 1928 paper [83] lead in this case to a vanishing gravitational field. The purpose of this paper is to investigate, for the same case, the nature of the gravitational field obtained from the field equations suggested by Einstein in his first 1929 paper [88].â
He also claims
âthat Wiener has shown in a paper to be published elsewhere soon that the Schwarzschild solution satisfies exactly the field equations suggested by Einstein in his second 1929 paper ([85]).â
Finally, Rosen and Vallarta [283] got together for a systematic investigation of the spherically symmetric, static field in Einsteinâs unified field theory of the electromagnetic and gravitational fields [93].
Further papers on Einsteinâs teleparallelism theory were written in Italy by Bortolotti in Cagliari, Italy [22, 23, 25, 24], and by Palatini [242].
In Princeton, people did not sleep either. In 1930 and 1931, T. Y. Thomas wrote a series of six papers on distant parallelism and unified field theory. He followed Einsteinâs example by also changing his field equations from the first to the second publication. After that, he concentrated on more mathematical problems , such as proving an existence theorem for the CauchyKowlewsky type of equations in unified field theory, by studying the characteristics and bicharacteristic, the characteristic Cauchy problem, and Huygenâs principle. T. Y. Thomas described the contents of his first paper as follows:
âIn a number of notes in the Berlin Sitzungsberichte followed by a revised account in the Mathematische Annalen, Einstein has attempted to develop a unified theory of the gravitational and electromagnetic field by introducing into the scheme of Riemannian geometry the possibility of distant parallelism. [âŠ] we are led to the construction of a system of wave equations as the equations of the combined gravitational and electromagnetic field. This system is composed of 16 equations for the determination of the 16 quantities h ^{kÌ}_{i} and is closely analogous to the system of 10 equations for the determination of the 10 components g_{ik} in the original theory of gravitation. It is an interesting fact that the covariant components h ^{kÌ}_{i} of the fundamental vectors, when considered as electromagnetic potential vectors, satisfy in the local coordinate system the universally recognised laws of Maxwell for the electromagnetic field in free space, as a consequence of the field equations.â [350]
This looks as if he had introduced four vector potentials for the electromagnetic field, and this, in fact, T. Y. Thomas does: âthe components h ^{kÌ}_{i} will play the role of electromagnetic potentials in the present theory.â The field equations are just the four wave equations \(\sum {{e_{\hat k}}h_{\hat \jmath,\hat k\hat k}^{\hat \imath}}\) where the summation extends over kÌ, with kÌ= 1,âŠ4, and the comma denotes an absolute derivative he has introduced. The gravitational potentials are still g_{ik}. In his next note, T. Y. Thomas changed his field equations on the grounds that he wanted them to give a conservation law.
âThis latter point of view is made the basis for the construction of a system of field equations in the present note â and the equations so obtained differ from those of note I only by the appearance of terms quadratic in the quantities \(h_{j,k}^{\hat \imath}\). It would thus appear that we can carry over the interpretation of the h ^{kÌ}_{i} as electromagnetic potentials; doing this, we can say that Maxwellâs equations hold approximately in the local coordinate system in the presence of weak electromagnetic fields.â [351]
The third paper contains a remark as to the content of the concept âunified field theoryâ:
âIt is the objective of the present note to deduce the general existence theorem of the CauchyKowalewsky type for the system of field equations of the unified field theory. [âŠ] Einstein (Sitzber. 1930, 18â23) has pointed out that the vanishing of the invariant \(h_{j,k}^{\hat \imath }\) is the condition for the fourdimensional world to be Euclidean, or more properly, pseudoEuclidean. From the point of view of our previous notes this fact has its interpretation in the statement that the world will be pseudoEuclidean only in the absence of electric and magnetic forces. This means that gravitational and electromagnetic phenomena must be intimately related since the existence of gravitation becomes dependent on the electromagnetic field. Thus we secure a real physical unification of gravitation and electricity in the sense that these concepts become but different manifestations of the same fundamental entity â provided, of course, that the theory shows itself to be tenable as a theory in agreement with experience.â [352].
In his three further installments, T. Y. Thomas moved away from unified field theory to the discussion of mathematical details of the theory he had advanced [353, 354, 355].
Unhindered by constraints from physical experience, mathematicians try to play with possibilities. Thus, it was only consequential that Valentin Bargmann in Berlin, after Riemann and Weyl, now engaged in looking at a geometry allowing a comparison âat a distanceâ of directions but not of lengths, i.e., only of the quotient of vector components, A^{i}/A^{k} [5]. In the framework of a purely affine theory he obtained a necessary and sufficient condition for this geometry,
with the homothetic curvature V_{kl} from Equation (31). Then Bargmann linked his approach to Einsteinâs first note on distant parallelism [84, 89], introduced a Dbein h ^{k}_{i} , and determined his connection such that the quotients A^{i}/A^{k} of vector components with regard to the Dbein remained invariant under parallel transport. The resulting connection is given by
where Ï_{m} corresponds to \({\Gamma _{lm}}^k\).
Schouten and van Dantzig also used a geometry built on complex numbers, and on Hermitian forms:
The Hermitian tensor referred to leads to a linear integrable connection that, in the special case that it âis real in the realâ, coincides with Einsteinâs teleparallel connection.
Distant parallelism was revived four decades later within the framework of PoincarĂ© gauge theory; the corresponding theories will be treated in the second part of this review.
Overdetermination and compatibility of systems of differential equations
In the course of Einsteinâs thinking about distant parallelism, his ideas about overdetermined systems of differential equations gradually changed. At first, the possibility of gaining hold on the paths of elementary particles â described as singular worldlines of point particles â was central. He combined this with the idea of quantisation, although Planckâs constant h could not possibly surface by such an approach. But somehow, for Einstein, discretisation and quantisation must have been too close to bother about a fundamental constant.
Then, after the richer constructive possibilities (e.g., for a Lagrangian) became obvious, a principle for finding the correct field equations was needed. As such, âoverdeterminationâ was brought into the game by Einstein:
âThe demand for the existence of an âoverdeterminedâ system of equations does provide us with the means for the discovery of the field equationsâ^{Footnote 249} ([90], p. 21)
It seems that Einstein, during his visit to Paris in November 1929, had talked to Cartan about his problem of finding the right field equations and proving their compatibility. Starting in December of 1929 and extending over the next year, an intensive correspondence on this subject was carried on by both men [50]. On 3 December 1929, Cartan sent Einstein a letter of five pages with a mathematical note of 12 pages appended. In it he referred to his theory of partial differential equations, deterministic and âin involution,â which covered the type of field equations Einstein was using and put forward a further field equation. He clarified the mathematical point of view but used concepts such as âdegree of generalityâ and âgenerality indexâ not familiar to Einstein ^{Footnote 250} . Cartan admitted^{Footnote 251}:
âI was not able to completely solve the problem of determining if there are systems of 22 equations other than yours and the one I just indicated [âŠ] and it still astonishes me that you managed to find your 22 equations! There are other possibilities giving rise to richer geometrical schemes while remaining deterministic. First, one can take a system of 15 equations [âŠ]. Finally, maybe there are also solutions with 16 equations; but the study of this case leads to calculations as complicated as in the case of 22 equations, and I was not fortunate enough to come across a possible system [âŠ].â ([50], pp. 25â26)
Einsteinâs rapid answer of 9 December 1929 referred to the letter only; he had not been able to study Cartanâs note. As the further correspondence shows, he had difficulties in following Cartan:
âFor you have exactly that which I lack: an enviable facility in mathematics. Your explanation of the indice de gĂ©nĂ©ralitĂ© I have not yet fully understood, at least not the proof. I beg you to send me those of your papers from which I can properly study the theory.â ([50], p. 73)
It would be a task of its own to closely study this correspondence; in our context, it suffices to note that Cartan wrote a special note^{Footnote 252}
â[âŠ] edited such that I took the point of view of systems of partial differential equations and not, as in my papers, the point of view of systems of equations for total differentials [âŠ]â^{Footnote 253}
which was better suited to physicists. Through this note, Einstein came to understand Cartanâs theory of systems in involution:
âI have read your manuscript, and this enthusiastically. Now, everything is clear to me. Previously, my assistant Prof. MĂŒntz and I had sought something similar â but we were unsuccessful.â^{Footnote 254} ([50], pp. 87, 94)
In the correspondence, Einstein made it very clear that he considered Maxwellâs equations only as an approximation for weak fields, because they did not allow for nonsingular exact solutions approaching zero at spacelike infinity.
âIt now is my conviction that for rigourous field theories to be taken seriously everywhere a complete absence of singularities of the field must be demanded. This probably will restrict the free choice of solutions in a region in a farreaching way â more strongly than the restrictions corresponding to your degrees of determination.â^{Footnote 255} ([50], p. 92)
Although Einstein was grateful for Cartanâs help, he abandoned the geometry with distant parallelism.
Geometrization of the Electron Field as an Additional Element of Unified Field Theory
After the advent of SchrĂ¶dingerâs and Diracâs equations describing the electron nonrelativistically and relativistically, a unification of only the electromagnetic and gravitational fields was considered unconvincing by many theoretical physicists. Hence, in the period 1927â1933, quite a few attempts were made to include SchrĂ¶dingerâs, Diracâs, or the KleinGordon equation as a classical oneparticle equation into a geometrical framework by relating the quantum mechanical wave function with some geometrical object. Such an approach then was believed to constitute a unification, up to a degree, of gravitation and/or electricity and quantum theory. In this section, we loosely collect some of these approaches.
The mathematicians Struik and Wiener found the task of an amalgamation of relativity and quantum theory (wave mechanics) attractive:
âIt is the purpose of the present paper to develop a form of the theory of relativity which shall contain the theory of quanta, as embodied in SchrĂ¶dingers wave mechanics, not merely as an afterthought, but as an essential and intrinsic part.â [338]
A further example for the new program is given by J. M. Whittaker at the University of Edinburgh [415] who wished to introduce the wave function via the matter terms:
âIn addition to the wave equations a complete scheme must include electromagnetic and gravitational equations. These will differ from the equations of Maxwell and Einstein in having âwaveâ terms instead of âparticleâ terms for the current vector and material energy tensor. The object of the present paper is to find these equations [âŠ].â ([415], p. 543)
Zaycoff, from the point of view of distant parallelism, found the following objection to unified field theory as the only valid one:
In fact, in a short note, Zaycoff presented his version of Whittakerâs theory with 8 coupled secondorder linear field equations for two âwave vectorsâ that, in a suitable combination, were to represent âthe Diracâs wave equationsâ; they contain the Ricci tensor and both the electromagnetic 4potential and field [432]. Thus, what is described is Diracâs equation in external gravitational and electromagnetic fields, not a unified field theory. Whittaker had expressed himself more clearly:
âEight wave functions are employed instead of Diracâs four. These are grouped together to form two fourvectors and satisfy wave equations of the second order. It is shown [âŠ] that these eight wave equations can be reduced, by addition and subtraction, to the four second order equations satisfied by Diracâs functions taken twice over; and that, in a sense, the present theory includes Diracâs.â ([415], p. 543)
Whittaker also had written down a variational principle by which the gravitational and electromagnetic field equations were also gained. However, as the terms for the various fields were just added up in his Lagrangian, the theory would not have qualified as a genuine unified field theory in the spirit of Einstein.
What fancy, if only shortlived, flowers sprang from the mixing of geometry and wave mechanics is shown by the example of H. Jehleâs â[âŠ] path leading, on the one hand, to electrical elementary particles and, on the other, to the explanation of cosmological problems by quantum theory.â [171] His adhoc modification of Einsteinâs equations was:
where Ï is the quantum mechanical state function. Although, two years later, Jehle withdrew his claims concerning elementary particles, he continued to apply
âwavemechanical methods to gravitational phenomena, by which the curious structure of the spiral nebulae and spherical star systems may be readily understood.â [172]
An eminent voice was Weylâs:
âIt seems to me that it is now hopeless to seek a unification of gravitation and electricity without taking material waves into account.â ([408], p. 325)
Now, this posed a problem because for the representation of the electrons in the form of Diracâs equation, the elements of spin space, i.e., spinors, had to be used. How to combine them with the vectors and tensors appearing in electromagnetic and gravitational theories? As the spinor representation is the simplest representation of the Lorentz group, everything may be played back to spin space. At the time, this was being done in different ways, in part by the use of number fields with which physicists were unacquainted such as quaternions and sedenions (cf. Schouten [315]). Others, such as Einstein and Mayer, liked vectors better and introduced socalled semivectors. Still others tried to write Diracâs equations in a vectorial form and took into account the doubling of variables and equations [213, 415]. Some less experienced, as e.g., âExhibition Research Studentâ G. Temple, even claimed that a tensorial theory was necessary to retain it relativistically:
âIt is an admitted fact that Diracâs wave functions are not the components of a tensor and that his wave equations are not in tensorial form. It is contended here that therefore his theory cannot be upheld without abandoning the theory of relativity.â ([343], p. 352)
While this story about geometrizing wave mechanics might not be a genuine part of unified field theory at the time, it seems interesting to follow it as a last attempt for binding together classical field theory and quantum physics. Even Einstein was lured into thinking about spinors by Diracâs equation; this equation promised more success for his program concerning elementary particles as solutions of differential equations (cf. Section 7.3. )
Unification of Maxwellâs and Diracâs equations, of electrons and light
The appearance of these socalled âwave equationsâ for some seemed to show a kinship between the photon and the electron; this led to attempts to obtain a common representation for both kind of particles (waves) [199, 200, 198, 286]. Some of the motivation for these papers came from formal considerations, i.e., the wish to replace the not yet wellunderstood new spinor representation of the Lorentz group by the old tensor representation; a working knowledge about noncommuting objects like matrices was not yet available to everyone:
âThere are probably readers who will share the present writerâs feeling that the methods of noncommutative algebra are harder to follow, and certainly much more difficult to invent, than are operations of types long familiar to analysis.â ([44], p. 654)
More interesting is Frenkelâs remark about Darwinâs presentation of Diracâs equations in a form analogous to Maxwellâs equations [44]:
âThis relation between the wavemechanical equations of a âquantum of electricityâ and the electromagnetic field equations, which may be looked at as wavemechanical equations for photons, ought to have a fundamental physical meaning. Therefore, I do not think it is superfluous to win the wave equation of the electron as a generalisation of M a x w e l lâs equations.â^{Footnote 257} ([140], p. 357)
H. T. Flint of Kingâs College in London aimed at describing the electron in a Maxwelllike way within a fivedimensional approach. He saw two âunsatisfactory pointsâ in Diracâs approach, the introduction of the operator \(\tfrac{h}{{2\pi i}}\tfrac{\partial }{{\partial {x^k}}}  e{\phi _k}\), and the mass term mc. In order to mend these thin spots he wrote down two Maxwellâs equations,
where \({{F^{\lambda \mu }}}\) and \({{G^{\lambda \mu }}}\) are two asymmetric field tensors, and \({J_\lambda } = \tfrac{{\partial \psi }}{{\partial {x^\lambda }}}\) and \({L_\lambda } = \tfrac{{\partial \theta }}{{\partial {x^\lambda }}}\) are two current vectors (ÎŒ, Î», Îœ= 1, 2, âŠ, 5). Ï is the electronâs wave function; although not provided by Flint, the interpretation of .. points to a second kind of wave function. Despite the plentitude of variables introduced, Flintâs result was meagre; his second order wave equation contained the correct mass term and two new terms; he did not write up all the equations occurring [129]. His approach for embedding wave mechanics into a Maxwelllike was continued in further papers, in part in collaboration with J. W. Fisher; to them it appeared
By adding four spinor equations at his choosing to Diracâs equation, Wisniewski in Poland arrived at a âsystem of equations similar to Maxwellâsâ. His conclusion sounds a bit strange:
âThese equations may be interpreted as equations for the electromagnetic field in an electron gas whose elements are electric and magnetic dipoles.â [388]
In this context, another unorthodox suggestion was put forward by A. Anderson who saw matter and radiation as two phases of the same substrate:
âWe conclude that, under sufficiently large pressure, even at absolute zero normal matter and blackbody radiation (gas of light quanta) become identical in every sense. Electrons and protons cannot be distinguished from quanta of light, gas pressure not from radiation pressure.â^{Footnote 258}
Anderson somehow sensed that charge conservation was in his way; he meddled through by either assuming neutral matter, i.e., a mixture of electrons and protons, or by raising doubt as to âwhether the usual quanta of light are strictly electrically neutralâ ([3], p. 441).
One of the German theorists trying to keep up with wave mechanics was Gustav Mie. He tried to reformulate electrodynamics into a SchrĂ¶dingertype equation and arrived at a linear, homogeneous wave equation of the Klein.Gordontype for the Ïfunction on the continuum of the components of the electromagnetic vector potential [232]. Heisenberg and Pauli, in their paper on the quantum dynamics of wave fields, although acknowledging Mieâs theory as an attempt to establish the classical side for the application of the correspondence principle, criticised it as a formal scheme not yet practically applicable [158].
Diracâs electron with spin, Einsteinâs teleparallelism, and Kaluzaâs fifth dimension
In the same year 1928 in which Einstein published his theory of distant parallelism, Dirac presented his relativistic, spinorial wave equation for the electron with spin. This event gave new hope to those trying to include the electron field into a unified field theory; it induced a flood of papers in 1929 such that this year became the zenith for publications on unified field theory. Although we will first look at papers which gave a general relativistic formulation of Diracâs equation without having recourse to a geometry with distant parallelism, Tetrodeâs paper seems to be the only one not influenced by Einsteinâs work with this geometry (cf. Section 6.4.5). Although, as we noted in Section 6.4.1, the technique of using nbeins (tetrads) had been developed by mathematicians before Einstein applied it, it may well have been that it became known to physicists through his work. Both Kaluzaâs fivedimensional space and fourdimensional projective geometry were also applied in the general relativistic formulation of Diracâs equation.
Spinors
This is a very sketchy outline with a focus on the relationship to unified field theories. An interesting study into the details of the introduction of local spinor structures by Weyl and Fock and of the early history of the general relativistic Dirac equation was given recently by Scholz [291].
For some time, the new concept of spinorial wave function stayed unfamiliar to many physicists deeply entrenched in the customary tensorial formulation of their equations^{Footnote 259}. For example, J. M. Whittaker was convinced that Diracâs theory for the electron
âhas been brilliantly successful in accounting for the âduplexityâ phenomena of the atom, but has the defect that the wave equations are unsymmetrical and have not the tensor form.â ([415], p. 543)
Some early nomenclature reflects this unfamiliarity with spinors. For the 4component spinors or Diracspinors (cf. Section 2.1.5) the name âhalfvectorsâ coined by Landau was in use^{Footnote 260}. Podolsky even purported to show that it was unnecessary to employ this concept of âhalfvectorâ if general curvilinear coordinates are used [259]. Although van der Waerden had written on spinor analysis as early as 1929 [368] and Weylâs [407, 408], Fockâs [133, 131], and Schoutenâs [306] treatments in the context of the general relativistic Dirac equation were available, it seems that only with van der Waerdenâs book [369], SchrĂ¶dingerâs and Bargmannâs papers of 1932 [319, 6], and the publication of Infeld and van der Waerden one year later [167] a better knowledge of the new representations of the Lorentz group spread out. Ehrenfest, in 1932, still complained^{Footnote 261}:
âYet still a thin booklet is missing from which one could leasurely learn spinor and tensorcalculus combined.â^{Footnote 262} ([68], p. 558)
In 1933, three publications of the mathematician Veblen in Princeton on spinors added to the development. He considered his first note on 2spinors âa sort of geometric commentary on the paper of Weylâ [378]. Veblen had studied Weylâs, Fockâs, and Schoutenâs papers, and now introduced a âspinor connection of the first kindâ \(\Lambda _{\beta \alpha }^A\), Î±=1,âŠ,4, with the usual transformation law under the linear transformation \({\bar \psi ^A} = T_D^A{\psi ^D}\;(A,D = 1,2)\) changing the spin frame:
where t ^{B}_{D} is the inverse matrix T^{1}. T ^{A}_{D} corresponds to A ^{A}_{B} of Equation (75); however, the transformation need not be unimodular. Thus, Veblen took up Schoutenâs concept of âspin densityâ [306] by considering quantities transforming like \({\bar \psi ^A} = {t^N}T_D^A{\psi ^D}\) (A, D=1, 2), with t:= det t ^{B}_{D} . Then, the covariant derivative of a spinor of weight N, Ï^{A} is considered; the expression^{Footnote 263}
gives âthe components of a geometric object which transforms like those of a spinor of weight N with respect to the index A and like those of a projective tensor with respect to Î±â. In his first paper [378], a further generalisation is introduced including âgaugetransformationsâ in an additional variable x^{0}: \({\bar \psi ^A} = {e^{k{x^0}}}{f^A}\) (A, D= 1, 2), with the gauge transformations
k is called the âindexâ of the spinor. In order to deal with 4spinors Veblen considered a complex projective 3space and defined 6 real homogeneous coordinates \({X^\sigma }\), with Ï=0, âŠ, 5, through Hermitian forms of the 4spinor components. The subspace X^{0}=0, X^{5}=0 of the quadratic \({({X^0})^2} + {({X^1})^2} + {({X^2})^2} + {({X^3})^2}  {({X^4})^2}  {({X^5})^2} = 0\) is then tangent to the Minkowski light cone ([377], p. 515).
Veblen imbedded spinors into his projective geometry [380]:
â[âŠ] The components of still other objects, the spinors, remain partially indeterminate after coordinates and gauge are fixed and become completely determinate only when the spin frame is specified. There are several ways of embodying this invariant theory in a formal calculus. The one which is here employed has its antecedents chiefly in the work ofWeyl, van derWaerden, Fock, and Schouten. It differs from the calculus arrived at by Schouten chiefly in the treatment of gauge invariance, Schouten (in collaboration with van Dantzig) having preferred to rewrite the projective relativity in a formalism obtainable from the original one by a sort of coordinate transformation, whereas I think the original form fits in better with the classical notations of relativity theory. [âŠ] The theory of spinors is more general than the projective relativity and is reduced to the latter by the specification of certain fundamental spinors. These spinors have been recognised by several students (Pauli and Solomon, Fock) of the subject but their role has probably not been fully understood since it has quite recently been thought necessary to give special proofs of invariance.â [380]
The transformation law for spinors is the same as before^{Footnote 264}:
In part, he also takes over van der Waerdenâs notation (dotted indices.) As to Veblenâs papers on 2 and 4spinors, my impression is that, beyond a more detailed presentation, alas with a less transparent notation, they do not really bring a pronounced advance with regard to Weylâs, Fockâs, van der Waerdenâs, and Pauliâs publications (cf. Sections 7.2.2 and 7.2.3). Veblen himself had a different opinion; for him the homogeneous coordinates used by Pauli seemed âto make things more complicatedâ (cf. the paragraphs on projective geometry in Section 2.1.3). Veblenâs inhomogeneous coordinates x^{i}, (i = 1, 2, 3, 4) and the homogeneous coordinates \({X^\mu }\), with ÎŒ=0, âŠ, 4, are connected by
According to Veblen,
âIn a fivedimensional representation the use of the homogeneous coordinates (X^{0}, âŠ ,X^{4}) amounts to representing the points of spacetime by the straight lines through the origin, whereas the use of x^{1}, âŠ , x^{4}, and the gauge variable amounts to using the system of straight lines parallel to the x^{0}axis for the same purpose. The transformation (192) given above carries the system of lines into the other.â [382]
General relativistic Dirac equation and unified field theory
After Tetrode and Wigner, whose contributions were mentioned in Section 6.4.5, Weyl also gave a general relativistic formulation of Diracâs equation. He gave up his original idea of coupling electromagnetism to gravitation and transferred it to the coupling of the electromagnetic field to the matter (electron) field: In order to keep quantum mechanical equations like Diracâs gauge invariant, the wave function had to be multiplied by a phase factor [407, 408]. Actually, Weyl had expressed the change in his outlook, so important for the idea of gaugesymmetry in modern physics ([424], pp. 13â19), already in 1928 in his book on group theory and quantum mechanics ([406], pp. 87â88). We have noted before his refutation of distant parallelism (cf. Section 6.4.4). In his papers, Weyl used a 2spinor formalism and a tetrad notation different from Einsteinâs and LeviCivitaâs: He wrote e^{p}(kÌ) in place of h ^{p}_{kÌ} , and o(l; kj) for the Ricci rotation coefficients Îł_{jkl}; this did not ease the reading of his paper. He partly agreed with what Einstein imagined:
âIt is natural to expect that one of the two pairs of components of D i r a câs quantity belongs to the electron, the other to the proton.â^{Footnote 265}
In contrast to Einstein, Weyl did not expect to find the electron as a solution of âclassicalâ spinorial equations:
âFor every attempt at establishing the quantumtheoretical field equations, one must not lose sight [of the fact] that they cannot be tested empirically, but that they provide, only after their quantization, the basis for statistical assertions concerning the behaviour of material particles and light quanta.â^{Footnote 266} ([407], p. 332)
For many years, Weyl had given the statistical approach in the formulation of physical laws an important role. He therefore could adapt easily to the BornJordanHeisenberg statistical interpretation of the quantum state. For Weyl and statistics, cf. Section V of Sigurdssonâs dissertation ([326], pp. 180â192).
At about the same time, Fock in May 1929 and later in the year wrote several papers on the subject of âgeometrizingâ Diracâs equation:
âIn the past two decades, endeavours have been made repeatedly to connect physical laws with geometrical concepts. In the field of gravitation and of classical mechanics, such endeavours have found their fullest accomplishment in E i n s t e i nâs general relativity. Up to now, quantum mechanics has not found its place in this geometrical picture; attempts in this direction (Klein, Fock) were unsuccessful. Only after Dirac had constructed his equations for the electron, the ground seems to have been prepared for further work in this direction.â^{Footnote 267} ([135], p. 798)
In another paper [134], Fock and Ivanenko took a first step towards showing that Diracâs equation can also be written in a generally covariant form. To this end, the matrixvalued linear form ds=Îł_{k}dx^{k} (summation over k=1, âŠ, 4) was introduced and interpreted as the distance between two points âin a space with four continuous and one discontinuous dimensionsâ; the discrete variable took only the integer values 1, 2, 3, and 4. Then the operatorvalued vectorial quantity Îł_{k}u^{k} with the vectorial operator u^{k} and its derivative \(\tfrac{{ds}}{{d\tau }} = {\gamma _k}{v^k}\) immediately led to Diracâs equation by replacing v_{k} by \(\tfrac{1}{m}\left( {\tfrac{h}{{2\pi i}}\tfrac{\partial }{{\partial {x^k}}} + \tfrac{e}{c}{A_k}} \right)\), where A_{k} is the electromagnetic 4potential, by also assuming the velocity of light c to be the classical average of the â4velocityâ v_{k}, and by applying the operator to the wave function Ï. In the next step, instead of the Dirac Îłmatrices with constant entries Îł ^{(0)}_{l} .. , the coordinatedependent beincomponents \({\gamma _{\hat k}} = {h_{\hat k}}^l\gamma _l^{(0)}\) are defined; ds^{2} then gives the orthonormality relations of the 4beins.
In a subsequent note in the Reports of the Parisian Academy, Fock and Ivanenko introduced Diracâs 4spinors under Landauâs name âhalf vectorâ and defined their parallel transport with the help of Ricciâs coefficients. In modern parlance, by introducing a covariant derivative for the spinors, they in fact already obtained the âgaugecovariantâ derivative \({\nabla _k}\psi : = (\tfrac{\partial }{{\partial {x^k}}}  \tfrac{{2\pi i}}{h}\tfrac{e}{c}{A_k})\psi\). Thus \(\delta \psi = \tfrac{{2\pi i}}{h}\tfrac{e}{c}{A_k}d{x^k}\psi\) is interpreted in the sense of Weyl:
âThus, it is in the law for the transport of a halfvector that Weylâs differential linear form must appear.â^{Footnote 268} ([134], p. 1469)
In order that gaugeinvariance results, Ï must transform with a factor of norm 1, innocuous for observation, i.e., \(\psi \to \exp (i\tfrac{{2\pi }}{h}\tfrac{e}{c}\sigma )\) if \({A_k} \to {A_k} + \tfrac{{\partial \sigma }}{{\partial {x^k}}}\). Another note and extended presentations in both a French and a German physics journal by Fock alone followed suit [133, 131, 132]. In the first paper Fock defined an asymmetric matter tensor for the spinor field,
where \({\Gamma _k} = \Sigma _{\hat l}{e_{\hat l}}{\alpha _{\hat l}}{h_{k\hat l}}{C_{\hat l}}\) is related to the matrixvalued spin connection in the expression for the parallel transport of a halfvector Ï:
The covariant derivative then is \({D_k} = \tfrac{\partial }{{\partial {x^k}}}  {\Gamma _k}\). Fock made clear that the covariant formulation of Diracâs equation did not need the special geometry of Einsteinâs theory of distant parallelism:
âBy help of the concept of parallel transport of a halfvector, Diracâs equations will be written in a generally invariant form. [âŠ] The appearance of the 4potential Ï_{l} besides the Riccicoefficients Îł_{ikl} in the expression for parallel transport, on the one hand provides a simple reason for the emergence of the term \({p_l}  \tfrac{e}{c}{\phi _l}\) in the wave equation and, on the other, shows that the potentials Ï_{l} have a place of their own in the geometrical worldview, contrary to Einsteinâs opinion; they need not be functions of the Îł_{ikl}.â^{Footnote 269} ([131], p. 261, Abstract)
For his calculations, Fock used Eisenhartâs book [119] and âthe excellent collection of the most important formulas and facts in the paper of LeviCivitaâ [207]. Again, Weylâs âprinciple of gauge invarianceâ as formulated in Weylâs book of 1928 [406] is mentioned, and Fock stressed that he had found this principle independently and earlier^{Footnote 270}:
âThe appearance of Weylâs differential form in the law for parallel transport of a half vector connects intimately to the fact, observed by the author and also by Weyl (l.c.), that addition of a gradient to the 4potential corresponds to multiplication of the Ï function with a factor of modulus 1.â^{Footnote 271} ([130], p. 266)
The divergence of the complex energymomentum tensor \(W_k^i = T_k^i + iU_k^i\) satisfies
with the electromagnetic field tensor F_{ik}, the Ricci tensor R_{ik}, and the Dirac current J^{k}. The French version of the paper preceded the German âcompleted presentationâ; in it Fock had noted:
âThe 4potential finds its place in Riemannian geometry, and there exists no reason for generalising it (Weyl, 1918), or for introducing distant parallelism (Einstein 1928). In this point, our theory, developed independently, agrees with the new theory by H. Weyl expounded in his memoir âgravitation and the electronâ.â^{Footnote 272} ([132], p. 405)
In both of his papers, Fock thus stressed that Einsteinâs teleparallel theory was not needed for the general covariant formulation of Diracâs equation. In this regard he found himself in accord with Weyl, whose approach to the Dirac equation he nevertheless criticised:
âThe main subject of this paper is âDiracâs difficultyâ^{Footnote 273}. Nevertheless, it seems to us that the theory suggested by Weyl for solving this problem is open to grave objections; a criticism of this theory is given in our article.â^{Footnote 274}
Weylâs paper is seminal for the further development of the gauge idea [407].
Although Fock had cleared up the generally covariant formulation of Diracâs equation, and had tried to propagate his results by reporting on them at the conference in Charkow in May 1929^{Footnote 275} [169], further papers were written. Thus, ReichenbĂ€cher, in two papers on âa wavemechanical 2component theoryâ believed that he had found a method different from Weylâs for obtaining Diracâs equation in a gravitational field. As was often the case with ReichenbĂ€cherâs work, after longwinded calculations a less than transparent result emerged. His mass term contained a square root, i.e., a Â± twovaluedness, which, in principle, might have been instrumental for helping to explain the mass difference of proton and electron. As he remarked, the chances for this were minimal, however [277, 278].
In two papers, Zaycoff (of Sofia) presented a unified field theory of gravitation, electromagnetism and the Dirac field for which he left behind the framework of a theory with distant parallelism used by him in other papers. By varying his Lagrangian with respect to the 4beins, the electromagnetic potential, the Dirac wave function and its complexconjugate, he obtained the 20 field equations for gravitation (of second order in the 4bein variables, assuming the role of the gravitational potentials) and the electromagnetic field (of second order in the 4potential), and 8 equations of first order in the Dirac wave function and the electromagnetic 4potential, corresponding to the generalised Dirac equation and its complex conjugate [426, 427].
In another paper, Zaycoff wanted to build a theory explaining the âequilibrium of the electronâ. This means that he considered the electron as extended. At this occasion, he fought with himself about the admissibility of the KaluzaKlein approach:
âRecently, repeated attempts have been made to raise the number of dimensions of the world in order to explain its strange lawfulness (H. Mandel, G. Rumer, the author et al.). No doubt, there are weighty reasons for such a seemingly paradoxical view. For it is impossible to represent PoincarĂ©âs pressure of the electron within the normal spacetime scheme. However, the introduction of such metaphysical elements is in gross contradiction with spacetime causality, although we may doubt in causality in the usual sense due to Heisenbergâs uncertainty relations. A multidimensional causality cannot be understood as long as we are unable to give the extra dimensions an intuitive meaning.â^{Footnote 276} [433]
Rumerâs paper is [285] (cf. Section 8). In the paper, Zaycoff introduced a sixdimensional manifold with local coordinates x_{0}, âŠ , x_{5} where x_{0}, x_{5} belong to the additional dimensions. His local 6bein comprises, besides the 4bein, four electromagnetic potentials and a further one called âeigenpotentialâ of the electromagnetic field. As he used a âsharpened cylinder condition, â no further scalar field is taken into account. For x^{0} to x^{4} he used the subgroup of coordinate transformations given in Kleinâs approach, augmented by x^{5âČ}=x^{5}.
Schouten seemingly became interested in Diracâs equation through Weylâs publications. He wrote two papers, one concerned with the fourdimensional and a second one with the fivedimensional approach [306, 307]. They resulted from lectures Schouten had given at the Massachusetts Institute of Technology from October to December 1930 and at Princeton University from January to March 1931; Weylâs paper referred to is in Zeitschrift fĂŒr Physik [407]. Schouten relied on his particular representation of the Lorentz group in a complex space, which later attracted SchrĂ¶dingerâs criticism. [305]. His comment on Fockâs paper [131] is^{Footnote 277}:
âFock has tried to make use of the indetermination of the displacement of spinvectors to introduce the electromagnetic vector potential. However the displacement of contravariant tensordensities of weight +1/2 being wholly determined and only these vectordensities playing a role, the idea of Weyl of replacing the potential vector by pseudovectors of class +1 and â1 seems much better.â ([306], p. 261, footnote 19)
Schouten wrote down Diracâs equation in a space with torsion; his iterated wave equation, besides the mass term, contains a contribution âŒâ1/4R if torsion is set equal to zero. Whether Schouten could fully appreciate the importance of Weylâs new idea of gauging remains open. For him an important conclusion is that
âby the influence of a gravitational field the components of the potential vector change from ordinary numbers into Diracnumbers.â ([306], p. 265)
Two years later, SchrĂ¶dinger as well became interested in Diracâs equation. We reproduce a remark from his publication [319]:
âThe joining of Diracâs theory of the electron with general relativity has been undertaken repeatedly, such as by Wigner [419], Tetrode [344], Fock [131], Weyl [407, 408], Zaycoff [434], Podolsky [259]. Most authors introduce an orthogonal frame of axes at every event, and, relative to it, numerically specialised Diracmatrices. This procedure makes it a little difficult to find out whether Einsteinâs idea concerning teleparallelism, to which [authors] sometimes refer, really plays a role, or whether there is no dependence on it. To me, a fundamental advantage seems to be that the entire formalism can be built up by pure operator calculus, without consideration of the Ïfunction.â^{Footnote 278} ([319], p. 105)
The Îłmatrices were taken by SchrĂ¶dinger such that their covariant derivative vanished, i.e., \({\gamma _{l\left\ m \right.}} = \tfrac{{\partial {\gamma _l}}}{{\partial {x^m}}}  \Gamma _{lm}^r(g){\gamma _r} + {\gamma _l}{\Gamma _m}  {\gamma _m}{\Gamma _l} = 0\), where Î_{l} is the spinconnection introduced by \({\psi _{\left\ l \right.}} = \tfrac{{\partial \psi }}{{\partial {x^l}}}  {\Gamma _l}\psi\). SchrĂ¶dinger took Îł_{0}, Îł^{i}, with i = 1, 2, 3, as Hermitian matrices. He introduced tensoroperators T ^{ik}_{lm} such that the inner product \(\psi *{\gamma _0}T_{lm}^{ik}\psi\) instead of \(\psi *T_{lm}^{ik}\psi\) stayed real under a âcomplemented pointsubstitutionâ.
In the course of his calculations, SchrĂ¶dinger obtained the wave equation
where ÎŒ=2Ïmc/h, f_{kl} is the electromagnetic field tensor, and \({s^{kl}}: = \tfrac{1}{2}{\gamma ^{[k}}{\gamma ^{l]}}\) with the Îłmatrices Îł^{k}, i.e., the spin tensor. As to the term with the curvature scalar R, SchrĂ¶dinger was startled:
âTo me, the second term seems to be of considerable theoretical interest. To be sure, it is much too small by many powers of ten in order to replace, say, the term on the r.h.s. For ÎŒ is the reciprocal Compton length, about 10^{11} cm^{1}. Yet it appears important that in the generalised theory a term is encountered at all which is equivalent to the enigmatic mass term.â^{Footnote 279} ([319], p. 128)
The coefficient 1/4 in front of the Ricci scalar in SchrĂ¶dingerâs (KleinGordon) wave equation differs from the 1/6 needed for a conformally invariant version of the scalar wave equation^{Footnote 280} (cf. [257], p. 395).
Bargmann in his approach, unlike SchrĂ¶dinger, did not couple âpointsubstitutions [linear coordinate transformations] and similarity transformations [in spin space]â[6]. He introduced a matrix Î± with \(\alpha + {\alpha ^\dag } = 0\) such that \({(\alpha {\gamma ^l})^\dag } = (\alpha {\gamma ^l})\), with l = 0, âŠ , 3.
LeviCivita wrote a letter to SchrĂ¶dinger in the form of a scientific paper, excerpts of which became published by the Berlin Academy:
âYour fundamental memoir induced me to develop the calculational details for obtaining, from Diracâs equations in a general gravitational field, the modified form of your four equations of second order and thus make certain the corresponding additional terms. These additional terms do depend in an essential way on the choice of the orthogonal tetrad in the spacetime manifold: It seems that without such a tetrad one cannot obtain Diracâs equation.â^{Footnote 281} [208]
The last, erroneous, sentence must have made Pauli irate. In this paper, he pronounced his anathema (in a letter to Ehrenfest with the appeal âPlease, copy and distribute!â):
âThe heap of corpses, behind which quite a lot of bums look for cover, has got an increment. Beware of the paper by LeviCivita: Dirac and SchrĂ¶dingertype equations, in the Berlin Reports 1933. Everybody should be kept from reading this paper, or from even trying to understand it. Moreover, all articles referred to on p. 241 of this paper belong to the heap of corpses.â^{Footnote 282} ([252], p. 170)
Pauli really must have been enraged: Among the publications banned by him is also Weylâs wellknown article on the electron and gravitation of 1929 [407].
SchrĂ¶dingerâs paper was criticised by Infeld and van der Waerden on the ground that his calculational apparatus was unnecessarily complicated. They promised to do better and referred to a paper of Schoutenâs [306]:
âIn the end, Schouten arrives at almost the same formalism developed in this paper; only that he uses without need nbein components and theorems on sedenions^{Footnote 283}, while afterwards the formalism is still burdened with auxiliary variables and pseudoquantities. We have taken over the introduction of âspin densitiesâ by Schouten.â^{Footnote 284} ([168], p. 4)
Unlike SchrĂ¶dingerâs, the wave equation derived from Diracâs equation by Infeld and Waerden contained a term +1/4R, with R the Ricci scalar.
It is left to an indepth investigation, how this discussion concerning teleparallelism and Diracâs equation involving Tetrode, Wigner, Fock, Pauli, London, SchrĂ¶dinger, Infeld and van derWaerden, Zaycoff, and many others influenced the acceptance of the most important result, i.e., Weylâs transfer of the gauge idea from classical gravitational theory to quantum theory in 1929 [407, 408].
Parallelism at a distance and electron spin
Einsteinâs papers on distant parallelism had a strong but shortlived impact on theoretical physicists, in particular in connection with the discussion of Diracâs equation for the electron,
where the 4spinor Ï and the Îłmatrices are used. At the time, there existed some hope that a unified field theory for gravitation, electromagnetism, and the âelectron fieldâ was in reach. This may have been caused by a poor understanding of the new quantum theory in SchrĂ¶dingerâs version: The new complex wave function obeying SchrĂ¶dingerâs, and, more interestingly for relativists, Diracâs equation or the ensuing KleinGordon wave equation, was interpreted in the spirit of de Broglieâs âonde piloteâ, i.e., as a classical matter wave, not â as it should have been â as a probability amplitude for an ensemble of indistinguishable electrons. One of the essential features of quantum mechanics, the noncommutativity of conjugated observables like position and momentum, nowhere entered the approaches aiming at a geometrization of wave mechanics.
Einstein was one of those clinging to the picture of the wave function as a real phenomenon in spacetime. Although he knew well that already for two particles the wave function no longer âlivedâ in spacetime but in 7dimensional configuration space, he tried to escape its statistical interpretation. On 5 May 1927, Einstein presented a paper to the Academy of Sciences in Berlin with the title âDoes SchrĂ¶dingerâs wave mechanics determine the motion of a system completely or only in the statistical sense?â. It should have become a 4page publication in the Sitzungsberichte. As he wrote to Max Born:
âLast week I presented a short paper to the Academy in which I showed that one can ascribe fully determined motions to SchrĂ¶dingerâs wave mechanics without any statistical interpretation. Will appear soon in Sitz.Ber. [Reports of the Berlin Academy].â^{Footnote 285} ([103], p. 136)
However, he quickly must have found a flaw in his argumentation: He telephoned to stop the printing after less than a page had been typeset. He also wanted that, in the Academyâs protocol, the announcement of this paper be erased. This did not happen; thus we know of his failed attempt, and we can read how his line of thought began ([183], pp. 134â135).
Each month during 1929, papers appeared in which a link between Einsteinâs teleparallelism theory and quantum physics was foreseen. Thus, in February 1929, Wiener and Vallarta stressed that
âthe quantities ^{s}h_{Î»}^{Footnote 286} of Einstein seem to have one foot in the macromechanical world formally described by Einsteinâs gravitational potentials and characterised by the index Î», and the other foot in a Minkowskian world of micromechanics characterised by the index s. That the micromechanical world of the electron is Minkowskian is shown by the theory of Dirac, in which the electron spin appears as a consequence of the fact that the world of the electron is not Euclidean, but Minkowskian. This seems to us the most important aspect of Einsteinâs recent work, and by far the most hopeful portent for a unification of the divergent theories of quanta and gravitational relativity.â [418]
The correction of this misjudgement of Wiener and Vallarta by Fock and Ivanenko began only one month later [134], and was complete in the summer of 1929 [134, 133, 131, 132].
In March, Tamm tried to show
Tamm added a torsion term \(i\hbar \sqrt {({S_i}{S^i})} \chi\) to the Dirac equation (197) and derived from it a general relativistic (SchrĂ¶dinger) wave equation in an external electromagnetic field with a contribution from the spin tensor coupled to a torsion term^{Footnote 288} \({\alpha ^{[i}}{\alpha ^{k]}}{S_{ik}}^l\). As Tamm assumed for the torsion vector \({S_k} = \pm \tfrac{{ie}}{{\hbar c}}\;{\phi _k}\), his tetrads had to be complex, with the imaginary part containing the electromagnetic 4potential Ï_{k}. This induced him to see another link to quantum physics; by returning to the first of Einsteinâs field equations (170) and replacing â in Equation (169) by \(i\tfrac{e}{c}\hbar\) in the limit Ä§â0, he obtained the laws of electricity and gravitation, separately. From this he conjectured that, for finite h, Einsteinâs field equations might correctly reproduce the quantum features of âthe microcosmâ ([341], p. 291); cf. also [340].
What remained after all the attempts at geometrizing the matter field for the electron, was the conviction that the quantum mechanical âwave equationsâ could be brought into a covariant form, i.e., could be dealt with in the presence of a gravitational field, but that quantum mechanics, spin, and gravitation were independent subjects as seen from the goal of reaching unified field theory.
Kaluzaâs theory and wave mechanics
For some, Kaluzaâs introduction of a fifth, spacelike dimension seemed to provide a link to quantum theory in the form of wave mechanics. Although he did not appreciate Kaluzaâs approach, ReichenbĂ€cher listed various possibilities: With the fifth dimension, Kaluza and Klein had connected electrical charge, Fock the electromagnetic potential, and London the spin of the electron [276]. Also, the idea of relating SchrĂ¶dingerâs matter wave function with the new metrical component g_{55} was put to work. Gonseth^{Footnote 289} and Juvet, in the first of four consecutive notes submitted in August 1927 [150, 148, 149, 147] stated:
âThe objective of this note is to formulate a fivedimensional relativity whose equations will give the laws for the gravitational field, the electromagnetic field, the laws of motion of a charged material point, and the wave equation of Mr. SchrĂ¶dinger. Thus, we will have a frame in which to take the gravitational and electromagnetic laws, and in which it will be possible also for quantum theory to be included.â^{Footnote 290} ([150], p. 543)
It turned out that from the R_{55}component of the Einstein vacuum equations \({R_{\alpha \beta }} = 0\), Î±, ÎČ=1, âŠ , 5, with the identification g_{55}=Ï made, and the assumption that Ï, \(\tfrac{{\partial \psi }}{{\partial {x^i}}}\) be âvery smallâ, while Ï, \(\tfrac{{\partial \psi }}{{\partial {x^5}}}\) be âeven smallerâ, the covariant dâAlembert equation followed, an equation that was identified by the authors with SchrĂ¶dingerâs equation. Their further comment is:
âWe thus can see that the fiction of a fivedimensional universe provides a deep reason for SchrĂ¶dingerâs equation. Obviously, this artifice will be needed if some phenomenon would force the physicists to believe in a variability of the [electric] charge.â^{Footnote 291} ([149], p. 450)
In the last note, with the changed identification g_{55}=Ï^{2} and slightly altered weakness assumptions, Gonseth and Juvet gained the relativistic wave equation with a nonlinear mass term.
Interestingly, a couple of months later, O. Klein had the same idea about a link between the g_{55}component of the metric and the wave function for matter in the sense of de Broglie and SchrĂ¶dinger. However, as he remarked, his hopes had been shattered [189]. Kleinâs papers were of import: Remember that Kaluza had identified the fifth component of momentum with electrical charge [181], and five years later, in his papers of 1926 [185, 184], Klein had set out to quantise charge. One of his arguments for the unmeasurability of the fifth dimension rested on Heisenbergâs uncertainty relation for position and momentum applied to the fifth components. If the elementary charge of an electron has been measured precisely, then the fifth coordinate is as uncertain as can be. However, Kleinâs argument is fallacious: He had compactified the fifth dimension. Consequently, the variance of position could not become larger than the compactification length l âŒ 10^{30}, and the charge of the electron thus could not have the precise value it has. In another paper, Klein suggested the idea that the physical laws in spacetime might be implied by equations in fivedimensional space when suitably averaged over the fifth variable. He tried to produce wavemechanical interference terms from this approach [187]. A little more than one year after his first paper on Kaluzaâs idea, in which he had hoped to gain some hold on quantum mechanics, Klein wrote:
âParticularly, I no longer think it to be possible to do justice to the deviations from the classical description of space and time necessitated by quantum theory through the introduction of a fifth dimension.â ([189], p. 191, footnote)
At about the same time, W. Wilson of the University of London rederived the SchrĂ¶dinger equation in the spirit of O. Klein and noted:
âDr. H. T. Flint has drawn my attention to a recent paper by O. Klein [189] in which an extension to five dimensions similar to that given in the present paper is described. The corresponding part of the paper was written some time ago and without any knowledge of Kleinâs work [âŠ].â ([420], p. 441)
Even Eddington ventured into the fifth dimension in an attempt to reformulate Diracâs equation for more than one electron; he used matrix algebra extensively:
âThe matrix theory leads to a very simple derivation of the first order wave equation, equivalent to Diracâs but expressed in symmetrical form. It leads also to a wave equation which we can identify as relating to a system containing electrons with opposite spin. [âŠ] It is interesting to note the way in which the existence of electrons with opposite spins locks the âfifth dimension,â so that it cannot come into play and introduce the absolute into a world of relation. The domain of either electron alone might be rotated in a fifth dimension and we could not observe any difference.â ([61], pp. 524, 542)
Eddingtonâs âpentadsâ built up from sedenions later were generalised by Schouten [307].
J. W. Fisher of Kingâs College reinterpreted KaluzaKlein theory as presented in Kleinâs third paper [187]. He proceeded from the special relativistic homogeneous wave equation in fivedimensional space and, after dimensional reduction, compared it to the KleinGordon equation for a charged particle. By making a choice different from Kleinâs for a constant he rederived the result of de Broglie and others that null geodesics in fivedimensional space generate the geodesics of massive and massless particles in spacetime [127].
Mandel of Petersburg/Leningrad believed that
âa consideration in five dimensions has proven to be well suited for the geometrical interpretation of macroscopic electrodynamics.â ([221], p. 567)
He now posed the question whether this would be the same for Diracâs theory. Seemingly, he also believed that a tensorial formulation of Diracâs equation was handy for answering this question and availed himself of âthe tensorial form given by W. Gordon [151], and by J. Frenkelâ^{Footnote 292} [140]. Mandel used, in fivedimensional space, the complexvalued tensorial wave function \({\Psi ^{ik}} = \psi {\gamma ^{ik}} + {\Psi ^{[ik]}}\) with a 5scalar Ï. Here, he had taken up a suggestion J. Frenkel had developed during his attempt to describe the ârotating electron,â i.e., Frenkelâs introduction of a skewsymmetric wave function proportional to the âtensor of magnetoelectric momentâ m_{ik} of the electron by m_{ik}Ï=m_{0}Ï_{ik} [141, 140]. Îš^{ik} may depend on x^{5}; by taking Îš periodic in x^{5}, Mandel derived a wave equation âwhich can be understood as a generalisation of the KleinFock fivedimensional wave equation [âŠ].â He also claimed that the vanishing of Ï made M_{5} cylindrical (in the sense of Equation (109) [221]). As he had taken notice of a paper of Jordan [173] that spoke of the electromagnetic field as describing a probability amplitude for polarised photons, Mandel concluded that the amplitude of his Îšfield might then represent polarised electrons as its quanta. However, he restricted himself to the consideration of classical oneparticle wave equations because
âin some cases one can properly speak of a quasimacroscopical onebody problem â think of a beam of monochromatic cathodrays in an arbitrary external forcefield.â^{Footnote 293}
In a later paper, Mandel came back to his wave equation with a skewsymmetric part and gave it a different interpretation [222].
Unlike Klein, Mandel tried to interpret the wave function as a new discrete coordinate, an idea going back to Pauli [247]. He took âDiracâs spin variableâ Î¶ and the spatial coordinate x^{5} as a pair of canonicallyconjugate operatorvalued variables; Î¶ is linked to positive and negative elementary charge (of proton and electron) as its eigenvalues. In Mandelâs fivedimensional space, the fifth coordinate, as a âchargeâ coordinate, thus assumed only 2 discrete values Î±e.
âThis completely corresponds to the procedure of the Dirac theory, with the only difference that for Dirac the coordinate Î¶ could assume not 2 but 4 values; from our point of view this remains unintelligible.â^{Footnote 294} ([222], p. 785)
In following Klein, Mandel concluded from the Heisenberg uncertainty relations that
â[âŠ] all possible values of this quantity [x^{5}] still remain completely undetermined such that all its possible values from inf to +inf are of equal probability.â
This made sense because, unlike Klein, Mandel had not compactified the fifth dimension. His understanding of quantum mechanics must have been limited, though: Only two pages later he claimed that the canonical commutation relations \([\mathbf{p},\mathbf{q}] = \tfrac{\hbar }{i}\mathbf{1}\) could not be applied to his pair of variables due to the discrete spectrum of eigenvalues. He then essentially went over to the Weyl form of the operators p, q in order to âsaveâ his argument [222].
Another one of the many versions of âDiracâs equationâ was presented, in December 1930, by Zaycoff who worked both in the framework of Einsteinâs teleparallel theory and of Kaluzaâs fivedimensional space. His Lagrangian is complicated^{Footnote 295},
where summation is implied and \({S_{kl}}^m\) is the torsion tensor, f_{m} the electromagnetic vector potential, and f_{ik} the electromagnetic field. Note that the Dirac current \({J_m}: = \tilde \psi {\gamma _m}\psi\) couples to both the torsion vector and the 4potential. The remaining variables in (198) are \({J_m}: = i\tilde \psi {\gamma _m}{\gamma _l}^\dag {\gamma _0}\psi\), with kâ l, and \({J_{klm}}: = i\tilde \psi {\gamma _k}{\gamma _l}^\dag {\gamma _m}\psi\), with kâ lâ m [434].
While Zaycoff submitted his paper, Schouten lectured at the MIT. and, among other things, showed âhow the massterm in the Dirac equations comes in automatically if we start with a fivedimensional instead of a fourdimensional Riemannian manifoldâ ([306], p. 272). He proved a theorem:
The Dirac equations for Riemannian spacetime with electromagnetic field and mass can be written in the form of equations without field or mass \({\alpha ^b}{\nabla _b}\psi = 0\) in an R_{5}.
Here Î±^{b} is the set of Dirac numbers defined by Î±_{(a}Î±^{b)}=g^{ab}, (Î±)^{a}Î±^{b})Î±^{c}=Î±^{a}(Î±^{b}Î±^{c}) with a, b, c = 0, âŠ , 4, and â_{b} the covariant spinor derivative defined by him.
As we mentioned above (cf. Section 6.3.2), another approach to the matter within projective geometry was taken by Pauli with his student J. Solomon [253]. After these two joint publications, marred by a calculational error, Pauli himself laid out his version of the projective theory in two installments with the first, as a service to the community, being a pedagogical presentation of the formalism connected with projective geometry [249]. The second paper, again, has the application to Diracâs equation as a prime motivation:
âThe following deductions are intended to show [âŠ] that the unifying combination of the gravitational and the electromagnetic fields, by projective differential geometry with the aid of five homogeneous coordinates, is a general method whose range reaches beyond classical fieldphysics and into quantum theory. Perhaps, the hope is not unjustified that the method will stand the test as a general framework for the laws of physics also with regard to a future physical and conceptual improvement of the foundations of Diracâs theory.â^{Footnote 296} ([250], pp. 837â838)
Pauli started with the observation that the group of orthogonal transformations in fivedimensional space had an irreducible, fourdimensional matrix representation satisfying
where \({\alpha _\mu }\) are 4Ă4 matrices given at the end of Section 2.1.5 in a different representation \( {\alpha _\mu } = \left( {\begin{array}{*{20}{c}} {{\sigma _\mu }}&0\\ 0&{  {\sigma _\mu }} \end{array}} \right) \) with ÎŒ = 1, 2, 3, \( {\alpha _4} = \left( {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right) \), and augmented by \( {\alpha _5} = \left( {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right) \). This had been known also to Eddington [61] and Schouten [303]. He then introduced projective spinors depending on five homogeneous coordinates without using beinquantities. He followed the methods of SchrĂ¶dinger and Bargmann [319, 6], i.e., used the existence of a matrix A such that \(\mathbf{A}{\alpha _\mu }\) is Hermitian. The transformation laws of 4spinors Îš and matrices \({\alpha _\mu }\) are coupled:
For transformations in the space of homogeneous coordinates \({X'^\mu } = {a^\mu }_\nu {X^\nu }\) such that \({\alpha _\nu } = {a^\mu }_\nu {\alpha '_\mu }\), the quantity \({a_\mu }: = {\Psi ^\star}A{\alpha _\mu }\Psi\) transforms, for fixed \({\alpha _\mu }\), under changes of the spin frame, Equation (200), as a covariant vector^{Footnote 297}.
Pauli criticised an analogous attempt at formulating Diracâs equation with the help of five homogeneous coordinates by Schouten and van Dantzig [317, 308, 309] as being âdifficult to understand and less than transparentâ^{Footnote 298}. A projective spinor is defined via
where Ï is a normal (âaffineâ) spinor (degree of homogeneity 0) and F a real scalar of (homogeneity) degree 1, i.e., \(F = {X^\mu }\tfrac{{\partial F}}{{\partial {X^\mu }}}\). There exist two (related) spinconnections Î_{k} for projective spinors Îš and \({\mathop \Lambda \limits^{\rm{R}} _k}\) for spinors Ï.
Pauliâs Dirac equation, derived from a Lagrangian, looked in five dimensions like
with \(k =  \tfrac{{imc}}{h}  \tfrac{{ie}}{{hc}}\tfrac{c}{{\sqrt \kappa }}\tfrac{1}{r}\), and \(l = + \tfrac{{ie}}{{hc}}\tfrac{c}{{\sqrt \kappa }}\tfrac{1}{r}\). The covariant derivative is formed with the spin connection Î_{k}. An involved calculation leads to Diracâs equation in four dimensions:
with the numerical factor r, the electrical 4potential ÎŠ_{k}, and the electromagnetic field tensor F_{kl}; furthermore, \({\alpha ^{\mu \nu }} = {\alpha ^{[\mu }}{\alpha ^{\nu ]}}\).
Pauli succeeded also in formulating a fivedimensional energymomentum tensor containing, besides the fourdimensional energymomentum tensor, the fourdimensional Dirac current vector. At the end of his paper Pauli stressed the
âmore provisional character of his 5dimensionalprojective form of Diracâs theory. [âŠ] In contrast to the joinder of the gravitational and electromagnetic fields, a direct logical coupling of the matterwavefield with these has not been achieved in the form of the theory developed here.â
Einstein, spinors, and semivectors
Ehrenfest, even after van der Waerdenâs paper on spinor analysis [368], in 1932 pressed Einstein to think about a simple geometric interpretation of spinors. To this end Einstein responded, together with his assistant Mayer, by introducing the concept of semivector seemingly more natural to him than a spinor, and also more general:
âIn spite of the great importance which the spinor concept, as introduced by P a u l i and D i r a c, has obtained in molecular physics, one cannot claim that the analysis of this concept up to now satisfies all justified demands. Our efforts have lead to a derivation corresponding, according to our opinion, to all demands for clarity and naturalness and avoiding completely any not so transparent artifice. Thereby, [âŠ] the introduction of novel quantities was shown to be necessary, the âsemivectorsâ, which include spinors but possess a clearly more transparent transformation character than spinors.â^{Footnote 299} ([110], p. 522)
In this first publication on the subject, Einstein and Mayer explicitly referred to the paper by Infeld and van der Waerden, of which they had received a copy several months before publication ([167], and [110], p. 25, footnote). Apparently, Einstein found the reconstruction of the spinor concept in his paper more âclear and naturalâ than Infeld and van der Waerdenâs. Nevertheless, the approach and notation of Infeld and van der Waerden became the accepted one by physicists.
About three months before the first paper on semivectors was published, Einstein wrote to Besso:
âI work with my Dr. Mayer on the theory of spinors. We already could clear up the mathematical relations. A grasp on the physics is far away, farther than one thinks at present. In particular, I still am convinced that the attempt at an essentially statistical theory will fail.â^{Footnote 300} ([99], p. 291)
Besides the aspiredto clarity and simplicity, Einsteinâs main hope was that, with his semivector system of equations replacing Diracâs equation, it might be possible to explain the existence of elementary particles with opposite charge and unlike mass, i.e., of electron and proton. As noted before, he had not been able to solve this problem by his affine field theory of 1923 (cf. Section 4.3.2), nor by the approaches to unified field theory that followed. As it turned out, the positron was discovered at about the same time, and the problem dissolved while Einstein and Mayer began to reformulate the spinor concept. Einstein again seems to have been fully convinced that his new concept of âsemivectorâ was superior to the spinor concept. On 7 May 1933, he wrote to De Haas:
âScientifically Mayer and I have found one very natural generalisation of Diracâs equation which makes it comprehensible, that there are two understandable elementary masses, while there is only one electric charge.â [69]
In the first paper in the reports of the Berlin Academy, the mathematical foundations of the semivector formalism are developed [110]. The basic idea of Einstein and Mayer is the possibility of a decomposition of any (proper) Lorentz transformation described by a real matrix D into a product BC of a pair of complexconjugate, commuting matrices^{Footnote 301} B and C. The transformations represented by B or C form a group isomorphic to the Lorentz group. In terms of infinitesimal Lorentz transformations given by an antisymmetric tensor Ï_{ik}, this amounts to the decomposition into a selfdual and an antiselfdual part: \({\omega _{ik}} = \tfrac{1}{2}\left( {{\omega _{ik}} + i\omega _{ik}^*} \right) + \tfrac{1}{2}\left( {{\omega _{ik}}  i\omega _{ik}^*} \right)\), with the dual^{Footnote 302} defined by \(\omega _{ik}^*: = \tfrac{1}{2}\sqrt g { \epsilon _{ijkl}}\;{\omega ^{kl}}\).
Contravariant semivectors of the first and second kind now are defined by their transformation laws: \({\rho ^{i'}} = {b^i}_k{\rho ^k}\) and \({\sigma ^{i'}} = {c^i}_k{\sigma ^k}\), where \({b^i}_k\), \({c^i}_k\) are the components of B, C. For real Lorentz transformation D, \({\bar b^i}_{\;\;k} = {c^i}_k\) must hold. As B, C are both also Lorentz transformations,
âthe metric tensor g_{ik} is also a semivector of 1st kind (and of 2nd kind) with transformationinvariant components.â
Thus it can be used for raising and lowering indices of semivectors ([110], p. 535).
The system of equations intended as a replacement of the Dirac equation appears only in the second publication [111]. A Lagrange function for the semivector is found and the generalised Dirac equations for the semivectors Ï, Ï look like^{Footnote 303}:
where Ï_{r} is the electromagnetic 4potential and \({{E^{r\sigma \tau }}}\) a numerically invariant tensor depending on 4 constants a_{(r)}:
Contrary to his idea of what a ârealâ unified field theory should look like, Einstein just added the Lagrangian for the semivector fields to the Lagrangians for the gravitational and electromagnetic fields.
In his âSpencer Lectureâ of 10 June 1933 in Oxford, Einstein embedded his point of view into the development of field theory: â[âŠ] Louis de Broglie guessed the existence of a wave field that could be used for the interpretation of certain quantum properties of matter. With the spinors, Dirac found novel field quantities whose simplest equations permitted the derivation of the properties of the electron to a great extent. With my collaborator, Dr. Walther Mayer, I now found that these spinors form a special case of a type of field, linked to fourdimensional space, which we called âsemivectorsâ. The simplest equations to be satisfied by such semivectors provide a key for the understanding of the existence of two elementary particles with different ponderable mass and like, but opposite, charge. These semivectors are, besides the usual vectors, the simplest mathematical fieldobjects possible in a fourdimensional metrical continuum; it appears that they naturally describe essential properties of the electrical elementary particles.â^{Footnote 304} ([100], p. 117)
That semivectors are in close connection with spinors and not always the simplest objects, had been noted by Schouten in a paper submitted to Zeitschrift fĂŒr Physik on 15 April 1933, even before Einsteinâs Spencer lecture:
âThe space of semivectors of first and second kind is the manifold of two simple bivectors in the local spacetimeworld, which lie on the null cone in two planes of the first and second system of planes, respectively. [âŠ] In the notprojective theory as well as in the projective without an electromagnetic field, semivectors have an advantage, [âŠ]. As soon as an electromagnetic field is present, in the projective theory, calculation with spinvectors is simpler than calculation with semivectors.â^{Footnote 305} [309]
In a previous paper, Schouten had geometrically explained the two approaches to spinor analysis followed by van der Waerden [368], Laporte and Uhlenbeck [203], and by himself. In the first approach, the vectors of the two invariant planes in spin space were identified; in the second, i.e., in Schoutenâs, they were taken as the basis for a fourdimensional vector space [308]. Schouten had been corresponding with Pauli who âwas so friendly as to allow me to quote this theorem from a not yet published manuscript.â^{Footnote 306} In his second paper, Schouten placed the semivectors of Einstein and Mayer into his geometrical setting. He showed that there exist two different complexconjugated threedimensional representations of the real Lorentz group as a 6parameter subgroup of the fourdimensional orthogonal group. With the help of these he constructed four âbuilding blocks: the special spin vectors of 1st and 2nd kind of spin space, which at the same time are special semivectors in the two preferred invariant planes in semispace. In both cases, always two vectors are joined which belong to different invariant planes, for semivectors two with same transformation law, for spin vectors two with conjugate complex transformations.â ([309], p. 106)
As Schouten used group theory, quaternions, sedenions, projective geometry â all not very familiar to physicists â Pauli must have thought of a much simpler disentanglement of semivectors and spinors. In fact, as Pauli pointed out in his letter to Einstein of 16 July 1933, a semivector does not form an irreducible representation of the rotation group while a spinor does^{Footnote 307} (see [252], p. 189). In this regard, the semivector concept is less ânaturalâ than the spinor concept. In contrast to what Schouten had shown, in his book on spinors Cartan flatly disclaimed any geometric definition for semivectors^{Footnote 308}. Pauli then set his doctoral student V. Bargmann on the problem of comparing Einsteinâs semivector approach to the spinor calculus as developed by van der Waerden. Bargmann acknowledged Schoutenâs paper [309], âwith which our presentation has quite a few points of contactâ. He then proved that
âto each semivector two 2component spinors correspond, which both satisfy the same transformation law.â ([7], p. 68)
and that the generalised Dirac equations of Einstein and Mayer (206)
âdecompose into two separate 4component systems of Diracâs type, which are distinguished only by the mass values.â ([7], p. 78)
This means that only one of these two systems is needed, and that it describes particles with opposite charge and the same mass. Thus, as van Dongen states curtly:
âIt is evident from Bargmannâs analysis that the most general semivector Dirac system of Einstein and Mayer is nothing more than just a linear superposition of two independent Dirac spinor systems and thus cannot give insight into the fundamental nature of electrons and protons.â ([371], p. 88, and [372])
In his letter to Einstein, Pauli had also mentioned his papers to be published in Annalen der Physik and discussed here in Section 7.2.3. Two more papers were written by Einstein and Mayer before Einstein quietly dropped the subject. The last paper considered semivectors as âusual vectors with a different differentiation characterâ [112, 113].
It seems that Einstein at the time had not followed quantum field theory intensively enough to be able to compete with that theory â leaving aside his rejection of âthe statistical fadâ. By continuing to pair âelectron and protonâ while others speculated already about âelectron and its antiparticleâ^{Footnote 309} or âproton and its antiparticleâ, he was bound to run into a dead end. About one year after the Spencer lecture, when Einstein was still publishing about semivectors, Pauli and Weisskopf quantised the scalar relativistic wave equation with an external field using BoseEinstein statistics:
âWithout further hypothesis the existence follows of particles with opposite charge and same rest mass, which can be produced or annihilated by absorption or emission of electromagnetic radiation. The frequency of these processes is shown to be of the same magnitude as the one for particles with the same charge and mass following from Diracâs hole theory.â^{Footnote 310} ([255], Abstract)
Of course, Einsteinâs problem was quickly solved; at about the same time the electronâs antiparticle was observed.
Less Than Unification
During the time span considered here, there also were those who did not adhere to the program of unified field theory, or changed their mind about it. The mathematician G. D. Birkhoff belonged to the first group. Instead of removing the matter tensor in favour of some general geometry, he just remained in the old setting of Einsteinâs theory: To the energymomentum tensors of a perfect fluid and of the electromagnetic field on the r.h.s. of Einsteinâs field equations, he added a contribution, named the âatomic potential tensorâ describing a scalar wave function Ï Ă la SchrĂ¶dinger: A_{ij}=Ï g_{ij} with the âpotentialâ Ï. He proposed to apply this theory
Another one was the Russian physicist G. Rumer, whose somewhat exotic suggestion, within a framework of Kantian philosophy, was to remain in Riemannian geometry and keep to Einsteinâs vacuum field equations but raise the number of the dimension of the underlying manifold. He named such a manifold an F_{n} and considered a threedimensional Riemannian subspace V_{3} embedded, locally and isometrically, into F_{n}. Then, by use of the Gauss.Codazzi and Ricci.Codazzi equations, he decomposed the Ricci tensor of F_{n} into its part in V_{3} and the rest. His classification went as follows:
âWe have seen that the V_{3} itself is either an F_{3} (in this case it is empty), or a subspace of an F_{4} (thus it contains a gravitational field), or a subspace of F_{5} (then also an electrical field is present). In the case in which âsomethingâ exists which is neither a gravitational nor an electrical field, the V_{3} must be a subspace of F_{n} (nâ„6). However, geometry shows that every V_{3} is a subspace of a particular F_{6}, i.e., the Euclidean or pseudoEuclidean space. This shows us that transition to F_{6} is the final step.â^{Footnote 311} ([285], p. 277)
The unidentified parts in the decomposition of the Ricci tensor into its piece in V_{3} and the rest Rumer ascribed to âmatter.â He acknowledged Bornâs
â[âŠ] stimulation and his interest extended toward the completion of this paper.â
^{Footnote 312} Born himself was mildly skeptical^{Footnote 313}:
â[âŠ] a young Russian surfaced here who brought with him a 6dimensional relativity theory. As I already felt frightened by the various 5dimensional theories, and had little confidence that something beautiful would result in this way, I was very skeptical.â^{Footnote 314}
After Lanczos had (mildly) criticised Einsteinâs parallelism at a distance [201], he seemed to have lost confidence in Einsteinâs program for unification and became a ârenegadeâ. He developed a theory by which
â[âŠ] the basic properties of the electromagnetic field may be derived effortlessly from the general properties of Riemannian geometry by use of a variational principle characterised by a very natural demand.â^{Footnote 315} ([202], p. 168)
For his Lagrangian, he took \({\mathcal L} = {R_{ik}}{R^{ik}} + C{({R^k}_k)^2}\), with C being a constant. He first varied with respect to the metric g_{ik} and the Ricci tensor R_{ik} as independent variables, and then expressed the variation ÎŽR_{ik} with ÎŽg_{ij}. The resulting variation is then set equal to zero. In the process âspontaneouslyâ a
âfree vector appears for which, later, a restraining equation of the type of the equation for the [electromagnetic] potential results â as a consequence of the conservation laws for energy and momentum.â
Also Rainichâs approach [265] mentioned in Section 6.1, which, in the case of Maxwellâs equations without sources, and for nonnull electromagnetic fields, did substitute a set of algebraic conditions on the Einstein tensor for Maxwellâs equations, might be seen as an alternative for the unification of gravity and electromagnetism. According to L. Witten:
âThe only criterion for a unified field theory that these equations do not satisfy is that they are not derived from a variational principle by means of a Lagrangeâs function involving geometric quantities alone.â ([422], p. 397)
Finally, van Dantzigâs program after 1934, which we briefly met in Section 1, might be considered. It aimed at showing, eventually, that the
âmetric should turn out finally to be a system of some statistical mean values of certain physical quantities.â ([363], p. 522)
This meant turning upside down Einsteinâs geometrization program for matter.
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 indepthstudy 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 metricaffine geometry received its original impetus from Einsteinâs general relativity and Weylâs 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. Weylâs influence was prominent among both parties, mathematicians (Cartan, Schouten, Struik, Eisenhart, Hlavaty, Wiener, etc.) and physicists (Eddington, Einstein, ReichenbĂ€cher, Mandel, Fock, Zaycoff, et al). In the review on differential geometry by the mathematician Berwald^{Footnote 316}, Weyl plays a prominent role while Einstein is mentioned only in passing in connection with âEinsteinmanifoldsâ (Einsteinspaces) ([13], p. 163). Berwald discusses Eddingtonâs affine theory as the most modern development with, again, Einsteinâs papers [77, 74, 76] 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 Pauliâs article in the Encyclopedia of Mathematical Sciences [246]. He notes papers by Bach (alias FĂ¶rster) [4] and G. Juvet [180] on Weylâs theory, and a paper of Kretschmann [196]. Unlike Berwald, WeitzenbĂ¶ck in his review article on the theory of invariants mentions Einstein directly:
âIn recent years, due to their use in modern physical theories, the theory of differential forms (tensors) was elaborated extensively. We mention Ricci, LeviCivita, Hessenberg, Einstein, Hilbert, [Felix] Klein, E. Noether, Weyl.â^{Footnote 317} [392]
Einstein is mentioned as the only theoretical physicist among seven mathematicians. The development of projective geometry did profit from the mathematician Kaluzaâs idea of a fivedimensional 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 coauthorship 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 ZĂŒrich, and the group of physicists (EinsteinPauliEddington, Kleinde Broglie, EinsteinReichenbĂ€cher, 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 communicationsnet 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 Einsteinâs 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 fĂŒr Physik. From the mutual references to their papers, among mathematicians Weyl, Cartan, Schouten, Eisenhart, Veblen, T. Y. Thomas, J. M. Thomas, LeviCivita, Berwald, WeitzenbĂ¶ck, and later HlavatĂœ and Vranceanu stand out. In the following, we give a few examples of interaction among mathematicians. From Schoutenâs acknowledgment,
âI 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.â^{Footnote 318} [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:
â[This note] suggested itself as a possible basis for a geometrical interpretation of Eddingtonâs 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 spinquantities which he was then developing.â ([306]; Schouten had Hlavaty as a coauthor [312])
Among Schoutenâs correspondents were theoretical physicists as well, such as Pauli. In a paper on âspace time and spin spaceâ Schouten acknowledged that
âMr. Pauli was so friendly as to permit me to quote this theorem from an unpublished manuscript.â^{Footnote 319} ([308], p. 406, footnote 4)
and
âA correspondence with Mr. Pauli induced me to investigate this invariance.â^{Footnote 320} ([308], p. 414, footnote 1)
We also noticed in Section 5 that Schouten wrote his papers on the classification of linear connections [297, 296] 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 ReichenbĂ€cher. In terms of his scientific output in the area of unified field theory, a more precise description of the balance between Einsteinâs being at the receiving end and his stimulating and creative role will have to be given in the future.
Public Reception of Unified Field Theory at the Time
We must distinguish between the acceptance of unified field theory in academic circles and its reverberations in the general press. Of course, Weyl could further his own theory through his well received book [396]. His unification of electricity and gravitation was also given room in Eddingtonâs monograph âSpace, Time, and Gravitationâ of 1920 [57]. In German scientific handbooks on physics and monographs on relativity theory at the time, there are long articles on special and general relativity, but unified field theory is mentioned only in passing, and with the names of Weyl, Eddington, and Kaluza stressed instead of Einsteinâs ([194], pp. 576â579, and [9], pp. 378â383). Perhaps, it was too early for Born to mention Weyl in his book on Einsteinâs relativity theory of 1920 [19]. In the third edition of 1922, he entered a single sentence after he had lauded Einstein for his derivation of the field equations of general relativity:
âHilbert, Klein [i.e., Felix Klein] and other mathematicians have taken part, have researched in depth and illuminated the formal structure of Einsteinâs formulas.â^{Footnote 321} ([20], p. 248)
After Pauliâs handbook article, Weyl hardly could have been overlooked. Nevertheless, the 2nd volume of Laueâs book did not even mention Weylâs theory, nor did LeviCivitaâs book in its English or German versions of 1927 and 1928 [387, 205]. In contrast, Schoutenâs and Eddingtonâs monographs treated Weyl extensively [300, 60]. Already in 1921, Fabreâs weak presentation of âLes thĂ©ories dâEinsteinâ [125] had an admiring but irrelevant appendix on Weylâs unified field theory, while, in 1922, J. Becquerel in Paris and E. Neumann in Marburg in their books on relativity had brief presentations of Weylâs theory [236, 10]. One or the other philosopher of science took into account some of the developments, mostly Weylâs theory, as did Reichenbach in 1928 [267]. In his book on âThe present world view according to the natural sciencesâ, Wenzl discussed Einsteinâs theory of distant parallelism but cautioned that this theory, as a physical theory was not on the same level as general relativity: It seemed unclear whether the new theory would predict new phenomena that could be empirically tested [394].
As to the impact on the public at large, Einsteinâs fame as the creator of the empirically tested general relativity continued to shine over his successive attempts at unified field theory. Chandarsekhar reports an afterdinner chat in 1933 in which Rutherford made Eddington responsible for Einsteinâs fame; he nailed it to the headlining, in the British papers, of the meeting of the Royal Society, at which the results of the British Solar Eclipse Expedition were reported:
â[âŠ] the typhoon of publicity crossed the Atlantic. From that point on, the American press played Einstein to the maximum.â ([42], p. 28)
Also, from 1930, Einsteinâs yearly travels to the United States and his sojourns there brought about increased publicity for his research on unified field theory. Thus, a âpreliminary announcement for the Josiah Macy, Jr., Foundationâ of his paper with his assistant Mayer on the 5vector formulation of Kaluzaâs theory [107] was printed in full in Science about a month before the publication appeared [94]. However, it seems safe to say that reports on Einsteinâs newest unified field theory in the dailies, whether seen as an educational affair or as part of entertainment, must have strained the general publicâs intellectual abilities.
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 Ricciâs 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 Einsteinâs path within unified field theories came from these mathematicians: FĂ¶rsterâs asymmetric metric^{Footnote 322}, Cartanâs distant parallelism, Kaluzaâs fivedimensional space, Weylâs, Schoutenâs, and Cartanâs completely general concept of connection, Veblenâs 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 reinvent mathematical developments made before. The ideas most fruitful for physics in the long run came from Weyl (âgauge conceptâ), Kaluza (âextension of number of space dimensionsâ), and O. Klein (âcompactificationâ) ^{Footnote 323}. Vizgin states that Einstein, around 1923 to 1930,
âbecame 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.â ([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, Einsteinâs 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 Einsteinâs zigzagging path through the wealth of constructive possibilities was followed by many in the (small) body of researchers in the field.
Cartan saw it positively:
âOne can see [âŠ] 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.â^{Footnote 324} ([36], p. 1184/1185)
If there is an enigma left in the scientific part of Einsteinâs 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 Einsteinâs endeavours â from the affine approach to the teleparallel theory  lead nowhere, but sometimes they were also quite behind what others knew already, as in KaluzaKlein 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 gravitoelectromagnetic 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 Einsteinâs 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:
âAt 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.â^{Footnote 325} ([92], p. 1)
Then, why did he so obstinately follow this line of research and isolate himself from most of his peers, except SchrĂ¶dinger and, perhaps, de Broglie? Was it an unfailing belief that âgeometrizable fieldsâ must play the fundamental role as compared to particles^{Footnote 326}? 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 hypotheticaldeductive approach alone, without assistance from any empirical input:
â[âŠ], 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.â ([86], pp. 114â115)
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 âthe derivation of the complicated field law of gravitation along a logical path,â Einstein went on to say:
âThe 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.â^{Footnote 327} [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 selfconfidence. 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:
âThis looks oldfashioned, and the dear colleagues and also you, my dear, will show me the tongue as long as they can. Because Planckâs 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 spacetime picture, and then these equations will form a starting point.â^{Footnote 328} ([99], p. 240)
Even seventy years later, with a limit to the âstatistical fadâ not yet reached, Einsteinâs 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 âgeometrizedâ^{Footnote 329}. The argument put forward in favor of such a continuation is now much the same as that advanced by K. Novobatzky in 1931:
âIn the present situation, neither from classical nor from quantum mechanical methods alone âallâ 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.â^{Footnote 330} ([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 Einsteinâs ZĂŒrich notebook of 1912 [279, 280].) 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.
Notes
Albert Einstein (1879â1955). Born in Ulm, WĂŒrttemberg (Germany). Studied physics and mathematics at the Swiss Federal Polytechnic School (ETH) Zurich and received his doctorâs degree in 1905. Lecturer at the University of Bern (Switzerland), Professor in Zurich, Prague (then belonging to Austria), Berlin (Germany) and Princeton (U.S.A.). Nobel Prize 1921 for his work on the lightelectric effect (photon concept). Best known for his special and general relativity theories. Important results in Brownian motion and the statistical foundations of radiation as a quantum phenomenon. Worked for more than 30 years on Unified Field Theory.
Wolfgang Ernst Pauli (1900â1958). Born in Vienna, Austria. Studied at the University of Munich with A. Sommerfeld who recognised his great gifts. Received his doctorate in 1921 for a thesis on the quantum theory of ionised molecular hydrogen. From October 1921 assistant of Max Born in GĂ¶ttingen. After a year with Bohr, Pauli, became a lecturer at the University of Hamburg in 1923. In 1928 he was appointed professor of theoretical physics at the Federal Institute of Technology in ZĂŒrich. From 1945â1950 guest professor at the Institute for Advanced Study, Princeton. He then returned to ZĂŒrich. Did important work in quantum mechanics, quantum field theory and elementary particle theory (fourth quantum number (spin), Pauli exclusion principle, prediction of neutrino). Fellow of the Royal Society. Nobel Prize winner in 1954.
Vizginâs book is the only one that covers the gamut of approaches during the period considered. Fortunately, he has made accessible contributions in the Russian language by scientists in the Soviet Union. Vizgin also presents and discusses attempts at unification prior to 1914.
The inclusion of the quantum corresponds to Vizginâs maximal unification problem [385], p. 169.
See Section 1 of Vizginâs book [385] for a treatment of the history of prerelativistic unified field theories and an exposition of Mieâs, Ishiwaraâs, and NordstrĂ¶mâs approaches.
In presentday interpretation, the first two fields are fields mediating the interactions while the third, the electron field, really is a matter field.
This definition corresponds, in a geometrical framework, to Vizginâs minimal unification problem ([385], p. 187).
Theodor Franz Eduard Kaluza (1885â1954). Born in Ratibor, Germany (now Raciborz, Poland). Studied mathematics at the University of Kânigsberg (now Kaliningrad, Russia) and became a lecturer there in 1910. In 1929 he received a professorship at the University of Kiel, and in 1935 was made full professor at the University of GĂ¶ttingen. He wrote only a handful of mathematical papers and a textbook on âHigher mathematics for the practicianâ (cf. [423]).
âEine Hoffnung ist aber nicht in ErfĂŒllung gegangen. Ich dachte, wenn es gelingt, dieses Gesetz aufzustellen, dass es eine brauchbare Theorie der Quanten und Materie bilden wĂŒrde. Aber das ist nicht der Fall. Die Konstruktion scheint am Problem der Materie und der Quanten zu scheitern.â
Jan Arnoldus Schouten (1883â1971). Born near Amsterdam in the Netherlands. Studied electrical engineering at the Technical University (Hogeschool) of Delft and then mathematics at the University of Leiden. His doctoral thesis of 1914 was on tensor analysis, a topic he worked on during his entire academic career. From 1914 until 1943 he held a professorship in mathematics at the University of Delft, and from 1948 to 1953 he was director of the Mathematical Research Centre at the University of Amsterdam. He was a prolific writer, applying tensor analysis to Lie groups, general relativity, unified field theory, and differential equations.
â[âŠ] AnhĂ€ufungen von positiver und negativer ElektrizitĂ€t, die wir in den positiven Wasserstoffkernen und in den negativen Elektronen antreffen. Die Ă€ltere Maxwellsche Theorie erklĂ€rt diese AnhĂ€ufungen nicht, aber auch den neueren Bestrebungen ist es bisher nicht gelungen, diese AnhĂ€ufungen als selbstverstĂ€ndliche Folgen der zugrundeliegenden Differentialgleichungen zu erkennen. Sollte aber eine solche ErklĂ€rung gefunden werden, so darf man vielleicht auch hoffen, dass die [âŠ] mysteriĂ¶sen Quantenbahnen in ein neues Licht gerĂŒckt werden.â
It is true that Dirac, in his first paper, in contrast to what his âholeâtheory implied, had identified the positively charged particle corresponding to the electron also with the proton [55]. However, after Weyl had pointed out that Diracâs hole theory led to equal masses [409], he changed his mind and gave the new particle the same mass as the electron [56].
Cornelius Lanczos (KornĂ©l LĂ¶wy) (1893â1974). Born in SzĂ©kesfehĂ©rvĂĄr (Hungary). Studied physics and mathematics at the University of Budapest with EĂ¶tvĂ¶s, FejĂ©r, and Lax. Received his doctorate in 1921, became scientific assistant at the University of Freiburg (Germany) and lecturer at the University of Frankfurt am Main (Germany). Worked with Einstein in Berlin 1928â1929, then returned to Frankurt. Became a visiting professor at Purdue University in 1931 and came back on a professorship in 1932. Worked mainly in mathematical physics and numerical analysis. After 1944 he held various posts in industry and in the National Bureau of Standards. Left the U.S.A. during the McCarthy era and in 1952 followed an invitation by SchrĂ¶dinger to become head of the Theoretical Physics Department of the Dublin Institute for Advanced Study.
âIch glaube darum, dass zwischen dem hier vertretenen âreactionĂ€ren Standpunktâ, der eine vollstĂ€ndige feldtheoretische Beschreibung auf Grund der normalen RaumZeitStruktur erstrebt, und dem wahrscheinlichkeitstheoretischen (statistischen) Standpunkt ein Kompromiss [âŠ] nicht mehr mĂ¶glich ist.â
âSollten sich die hier vorausgeahnten MĂ¶glichkeiten als wirklich lebensfĂ€hig erweisen, so wĂŒrde die Quantenmechanik aufhĂ¶ren, eine selbstĂ€ndige Disziplin zu sein. Sie wĂŒrde verschmelzen mit einer vertieften âTheorie der Materieâ, die auf regulĂ€re LĂ¶sungen von nichtlinearen Differentialgleichungen aufzubauen hĂ€tte, â in letztem Zusammenhang also aufgehen in den âWeltgleichungenâ des Universums. Der Dualismus âMaterieFeldâ wĂŒrde dann ebenso ĂŒberwunden sein, wie der Dualismus âKorpuskelWelleâ.â
David van Dantzig (1900â1959). Born in Rotterdam, Netherlands. Studied mathematics at the University of Amsterdam. Worked first on differential geometry, electrodynamics and unified field theory. Known as cofounder, in 1946, of the Mathematical Centre in Amsterdam and by his role in establishing mathematical statistics as a subdiscipline in the Netherlands.
Thus, in a paper of 1934 really belonging to the 2nd part of this review, Schouten and Haantjes exchanged the previously assigned âinduced geodesic linesâ for âautogeodesical linesâ [311].
Ernst ReichenbĂ€cher (1881â1944). Studied mathematics and received his doctorate from the University of Halle in 1903 under the guidance of AlbertWangerin (a student of Franz Neumann in KĂ¶nigsberg). At first, ReichenbĂ€cher did not enter an academic career, but started teaching in a Gymnasium in Wilhelmshaven in North Germany, then in KĂ¶nigsberg on the Baltic Sea. In 1929 he became a Privatdozent (lecturer) at the University of KĂ¶nigsberg (now Kaliningrad, Russia). His courses covered special and general relativity, the physics of fixed stars and galaxies with a touch on cosmology, and quantum mechanics. In the fifth year of World War II he finally received the title of professor at the University KĂ¶nigsberg, but in the same year was killed during a bombing raid on the city.
Hermann Klaus Hugo Weyl (1885â1955). Born in Elmshorn, Germany. Studied at the Universities of Munich and GĂ¶ttingen where he received his doctorate in 1908 (Hilbert was his supervisor). From 1913 he held the Chair of Mathematics at the Federal Institute of Technology in ZĂŒrich, and from 1930 to 1933 a corresponding Chair at the University of GĂ¶ttingen. Then until retirement he worked at the Institute for Advanced Study in Princeton. Weyl made important contributions in mathematics (integral equations, Riemannian surfaces, continuous groups, analytic number theory) and theoretical physics (differential geometry, unified field theory, gauge theory). For his papers, cf. also the Collected Works [411]
Arthur Stanley Eddington (1882â1944). Born in Kendal, England. Studied mathematics at Owens College, Manchester and Trinity College, Cambridge. After some work in physics, moved into astronomy in 1905 and was appointed to the Royal Observatory at Greenwich. From 1914 director of the Cambridge Observatory. Fellow of the Royal Society. As a Quaker he became a conscientious objector to military service during the First World War. Eddington made important contributions to general relativity and astrophysics (internal structure of stars). In 1918, he led an eclipse expedition from which the first indications resulted that Einsteinâs general relativity theory was correct. Wrote also on epistemology and the philosophy of science.
Oskar Klein (1894â197). Born in MĂ¶rby, Sweden. After work with Arrhenius in physical chemistry, he met Kramers, then a student of Bohr, in 1917. Klein worked with Bohr in the field of molecular physics and received his doctorate in 1921 at Stockholm HĂ¶gskola. His first research position was at the University of Michigan in Ann Arbor, where he worked on the Zeeman effect. Back in Europe from 1925, he taught at Lund University and tried to connect Kaluzaâs work with quantum theory. In 1930 he became professor for mathematical physics at Stockholm HĂ¶gskola until retirement. His later work included quantum theory (KleinNishina formula), superconductivity, and cosmology.
Luther Pfahler Eisenhart (1876â1965). Born in York, Pennsylvania, U.S.A. Studied mathematics at John Hopkins University, Baltimore and received his doctorate in 1900. Eisenhart taught at the University of Princeton from 1900, was promoted to professor in 1909 and remained there (as Dean of the mathematical Faculty and Dean of the Graduate School) until his retirement in 1945. All his work is in differential geometry, including Riemannian and nonRiemannian geometry and in group theory.
Oswald Veblen (1880â1960). Born in Decorah, Iowa, U.S.A. Entered the University of Iowa in 1894, receiving his B.A. in 1898. He obtained his doctorate from the University of Chicago on âa system of axioms in geometryâ in 1903. He taught mathematics at Princeton (1905â1932), at Oxford in 1928â1929, and became a professor at the Institute for Advanced Study in Princeton in 1932. Veblen made important contribution to projective and differential geometry, and to topology. He gave a new treatment of spin.
Elie Joseph Cartan (1869â1951). Born in Dolomien near ChambĂ©ry, France. Student at lâĂcole Normale since 1888, he received his Ph.D. in 1894 with a thesis in which he completed Killingâs classification of semisimple algebras. He lectured at Montpellier (1894â1903), Lyon (1896â1903), Nancy (1903â1909), and Paris (1909â1940). His following work on the representation of semisimple Lie groups combines group theory, classical geometry, differential geometry, and topology. From 1904 he worked on differential equations and differential geometry, and developed a theory of moving frames (calculus of differential forms). He also contributed to the geometry of symmetric spaces and published on general relativity and its geometric extensions as well as on the theory of spinors. For his Collected Works, cf. [38].
Dirk J. Struik (1894â2000). Born in Rotterdam in the Netherlands. Studied mathematics and physics at the University of Leiden with Lorentz and de Sitter. Received his doctorate in 1922. Then worked with Schouten at the University of Delft and, with a Rockefeller International Education Fellowship, moved to Rome and GĂ¶ttingen. After a collaboration with Wiener, in 1926 he received a lectureship at the Massachusetts Institute of Technology (MIT) in Cambridge, Ma. where he became full professor in 1940. He stayed on the MIT mathematics faculty until 1960. As a professed Marxist he was suspended from teaching duties during the McCarthy period but was reinstated in 1956. In 1972, he became an honorary research associate in the History of Science Department of Harvard University.
for the following, it is assumed that readers have some prior knowledge of the mathematics underlying General Relativity.
for the precise definition of âgeometrical objectâ, cf. Yanoâs book [425].
The second fundamental form comes into play when local isometric embedding is considered, i.e., when .... is taken as a submanifold of a larger space such that the metrical relationships are conserved. In the following, all geometrical objects are supposed to be differentiable as often as is needed.
Here, the Kroneckersymbol ÎŽ^{i}_{k} with value +1 for i=k, and value 0 for iâ k is used. ÎŽ^{i}_{k} keeps its components unchanged under arbitrary coordinate transformations.
Latin indices i, j, k, âŠ run from 1 to D, or from 0 to Dâ1 to point to the single timelike direction. We have used the symmetrisation bracket defined by \({A_{(ij)}}: = 1/2({A_{ij}} + {A_{ji}})\).
In physical applications, special conditions for Îł and Ï might be needed in order to guarantee that g is a Lorentz metric.
We have \({({L_X}Y)^k}: = {([X,Y])^k} = {X^i}{\partial _i}{Y^k}  {Y^i}{\partial _i}{X^k}\).
Eisenhart called his object âennupleâ instead of nbein [119]. In French, the expressions ânpodeâ, ânĂšdreâ, and âpolyaxe ĂĄ n dimensionsâ were also used.
In an arbitrary basis for the differential forms (cotangent space), the connection may be represented by a 1form.
In the literature, different notations and conventions are used. Tonnelat [356] writes \({A_{\mathop k\limits_ + ;j}}: = {A_{k,j}}  {L_{kj}}^l{A_l}\) and \({A_{\mathop {{\rm{ }}k}\limits_  ;j}}: = {A_{k,j}}  {L_{jk}}^l{A_l}\).
Many authors replace âautoparallelâ by âgeodesicâ. We will reserve the name geodesic for curves of extreme length; cf. Riemannian geometry.
In the following we will note a symmetrical connection by \({\Gamma _{ij}}^k = {L_{(ij)}}^k\).
Schouten denotes the curvature tensor by \({K_{jkl}}^i\).
Again, we have two kinds of homothetic curvatures deriving from \(\mathop K\limits^ +\) and \(\mathop K\limits^ \). In the following, we will mostly use the \(\mathop X\limits^ +\)quantities and drop the + sign.
for a threeindextensor the symmetrisation bracket is defined by
$$ {A_{(ikl)}}: = \frac{1}{{3!}}({A_{ikl}} + {A_{lik}} + {A_{kli}} + {A_{kil}} + {A_{lki}} + {A_{ilk}}). $$Tullio LeviCivita (1873â1941). Born in Padua, Italy. Studied mathematics and received his doctorate at the University of Padua. Was given the Chair of Mechanics there and, in 1918, went to the University of Rome in the same position. Together with Ricci, he developed tensor calculus and introduced covariant differentiation. He worked also in the mechanical manybody problem, in hydrodynamics, general relativity theory, and unified field theory. Strongly opposed to Fascism in Italy and dismissed from his professorship in 1938.
This definition differs from Schoutenâs by an overall minus sign.
See [157]; his sign conventions are different, though.
In the case of an asymmetric metric, this relation must be satisfied separately by the symmetric and antisymmetric parts of g:
$$ \begin{array}{*{20}{c}} {{\nabla _k}{h_{ij}} = 0,\;\;\;}&{{\nabla _k}{\phi _{ij}} = 0.} \end{array} $$Joseph Miller Thomas (1898â1979). Studied mathematics in Philadelphia at the University of Pennsylvania. Received his doctorate in 1923. From 1927 assistant professor at the University of Pennsylvania, from 1930 assistant and in 1935 full professor of mathematics at the Duke University in Durham, North Carolina. His fields were differential geometry and partial differential equations. He was the principle founder of Duke Mathematical Journal.
The index 0 does not refer to a timecoordinate.
Note that \({X^\alpha }\) and \(\lambda {X^\alpha }\) correspond to the same point of M_{D}; Veblen uses \(\lambda = \exp (N{x^0})\) (see [379], Section II).
For an asymmetric connection, this corresponds to the + derivative.
The external derivative d of linear forms Ï, ÎŒ satisfies the following rules:

(1)
\(d(a\omega + b\mu ) = ad\omega + bd\mu\),

(2)
\(d(\omega \wedge \mu ) = d\omega \wedge \mu  \omega \wedge d\mu\),

(3)
ddÏ=0.

(1)
A representation of a group is defined as a map to the vectors of a linear space that is homomorphic in the group operation.
The full Lorentz group contains as further elements the temporal and spatial reflections.
\({\mathbf{X}^ + }_{AB} = {\mathbf{X}_{BA}};A,B = 1,2\).
2component spinors are also called Weylspinors.
For a contemporary exposition of the use of spinors in spacetime, see the book of Penrose and Rindler [256].