On the History of Unified Field Theories

The second structure to be introduced is a linear connectionL with D3 components L_ij^k; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations³⁵. The connection is a device introduced for establishing a comparison of vectors in different points of the manifold. By its help, a tensorial derivative ∇, 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 ω 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 Xⁱ_,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 others³⁶.

For a vector density (cf. Section 2.1.5), the covariant derivative of X̂ contains one more term:

A smooth vector field Y is said to be parallely transported along a parametrised curve λ(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 point³⁷: By a particular choice of the curve73x2019;s parameter, σ=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].

From the connection L_ij^k further connections may be constructed by adding an arbitrary tensor field T to its symmetrised part³⁸: 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 \({S_i} = \tfrac{1}{2}({L_i} - {\tilde L_i})\).

In this section, we briefly present Cartan’s one-form formalism in order to make understandable part of the literature. Cartan introduces one-forms \({\theta ^{\hat a}}(\hat a = 1, \ldots ,4)\) 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 bein-components X^k̂ is defined via Cartan’s first structure equations, where \({\omega ^{\hat \imath }}_{\hat k}\) is the connection-1-form, and \({\Theta ^{\hat \imath}}\) is the torsion-2-form, \({\Theta ^{\hat \imath }} = - {S_{\hat l\hat m}}^{\hat \imath}{\theta ^{\hat l}} \wedge {\theta ^{\hat m}}\). 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 \({\omega ^{\hat \imath}}_{\hat k} = h_l^{\hat \imath}h_{\hat k}^m{L_{\hat rm}}^l{\theta ^{\hat r}}\)⁵⁰. The covariant derivative of a tangent vector with bein-components X^k̂ then is

By further external derivation ⁵¹ on Θ we arrive at the second structure relation of Cartan, In Equation (65) the curvature-2-form \({\Omega ^{\hat k}}_{\hat l} = \tfrac{1}{2}{R^{\hat k}}_{\hat l\hat m\hat n}{\theta ^{\hat m}} \wedge {\theta ^{\hat n}}\) appears, which is given by \({\Omega ^{\hat k}}_{\hat k}\) is the homothetic curvature.

Weyl’s 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:

“If 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.” 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.

“However, the possibility of such a comparison ‘at a distance’ in no way can be admitted in a pure infinitesimal geometry.”⁷⁷ ([397], p. 397)

In order to invent a purely “infinitesimal” geometry, Weyl introduced the 1-dimensional, 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., δl=λl. If the non-metricity tensor is assumed to have the special form Q_ijk=Q_kg_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 Maxwell’s 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=Qⁱdxⁱ. With regard to the gauge transformations (98), \({\Gamma _{ij}}^k\) remains invariant. From the 1-form dQ, by exterior derivation a gauge-invariant 2-form \(F = {F_{ij}}d{x^i}\,\Lambda \;d{x^j}\) with \(F = {Q_{i,j}} - {Q_{j,i}}\) follows. It is named “Streckenkrümmung” (“line curvature”) by Weyl, and, by identifying Q with the electromagnetic 4-potential, 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 curve’s parameter, we may write (101) in the form \(d\left| X \right| = - {Q_k}{X^k}\left| X \right|du\) 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 4-potential becomes a gradient (“pure gauge”), 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 Weyl’s 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 Mie’s ideas of an electromagnetic world view and Hilbert’s approach to unification, in the first edition of his book, Weyl remained reserved:

“Again 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 mass-points based on Newton’s Principia did triumph. Yet, also today, the circumstances are such that our trees do not grow into the sky.”⁷⁸ ([396], p. 170; preface dated “Easter 1918”)

However, a little later, in his paper accepted on 8 June 1918, Weyl boldly claimed:

“I am bold enough to believe that the whole of physical phenomena may be derived from one single universal world-law of greatest mathematical simplicity.”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 Weyl’s theory can be derived. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2.1.5⁸⁰. As the Lagrangian \({\mathcal L} = \sqrt { - g} L\) must have gauge-weight w=0, we are looking for a scalar L of gauge-weight -2. Weitzenböck⁸¹ 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 Maxwell’s 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 result⁸²; it is given by Schouten ([310], p. 142) and follows from Equation (51): If the metric field g and the 4-potential Q_i≡A_i are varied independently, from each of the curvature-dependent scalar invariants we do get contributions to Maxwell’s equations.

Perhaps Bach (alias Förster) was also dissatisfied with Weyl’s calculations: He went through the entire mathematics of Weyl’s 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 4-potential [4].

In spring 1918, the first edition of Weyl’s famous book on differential geometry, special and general relativity Raum-Zeit-Materie appeared, based on his course in Zürich 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

“As 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 Maxwell’s equations in first order. In the most general case, the equations will be of 4th order, though.”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. 663–664). 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 — with a tone varying between praise and criticism. His first response of 6 April 1918 on a postcard was enthusiastic:

“Your 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.”⁸⁴ ([321], Volume 8B, Document 498, 710)

Einstein’s “objection” is formulated in his “Addendum” (“Nachtrag”) to Weyl’s 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 Weyl’s 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 pre-history 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

“Regrettably, 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.”⁸⁵ ([395], Addendum, p. 478)

Weyl answered Einstein’s comment to his paper in a “reply of the author” affixed to it. He doubted that it had been shown that a clock, if violently moved around, measures proper time ∫ ds. Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:

“The 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 Einstein’s theory]; the task remains, in my theory as well as in Einstein’s, to derive this fact by a dynamics carried through explicitly.”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 Dällenbach, he wrote (after 15 June 1918):

“[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 pre-history, 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 well-defined chemical elements would not exist. […] 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.”⁸⁷ ([321], Volume 8B, Document 565, 803)

On a postcard to Weyl on 8 April 1918, Einstein reaffirmed his admiration for Weyl’s theory, but remained firm in denying its applicability to nature. Weyl had given an argument for dimension 4 of space-time that Einstein liked: As the Lagrangian for the electromagnetic field F_ikF^ik is of gauge-weight -2 and \(\sqrt { - g}\) has gauge-weight 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: “Apart from the [lacking] agreement with reality it is in any case a grandiose intellectual performance”⁸⁸ ([321], Vol. 8B, Doc. 499, 711). Weyl did not give in:

“Your rejection of the theory for me is weighty; […] 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 ‘in the near’, 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.”⁸⁹ ([321], Volume 8B, Document 544, 767)

After Weyl’s next paper on “pure infinitesimal geometry” had been submitted, Einstein put forward further arguments against Weyl’s theory. The first was that Weyl’s 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 Weyl’s theory:

“[Weyl’s] 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) world-points exists. Then, Weyl’s 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 […].”⁹⁰ ([99], p. 133)

Einstein’s remark concerning “affine geometry” 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 Einstein’s viewpoint, in Weyl’s theory the line element ds is no longer a measurable quantity — the electromagnetical 4-potential never had been one. Writing from his vacations on 18 September 1918, Weyl presented a new argument in order to circumvent Einstein’s objections. The quadratic form Rg_ikdxⁱdx^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

“[…] by the prefixing of this factor, so to speak, the absolute norming of the unit of length is accomplished after all”⁹¹ ([321], Volume 8B, Document 619, 877–879)

Einstein was unimpressed:

“But the expression Rg_ikdxⁱdx^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.”⁹²

Einstein’s 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 Weyl’s theory, the connection between R and the matter tensor again might become less direct. Einstein added:

“Of 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 λ-terms undeniably is a result of resignation.[…] In the end, things must arrange themselves such that action-densities need not be glued together additively.”⁹³ ([321], Volume 8B, Document 626, 893–894)

The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.

Sommerfeld seems to have been convinced by Weyl’s theory, as his letter to Weyl on 3 June 1918 shows:

“What 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.”⁹⁴ [327]

In the section on Weyl’s theory in his article for the Encyclopedia of Mathematical Sciences, Pauli described the basic elements of the geometry, the loss of the line-element 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 Eddington’s and Einstein’s 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 time-reflection symmetry (see [246], pp. 774–775; [244]). For Einstein, this criticism was not only directed against Weyl’s theory

“but also against every continuum-theory, 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?” ([103], p. 43)

In a letter to Besso on 26 July 1920, Einstein repeated an argument against Weyl’s theory which had been removed by Weyl — if only by a trick to be described below; Einstein thus said:

“One 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.”⁹⁵ ([99], p. 153)

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 Weyl’s theory in his monograph “The mathematical theory of relativity”

“from the wrong end — 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 Einstein’s work.” ([59], p. 198)

Of course, “wrong end” meant that Eddington took Weyl’s theory such

“that his non-Riemannian geometry is not to be applied to actual space-time; it refers to a graphical representation of that relation-structure which is the basis of all physics, and both electromagnetic and metrical variables appear in it as interrelated.” ([59], p. 197)

Again, Eddington liked Weyl’s natural gauge encountered in Section 4.1.5, which made the curvature scalar a constant, i.e., K=4λ; it became a consequence of Eddington’s own natural gauge in his affine theory, K_ij=λg_ij (cf. Section 4.3). For Eddington, Weyl’s theory of gauge-transformation was a hybrid:

“He admits the physical comparison of length by optical methods […]; but he does not recognise physical comparison of length by material transfer, and consequently he takes λ to be a function fixed by arbitrary convention and not necessarily a constant.“ ([59], pp. 220–221)

In the depth of his heart Weyl must have kept a fondness for his idea of “gauging” 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 (“matter wave”) in Schrödinger’s and Dirac’s 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:

“While it was not difficult to adapt also Maxwell’s 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.[…] My book describes an attempt to attain this goal by a new principle which I called gauge invariance. (Eichinvarianz). This attempt has failed.” ([410], p. V)

Weyl himself continued to develop the dynamics of his theory. In the third edition of his Space-Time-Matter [398], at the Naturforscherversammlung in Bad Nauheim in 1920 [399], and in his paper on “the foundations of the extended relativity theory” in 1921 [402], he returned to his new idea of gauging length by setting R=λ= const. (cf. Section 4.1.3); he interpreted λ to be the “radius of curvature” of the world. In 1919, Weyl’s Lagrangian originally was \({\mathcal L} = \tfrac{1}{2}\sqrt g {K^2} + \beta {F_{ik}}{F^{ik}}\) together with the constraint K=2λ, with constant λ ([398], p. 253). As an equivalent Lagrangian Weyl gave, up to a divergence⁹⁶ with the 4-potential φ 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 ∊ a factor in Weyl’s connection (100), In both presentations, he considered as an advantage of his theory:

“Moreover, this theory leads to the cosmological term in a uniform and forceful manner, [a term] which in Einstein’s theory was introduced ad hoc”⁹⁷ ([402], p. 474)

Reichenbächer seemingly was unhappy about Weyl’s taking the curvature scalar to be a constant before the variation; in the discussion after Weyl’s talk in 1920, he inquired whether one could not introduce Weyl’s “natural gauge” after the variation of the Lagrangian such that the field equations would show their gauge invariance first ([399], p. 651). Eddington criticised Weyl’s choice of a Lagrangian as speculative:

“At the most we can only regard the assumed form of action […] as a step towards some more natural combination of electromagnetic and gravitational variables.” ([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 “matter whose characteristical traits are charge, mass, and motion generates the field”, a question which was considered as unanswered by Weyl, he also mentioned a publication of Reichenbächer [272]. For Weyl, knowledge of the charge and mass of each particle, and of the extension of their “world-channels” were insufficient to determine the field uniquely. Weyl’s hint at a solution remains dark; nevertheless, for him it meant

“to reconciliate Reichenbächer’s idea: matter causes a ‘deformation’ of the metrical field and Einstein’s idea: inertia and gravitation are one.” ([400], p. 561, footnote)

Although Einstein could not accept Weyl’s theory as a physical theory, he cherished “its courageous mathematical construction” and thought intensively about its conceptual foundation: This becomes clear from his paper “On a complement at hand of the bases of general relativity” 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 “distance”, i.e., without using rulers and clocks. He then embarked on conformal invariants and tensors of gauge-weight 0, and gave the one formed from the square of Weyl’s conformal curvature tensor (59), i.e. \({C^i}_{jkl}C_i^{jkl}\). His colleague in Vienna, Wirtinger⁹⁸, had helped him in this⁹⁹. Einstein’s conclusion was that, by writing down a metric with gauge-weight 0, it was possible to form a theory depending only on the quotient of the metrical components. If J has gauge-weight -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.

Einstein’s rejection of the physical value of Weyl’s theory was seconded by Dienes¹⁰⁰, if only with a not very helpful argument. He demanded that the connection remain metric-compatible from which, trivially, Weyl’s gauge-vector must vanish. Dienes applied the same argument to Eddington’s generalisation of Weyl’s theory [51]. Other mathematicians took Weyl’s theory at its face value and drew consequences; thus M. Juvet calculated Frenet’s formulas for an “n-èdre” in Weyl’s 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. John’s 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 K² and concluded:

“The part independent of the ‘electrical’ vector φ_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.” ([237], p. 623)

The third main idea that emerged was Eddington’s 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 Γ in order to avoid an “infinitely crinkled” world [58]. His motivation went beyond the unification of gravitation and electromagnetism:

“In 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 non-Maxwellian 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 solution” ([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 “geometry of the world-structure” from Riemannian geometry as “the natural geometry of the world”. He starts by calculating both the curvature and Ricci tensors from the symmetric connection according to Equation (39). The Ricci tensor K^ij(Γ):=*G^ij is asymmetric¹⁰⁷, with R_kl(Γ) being the symmetric and F_kl(Γ) the antisymmetric part. According to Equation (31) F_kl derives from a “vector potential”, 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 Maxwell’s 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:

“The divergence of j^k will vanish identically if j^k is itself the divergence of any antisymmetrical contravariant tensor.” ([64], p. 223; cf. also [58], p. 113)

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 “introducing a universal constant λ, for convenience, in order to remain free to use the centimetre instead of the natural unit of length”. This is called “Einstein’s gauge” by Eddington; he is delighted that

“Our gauging-equation 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?” ([58], p. 114)

While this remark certainly is true, there is no guarantee in Eddington’s 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≠0.

In connection with cosmological considerations, Eddington cherished the λ-term in Equation (118):

“I would as soon think of reverting to Newtonian theory as of dropping the cosmic constant.” ([63], p. 35)

Eddington’s main goal in this paper was to include matter as an inherent geometrical structure:

“What we have sought is not the geometry of actual space and time, but the geometry of the world-structure which is the common basis of space and time and things.” ([58], p. 121)

By “things” he meant 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 “a region of abnormal world-curvature”, i.e., of abnormally large curvature.

Lorentz did not like the large number of variables in Eddington’s theory; there were 4 components of the electromagnetic potential, 10 components of the metric and 40 components of the connection:

“It 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.” ([211], p. 382)

Eddington’s publication early in 1921, generalising Einstein’s and Weyl’s 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. 453–475; here p. 466)), but one and a half years later he became attracted by Eddington’s idea. To Bohr, Einstein wrote from Singapore on 11 January 1923:

“I believe I have finally understood the connection between electricity and gravitation. Eddington has come closer to the truth than Weyl.” ([139], p. 274)

He now tried to make Eddington’s theory work as a physical theory; Eddington had not given field equations:

“I must absolutely publish since Eddington’s idea must be thought through to the end.” (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: “[…] Over it lingers the marble smile of inexorable nature, which has bestowed on us more longing than brains.”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 “Haruna Maru” of the Japanese Nippon Yushen Kaisha line¹⁰⁹ [77].

“In past years, the wish to understand the gravitational and electromagnetic field as one in essence has dominated the endeavours of theoreticians. […] 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 […] Eddington has done this.”¹¹⁰ ([77], p. 32)

Like Eddington, Einstein used a symmetric connection and wrote down the equation¹¹¹ where g_kl=g_(kl) and φ_kl=φ_[kl], and λ is a “large number”. By this, the metric was defined as the symmetric part of the Ricci tensor. Due to one half of Maxwell’s equations is satisfied if φ_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 4-potential under gauge transformations.

The field equations are obtained from the Lagrangian by variation with regard to the connection \({\Gamma _{kj}}^l\) and are (Einstein worked in space-time) 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_klj^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),

For non-vanishing 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:

“But the extraordinary smallness of 1/λ² implies that finite φ_kl are possible only for tiny, almost vanishing current density. Except for singular positions, the current density is practically vanishing.”¹¹²

Einstein went on to show that Maxwell’s 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 “electrical elementary elements”, i.e., nonsingular electrons, are possible in this theory; they might be. He found it remarkable “[…] that, according to this theory, positive and negative electricity cannot differ just in sign”¹¹³ ([77], p. 38). His final conclusion was:

“that EDDINGTON’S 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.”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 ŝ^kl and f̂^kl. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skew-symmetric 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:

“Therefore, the theory may not account for the difference in mass of positive and negative electrons.”¹¹⁵ ([74], p. 77)

In the third paper [76], apart from changing notations¹¹⁶, Einstein set λ=1. He also dropped the assumption (119) and replaced it by allowing his Lagrangian (Hamiltonian) Ĥ 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 δγ_kl and δφ_kl were expressed by δΓ_ik^l via (127) and inserted into δĤ=0. The ensuing equation could be solved for Γ_ik^l and led to Equation (123). In the third step, the Lagrangian Ĥ* is taken as a functional of the variables introduced in the first step, i.e., of ĝ^kl, f̂^kl such that in place of Equation (128) the relations hold. Einstein then took “the expression most natural vis-a-vis our present knowledge”, 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_ki_l on the side of the energy-momentum tensor of the electromagnetic field and Equation (125) with a changed l.h.s. now reading -βf_ik.

After a field rescaling, he then took a third expression to become his Lagrangian where α and β are arbitrary constants, and κ is the gravitational constant. i_l is defined to be proportional to the electromagnetic 4-potential f_k, i.e., 1/βi_l=-f_k, and f_ij corresponds to φ_ij, \(\sqrt { - g} {f^{ij}}\) to f̂^ij. After the field equations had been obtained by this longwinded procedure, it became obvious that they could also be derived from H̄=H. taken as an “effective” Lagrangian varied with respect to g_ik and f_ik. In Einstein’s words: “R is the Riemannian curvature scalar formed from g_ij”¹¹⁷. In the third paper as well, Einstein’s desire to create a unified field theory satisfying all his criteria still was not fulfilled: His equations, again, did not give a singularity-free electron. In a paper on Hilbert’s vision of a unified science, Sauer and Majer recently have found out from lectures of Hilbert given in Hamburg and Zürich in 1923, that Hilbert considered Einstein’s work in affine theory a return to his own results of 1915 by “[…] a colossal detour via Levi-Civita, Weyl, Schouten, Eddington […]” [215]. It seems that, in this evaluation, Hilbert was influenced by Einstein’s proportionality between the 4-potential and the electrical current which Hilbert had assumed as early as in 1915 [161]¹¹⁸.

While, in the meantime, mathematicians had taken over the conceptual development of affine theory, some other physicists, including the perpetual pièce de resistance Pauli, kept a negative attitude:

“[…] 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. […] 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 mathematician’s discovery of the possibility to found a geometry on an affine connection without a metric as meaningless for physics, in the first place.”¹¹⁹ (Pauli to Eddington on 20 September 1923; [251], pp. 115–119)

Likewise, Eddington himself did not appreciate much Einstein’s followership. In Note 14, § 100 appended to the second edition of his book, he laid out Einstein’s theory but not without first having warned the reader:

“The theory is intensely formal as indeed all such action-theories 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.” ([64], pp. 257–261)

In fact, when Eddington’s 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:

“For me, the final result of this consideration regrettably consists in the impression that the deepening of the geometrical foundations by Weyl-Eddington is unable to bring progress for our physical understanding; hopefully, future developments will show that this pessimistic opinion has been unjustified.”¹²⁰ ([60], p. 371)

An echo of this can be found in Einstein’s letter to Besso of 5 June 1925:

“I am firmly convinced that the entire chain of thought Weyl-Eddington-Schouten does not lead to something useful in physics, and I now have found another, physically better founded approach. To me, the quantum-problem seems to require something like a special scalar, for the introduction of which I have found a plausible way.”¹²¹ ([99], p. 204)

This remark shows that Einstein must have taken some notice of Schouten’s work in affine geometry. What the “special scalar” was, remains an open question.

Einstein spent much time in thinking about the “quantum problem”, as he confessed to Born:

“I 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 quantum-structure through an overdetermination by differential equations.”¹²² ([103], pp. 48–49)

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 “overdetermination” of systems of differential equations could solve the “quantum problem”.

“According 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 Bohr’s 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.”¹²³ ([75], pp. 360–361)

He then ventured the hope that a system of overdetermined differential equations is able to determine

“also 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. […] If it is possible at all to solve the quantum problem by differential equations, we may hope to reach the goal in this direction.”

We note here Einstein’s 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:

“The 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 Einstein-Maxwell equations].”¹²⁴

This attitude can also be found in a letter to M. Besso from 5 January 1924:

“The 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 un-ambiguity of the initial conditions ought to be understood without leaving field theory. […] The equations of motion of material points (electrons) will be given up totally; their motion ought to be co-determined by the field laws.”¹²⁵ ([99], p. 197)

In his answer, Besso asked for more information concerning the quantum aspect of the concept of “overdetermination”, because:

“On 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.”¹²⁶ ([99], p. 199)

Einstein became interested in Kaluza’s theory again due to O. Klein’s paper concerning a relation between “quantum theory and relativity in five dimensions” (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: “Subject Kaluza, Schroedinger, general relativity”, and, again on 3 September 1926: “Klein’s paper is beautiful and impressive, but I find Kaluza’s principle too unnatural.” However, less than half a year later he had completely reversed his opinion:

“It appears that the union of gravitation and Maxwell’s theory is achieved in a completely satisfactory way by the five-dimensional theory (Kaluza-Klein-Fock).” (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 gauge-group, wrote down the geodesic equation, and derived exactly the Einstein.Maxwell equations — 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:

“Mr. 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.”¹⁵⁸

He then referred to the papers of Klein [185, 186] and to “Fochs Arbeit” which is a paper by Fock¹⁵⁹ 1926 [130], submitted three months later than Klein’s paper. That Klein had published another important clarifying note in Nature, in which he closed the fifth dimension, seems to have escaped Einstein¹⁶⁰ [184]. Unlike in his paper with Grommer, but as in Klein’s, Einstein, in his notes, applied the “sharpened cylinder condition”, i.e., dropped the scalar field. Thus, the three of them had no chance to find out that Kaluza had made a mistake: For g₅₅≠const., even in first approximation the new field will appear in the four-dimensional Einstein.Maxwell equations ([145], p. 5).

Mandel¹⁶¹ of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. Klein’s results [216]. In a footnoote, Mandel stated that he had learned of Kaluza’s (whom he spelled “Kalusa”) paper only through Klein’s article. He started by embedding space-time as a hypersurface x⁵=const. into M₅, and derived the field equations in space-time by assuming that the five-dimensional curvature tensor vanishes; by this procedure he obtained also a matter-energy tensor “closely linked to the second fundamental form of this hypersurface”. From the geodesics in M₅ 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

“that the experimental discovery of the second term appears difficult, yet perhaps not entirely impossible.” ([216], p. 145)

As to Fock’s paper, it is remarkable because it contains, in nuce, the coupling of the Schrödinger wave function ψ and the electromagnetic potential by the gauge transformation ψ=ψ0 e^2πip/h, where h is Planck’s constant and p “a new parameter with the unit of the quantum of action” [130]. In Fock’s words:

“The 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 4-potential.”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 M₅. Neither Mandel nor Fock used the “sharpened cylinder condition” (110).

A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation and by neglecting the gravitational field, he arrived at the four-dimensional Schrödinger equation after insertion of the quantum mechanical differential operators \(- \tfrac{{ih}}{{2\pi }}\tfrac{\partial }{{\partial {x^i}}}\). It was Klein’s papers and the magical lure of a link between classical field theory and quantum theory that raised interest in Kaluza’s idea — seven years after Kaluza had sent his manuscript to Einstein. Klein acknowledged Mandel’s 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 Einstein’s papers [189]. Likewise, Einstein did not comment on Klein’s new idea of “dimensional reduction” as it is now called and which justifies Klein’s name in the “Kaluza-Klein” theories of our time. By this, the reduction of five-dimensional equations (as e.g., the five-dimensional wave equation) to four-dimensional equations by Fourier decomposition with respect to the new 5th spacelike coordinate x⁵, 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 “compactification”. By adding to this picture the idea of de Broglie waves, Klein brought in Planck’s constant and determined the linear scale of x⁵ to be unmeasurably small (∼10^-30). From this, the possibility of “forgetting” the fifth dimension arose which up to now has not been observed.

In his papers, Einstein took over Klein’s condition g₅₅=1, which removed the additional scalar field admitted by the theory. It was Reichenbächer who apparently first tried to perform the projection into space-time of the most general five-dimensional metric, and without using the cylinder condition (109):

“Now, a rather laborious calculation of the five-dimensional curvature quantities in terms of a four-dimensional submanifold contained in it has shown to me also in the general case (g₅₅≠const., dependence of the components of the f u n d a m e n t a l [tensor] of x⁵ 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 sⁱ do not vanish. […] 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 charge¹⁶³.” ¹⁶⁴ ([276], p. 426) Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail [102].

Klein’s lure lasted for some years. In 1930, N. R. Sen claimed to have investigated the “Kepler-problem for the five-dimensional wave equation of Klein”. What he did was to calculate the energy levels of the hydrogen atom (as a one particle-system) with the general relativistic wave equation in space-time (148) with a^ik=γ^ik, where \({\gamma _{ik}} = {g_{ik}} + {\alpha ^2}{\gamma _{55}}\) is the metric on space-time following from the 5-metric \({\gamma _{\alpha \beta }}\) by dx⁵=0. For g_ik he took the Reissner-Nordström solution and did not obtain a discrete spectrum [324]. He continued his approach by trying to solve Schrödinger’s wave equation [325]

In a paper with his assistant Mayer, Einstein now presented Kaluza’s approach in the form of an implicit projective four-dimensional theory, although he did not mention the word “projective” [107]:

“Psychologically, the theory presented here connects to Kaluza’s well-known theory; however, it avoids extending the physical continuum to one of five dimensions.”¹⁶⁵

In the eyes of Einstein, by avoiding the artificial cylinder condition (109), the new method removed a serious objection to Kaluza’s theory.

Another motivation is also put forward: The linearity of Maxwell’s equations “may not correspond to reality”; thus, for strong electromagnetic fields, Einstein expected deviations from Maxwell’s equations. After a listing of all the shortcomings of Kaluza’s theory, the new approach is introduced: At every event a five-dimensional vector space V₅ is affixed to space-time V₄, and “mixed” tensors \({\gamma _\imath }^{\;k}\) are defined linking the tangent space of space-time V₄ with a V₅ such that where g_lm is the metric tensor of V₄, and \({g_{\iota \kappa }}\) a non-singular, symmetric tensor on V₅ with ι, κ=1, … , 5, and k, l=1, … , 4¹⁶⁶. Indices are raised and lowered with the metrics of V₅ or V₄, respectively. There exists a “preferred direction of V₅” defined by \({\gamma _\iota }^{\;k}{A^\iota } = 0\), and which is the normal to a “preferred plane” \({\gamma _\iota }^{\;k}{\omega _k} = 0\)¹⁶⁷. A consequence then is A covariant derivative for five-vectors in V₄ is defined with a “three-index-symbol” \({\Gamma ^\iota }_{\pi l}\) with two indices in V₅, and one in V₄ standing in for the connection coefficients: The covariant derivative of 4-vectors is defined as usual, where {_jlⁱ} is calculated from the metric of V₄ 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 V₅ and those referring to V₄, is formed correspondingly. In this context, Einstein and Mayer mention an extension of absolute differential calculus by “WAERDEN and BARTOLOTTI” without giving any reference to their respective papers. They may have had in mind van der Waerden’s [368] and Bortolotti¹⁶⁸’s [24] papers. The autoparallels of V₅ lead to the exact equations of motion of a charged particle, not the geodesics of V₄.

Einstein and Mayer made three basic assumptions: where \({A^\iota }\) is the preferred direction and F_kl an arbitrary 2-form, 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 V₄ into V₅.

Einstein and Mayer introduced what they called “Fünferkrümmung” (5-curvature) via the three-index symbol given above by It is related to the Riemannian curvature \({R^r}_{mlk}\) of V₄ by and From (154), by transvection with \({\gamma ^{\tau k}}\), the 5-curvature 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: 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 Einstein-Maxwell vacuum field equations. Thus, by another formalism, Einstein and Mayer rederived what Klein had obtained in his first paper on Kaluza’s theory [185].

The authors’ conclusion is:

“From 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.”¹⁷⁰ ([107], p. 19)

After this paper Einstein wrote to Ehrenfest in a letter of 17 September 1931 that this theory “in my opinion definitively solves the problem in the macroscopic domain” ([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 5-vector approach. In talking about the failed endeavours to reconcile classical field theory and quantum theory (“a cemetery of buried hopes”) he is reported to have said:

“Since 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. […] According to my and Mayer’s opinion, the fifth dimension will not show up. […] according to which relationships between a hypothetical five-dimensional space and the four-dimensional can be obtained. In this way, we succeeded to recognise the gravitational and electromagnetic fields as a logical unity.”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 Kaluza’s unitary field theory:

“The only result of our investigation is the unification of gravitation and electricity, whereby the equations for the latter are just Maxwell’s equations for empty space. Hence, no physical progress is made, [if at all] at most only in the sense that one can see that Maxwell’s equations are not just first approximations but appear on as good a rational foundation as the gravitational equations of empty space. Electrical and mass-density are non-existent; 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 5-vectors \({a^\sigma }\) in fourdimensional space, which are bound to space by a linear mechanism. Let a^s be the 4-vector 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 5-curvature from which spring the field equations.”¹⁷² ([99], pp. 274–25)

In his report for the Macy-Foundation, which appeared in Science on the very same day in October 1931, Einstein had to be more optimistic:

“This 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.” ([94], p. 439)

Unfortunately, as in the case of his previous papers on Kaluza’s 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 Einstein’s and Mayer’s publication, with his student Hoffmann¹⁷³, he had suggested an application to physics equivalent to the Kaluza-Klein theory [381, 163]. However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:

“But 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 Kaluza-Klein theory by means of the cylinder condition […].”¹⁷⁴ ([249], p. 307)

By using the idea that an affine (n+1)-space can be represented by a projective n-space [413], Veblen and Hoffmann avoided the five dimensions of Kaluza: There is a one-to-one correspondence between the points of space-time and a certain congruence of curves in a five-dimensional space for which the fifth coordinate is the curves’ parameter, while the coordinates of space-time are fixed. The five-dimensional space is just a mathematical device to represent the events (points) of space-time by these curves. Geometrically, the theory of Veblen and Hoffmann is more transparent and also more general than Einstein and Mayer’s: It can house the additional scalar field inherent in Kaluza’s 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 Dirac’s equations (without gravitation), although some technical difficulties remained. Nevertheless, Hoffman remained optimistic:

“There 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.” ([163], p. 89)

In his book, Veblen emphasised

“[…] that our theory starts from a physical and geometrical point of view totally different from KALUZA’s. In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly four-dimensional.”¹⁷⁵ [379]

In a second note, Einstein and Mayer extended the 5-vector-formalism to include Maxwell’s equations with a non-vanishing 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 5-curvature 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 ∼V^V_prqF^rq.

In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:

“We 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 ‘compatibility’ in this paper and in previous papers.”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 α= 1, …, 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]¹⁷⁷, which, according to Pauli, combine

“all advantages of the formulations of Kaluza-Klein and Einstein-Mayer while avoiding all their disadvantages.” ([249], p. 307)

Both the Einstein-Mayer theory and Veblen and Hoffmann’s approach turned out to be subcases of the more general scheme of Schouten and van Dantzig intending

“to give a unification of general relativity not only with Maxwell’s electromagnetic theory but also with Schrödinger’s and Dirac’s theory of material waves.” ([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 kind¹⁷⁸. 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. Solomon¹⁷⁹, generalised Klein, and Einstein and Mayer by allowing for an arbitrary signature in an investigation concerning “the form that take Dirac’s equations in the unitary theory of Einstein and Mayer”¹⁸⁰ [253]. In a note added after proofreading, the authors showed that they had noted Schouten and Dantzig’s papers [316, 317]. The authors pointed out that

“[…] even in the absence of gravitation we must pay attention to a difference between Dirac’s equation in the theory of Einstein and Mayer, and Dirac’s equation as it is written out, usually.”¹⁸¹ ([253], p. 458)

The second order wave equation iterated from their form of Dirac’s equation, besides the spin term contained a curvature term -1/4R, with the numerical factor different from Veblen’s and Hoffmann’s. In a sequel to this publication, Pauli and Solomon corrected an error:

“We examine from a general point of view the theory of spinors in a five-dimensional space. Then we discuss the form of the energy-momentum tensor and of the current vector in the theory of Einstein-Mayer.[…] Unfortunately, it turned out that the considerations of §in the first part are marred by a calculational error… This has made it necessary to introduce a new expression for the energy-momentum tensor and […] likewise for the current vector […].”¹⁸² ([254], p. 582)

In the California Institute of Technology, Einstein’s and Mayer’s new mathematical technique found an attentive reader as well; A. D. Michal and his co-author generalised the Einstein-Mayer 5-vector-formalism:

“The geometry considered by Einstein and Mayer in their ‘Unified field theory’ leads to the consideration of an n-dimensional Riemannian space V_n with a metric tensor g_ij, to each point of which is associated an m-dimensional linear vector space V_m, (m>n), for which vector spaces a general linear connection is defined. For the general case (m−n≠1) we find that the calculation of the m−n ‘exceptional directions’ 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 Einstein-Mayer case.“ [228]

Michal had come from Cartan and Schouten’s 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 Einstein’s and Mayer’s spherically symmetric exact solution [282] (cf. Section 6.4.3).

As concerns the geometry of “Fernparallelism”, it is a special case of a space with Euclidean connection introduced by Cartan in 1922/23 [31, 30, 32]. Pais let Einstein “invent” and “discover” distant parallelism, and he states that Einstein “did not know that Cartan was already aware of this geometry” ([241], pp. 344–345). 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

“appeared at the moment at which you gave your talks at the Collège de France. I even remember having tried, at Hadamard’s 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.”¹⁸⁴ (letter of Cartan to Einstein on 8 May 1929; cf. [50], p. 4)

This remark refers to Einstein’s 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, Einstein’s motivation came from the wish to generalise Riemannian geometry such that the electromagnetic field could be geometrized:

“Therefore, 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.”¹⁸⁵ ([84], p. 217)

Anyhow, in his response (Einstein to Cartan on 10 May 1929, [50], p. 10), Einstein admitted Cartan’s priority and referred also to Eisenhart’s book of 1927 and to Weitzenböck’s paper [393]. He excused himself byWeitzenböck’s likewise omittance of Cartan’s papers among his 14 references. In his answer, Cartan found it curious that Weitzenböck was silent because

“[…] he indicates in his bibliography a note by Bortolotti in which he several times refers to my papers.”¹⁸⁶ (Cartan to Einstein on 15 May 1929; [50], p. 14)

The embarrassing situation was solved by Einstein’s suggestion that he had submitted a comprehensive paper on the subject to Zeitschrift für 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:

“I am now writing up the work for the Mathematische Annalen and should like to add yours […]. The publication should appear in the Mathematische Annalen because, at present, only the mathematical implications are explored and not their applications to physics.”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 Weitzenböck who had introduced the concept of distant parallelism, as valuable as his results were after the concept had become known. Also, he took Einstein’s treatment of Fernparallelism as a special case of his more general considerations. Interestingly, he permitted himself to interpet the physical meaning of geometrical structures¹⁸⁸:

“Let 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.”¹⁸⁹ ([35], p. 703)

Einstein explained:

“In particular, I learned from Mr. Weitzenböck and Mr. Cartan that the treatment of continua of the species which is of import here, is not really new.[…] 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.”¹⁹⁰ ([89], p. 685)

For Einstein, the attraction of his theory consisted

“For 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 Newton-Poisson theory of gravitation and to Maxwell’s 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.”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.

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:

“For 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.”

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:

“Now, 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.” [2]

Whether Einstein used this as a metaphorical language or, whether he at this time still believed that the system “nucleus and electrons” is dominated by the electromagnetic force, remains open.

The paper announced by Nature is Einstein’s “Zur einheitlichen Feldtheorie”, 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, “written as an explanation of his thesis for readers who do not possess an expert knowledge of mathematics”. 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 two-dimensional 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 — without using that name. His closing sentences are¹⁹²:

“Which 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.” ([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 “Einstein’s and other Unitary Field Theories”. 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:

“Of 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 ‘graph’ and so outside the ken of the ordinary physicist.” ([258], p. 879)

The use of the concept “graph” had its origin in Eddington’s interpretation of his and other peoples’ unified field theories to be only graphs of the world; the true geometry remained the Riemannian geometry underlying Einstein’s general relativity.

Even the French-Belgian writer and poet Maurice Maeterlinck had heard of Einstein’s latest achievement in the area of unified field theory. In his poetic presentation of the universe “La grande féerie”¹⁹³ we find his remark:

“Einstein, 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.”¹⁹⁴ ([214], p. 68)

In his first note [84], dynamics was absent; Einstein made geometrical considerations his main theme: Introduction of a local “n-bein-field” \(h_{\hat \imath}^k\) at every point of a differentiable manifold and the related object \({h_{k\hat \imath}}\) defined as the collection of the “normed subdeterminants of the \(h_{\hat \imath}^k\)”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 assumed¹⁹⁷.

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 bein-vector vanishes, by use of Equation (161). Also, the metric is covariantly constant Neither fact is mentioned in Einstein’s note. Also, no reference is given to Eisenhart’s paper of 1925 [118], in which the connection (161) had been given (Equation (3.5) on p. 248 of [118]), as noted above, its metric-compatibility shown, and the vanishing of the curvature tensor concluded.

At the end of the note Einstein compared his new approach to Weyl’s and Riemann’s:

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 note¹⁹⁹. He introduced \({\phi _k}: = {\Lambda _{kl}}^l\), and took the case φ=0 to describe a “purely gravitational field”. 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 “electrical potential” or “electrical field”. He then showed that in a first-order approximation starting from \({h_{i\hat \jmath}} = {\delta _{i\hat \jmath}} + {j_{i\hat \jmath}} + \ldots\), both the Einstein vacuum field equations and Maxwell’s 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

“The separation of the gravitational and the electromagnetic field appears artificial in this theory. […] Furthermore, it is remarkable that, according to this theory, the electromagnetic field does not enter the field equations quadratically.”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 tensor²⁰¹. He thus arrived at the identity given by Equation (29), i.e., (Einstein’s 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 identity²⁰²: 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 (Einstein’s equation (5), p. 5): As we have seen in Section 2, if he had read it, Einstein could have taken these identities from Schouten’s book of 1924 [300].

By replacing Â^jk by V̂^k̂l̂j 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 ∊=0, cf. below) and Maxwell’s 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:

“The 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_i^k̂ By ‘independent’ we understand that none of these equations can be derived from the remaining ones, even if there exist 8 identical (differential) relations among them.”²⁰³ ([88], p. 8)

He still was not entirely sure that the theory was physically acceptable:

“A 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.”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 Müntz²⁰⁵ 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 compatibility-problem 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: 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 ∊₁, ∊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 Maxwell’s 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 torsion²⁰⁶.

In a Festschrift for his former teacher and colleague in Zürich, A. Stodola, Einstein summed up what he had reached²⁰⁷. 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

“[…] that coincide in first approximation with the known laws for the gravitational and electromagnetic field […]”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 Müntz, he had shown that for an uncharged mass point the Schwarzschild solution again obtained [87].

Einstein’s next publication was the note preceding Cartan’s 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 are²⁰⁹. where S^ikl=g^img^knS_mn^l. There exist 4 identities among the 16+6 field equations As Cartan remarked, Equation (172) expresses conservation of torsion under parallel transport:

“In 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.”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:

“But 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. […] In a new presentation in the Sitzungsberichten, I used this identity while taking the liberty of pointing to you as its source.” ²¹¹ (letter of Einstein to Cartan from 18 December 1929, Document X of [50], p. 72)

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, Zaycoff²¹², tried to make the best out of them:

“Recently, A. Einstein ([89]), following investigations by E. Cartan ([35]), has considerably modified his teleparallelism theory such that former shortcomings (connected only to the physical identifications) vanish by themselves.“²¹³ ([433], p. 410)

In November 1929, Einstein gave two lectures at the Institute Henri Poincaré in Paris which had been opened one year earlier in order to strengthen theoretical physics in France ([14], pp. 263–272). 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 Cartan’s name is mentioned in the text:

“It 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.”²¹⁴ ([92], p. 4)

Note that Einstein does not say that it was Cartan who first “envisaged” these spaces before. Later in the paper, he comes closer to the point:

“This type of space had been envisaged before me by mathematicians, notably by WEITZENBÖCK, EISENHART et CARTAN […].”²¹⁵ [92]

Again, he held back in his support of Cartan’s priority claim.

A. Proca, who had attended Einstein’s lectures, gave an exposition of them in a journal of his native Romania. He was quite enthusiastic about Einstein’s new theory:

“A great step forward has been made in the pursuit of this total synthesis of phenomena which is, right or wrong, the ideal of physicists. […] 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.”²¹⁶ [260, 261]

Then Einstein presented the same field equations as in his paper in Annalen der Mathematik, which he demanded to be 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.

“Therefore 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.”²¹⁷ ([90], p. 21)

He then gave a proof of the compatibility of his field equations:

“The 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. §3, [16]).”²¹⁸

The reader had to make out for himself what Cartan’s contribution really was.

In order to test the field equations by exhibiting an exact solution, a simple case would be to take a spherically symmetric, asymptotically (Minkowskian) 4-bein. This is what Einstein and Mayer did, except with the additional assumption of space-reflection symmetry [106]. Then the 4-bein contains three arbitrary functions of one parameter s: where α, \(\hat \imath = 1,2,3\). 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 “ponderomotive mass,” respectively. A further exact solution for uncharged point particles was also derived; it is static and corresponds to “two or more unconnected electrically neutral masses which can stay at rest at arbitrary distances”. 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 singularity-free 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 Einstein’s latest publications, told the readers of Science:

“It 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. […] 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.” ([339], p. 391)

Two days before the paper by Einstein and Mayer became published by the Berlin Academy, Einstein wrote to his friend Solovine:

“My field theory is progressing well. Cartan has already worked with it. I myself work with a mathematician (S. Mayer²¹⁹ from Vienna), a marvelous chap […].“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 Einstein’s program:

“The problem is nearly solved; and to the first approximations he gets laws of gravitation and electro-magnetics. 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 ‘quasistatical motions’, but he also wants to derive elements of matter (electrons and protons) out of the metric structure of space.” ([91], p. 610)

With his “assistant” 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: For the field equations, the following ansatz was made: where R^ik is a collection of terms quadratic in torsion S, and p, q, a₁, a₂, a₃ are constants. They must be determined in such a way that the “divergence-identity” is satisfied. Here, 8 new constants A, B, c_r with r=1, …, 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:

“Two of these are (non-trivial) 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 […].”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 content²²². 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:

“[…] 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 […].” ([50], p. 209)

What Cartan might have felt, after investing the forty odd pages of his calculations printed in Debever’s book, is unknown. However, the correspondence on the subject came to an end in May 1930 with a last letter by Cartan. ²²³.

As concerns parallelism at a distance, Reichenbach was not enthusiastic about Einstein’s new approach:

“[…] it is the aim of Einstein’s 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.”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 Einstein’s 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:

“For 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 Weyl’s theory and its affine generalisation afford considerable enlightenment, may be excused for doubting whether the new theory offers sufficient inducement to make an exchange.” [62]

Weyl was the next unhappy colleague; in connection with the redefinition of his gauge idea he remarked (in April/May 1929):

“[…] my approach is radically different, because I reject distant parallelism and keep to Einstein’s general relativity. […] 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 […] only by this loosening [of the relationship between the local frames] the existing gauge-invariance becomes intelligible. Second, the possibility to rotate the frames independently, in the different events, […] 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.”²²⁷ ([407], pp. 330–332.)

As usual, Pauli was less than enthusiastic; he expressed his discontent in a letter to Hermann Weyl of 26 August 1929:

“First let me emphasize that side of the matter about which I fully agree with you: Your approach for incorporating gravitation into Dirac’s theory of the spinning electron […] I am as adverse with regard to Fernparallelismus as you are […] (And here I must do justice to your work in physics. When you made your theory with g′_ik=λg_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].)”²²⁸ ([251], pp. 518–519)

Another confession of Pauli’s went to Paul Ehrenfest:

“By the way, I now no longer believe in one syllable of teleparallelism; Einstein seems to have been abandoned by the dear Lord.”²²⁹ (Pauli to Ehrenfest 29 September 1929; [251], p. 524)

Pauli’s remark shows the importance of ideology in this field: As long as no empirical basis exists, beliefs, hopes, expectations, and rationally guided guesses abound. Pauli’s letter to Weyl from 1 July 1929 used non-standard language (in terms of science):

“I share completely your skeptical position with regard to Einstein’s 4-bein geometry. During the Easter holidays I have visited Einstein in Berlin and found his opinion on modern quantum theory reactionary.”²³⁰ ([251], p. 506)

while the wealth of empirical data supporting Heisenberg’s and Schrödinger’s quantum theory would have justified the use of a word like “uninformed” or even “not up to date” for the description of Einstein’s position, use of “reactionary” 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:

“I 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 […]. 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 […]. It just remains […] 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.”²³¹ (letter to Einstein of 19 December 1929; [251], 526–527)

Einstein answered on 24 December 1929:

“Your 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. […] 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 — more so because, in the meantime, the proof of the compatibility of the equations could be simplified.”²³² ([251], p. 582)

Before he had written to Einstein, Pauli, with lesser reservations, complained vis-a-vis Jordan:

“Einstein 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 Maxwell’s 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.”²³³ (letter of 30 November 1929, [251], p. 525)

Of course, Pauli’s spells of rudeness are well known; in this particular case they might have been induced by Einstein’s unfounded hopes for eventually replacing the Schrödinger-Heisenberg-Dirac quantum mechanics by one of his unified field theories.

Born’s voice was the lonely approving one (Born to Einstein on 23 September 1929)²³⁴:

“Your 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 involved”²³⁵ ([154], p. 307)

Born, however, was not yet a player in unified field theory, and it turned out that Einstein’s theory of distant parallelism became as involved as the previous ones.

Einstein’s collaborator Lanczos even wrote a review article about distant parallelism with the title “The new field theory of Einstein” [201]. In it, Lanczos cautiously offers some criticism after having made enough bows before Einstein:

“To 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.[…] 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.”²³⁶ ([201], p. 126)

When Pauli reviewed this review, he started with the scathing remark

“It is indeed a courageous deed of the editors to accept an essay on a new field theory of Einstein for the ‘Results in the Exact Sciences’ [literal translation of the journal’s title]. His never-ending 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 ‘definite solution’. Hence, […] one could cry out: ’Einstein’s new field theory is dead. Long live Einstein’s new field theory!’”²³⁷ ([248], p. 186)

For the remainder, Pauli engaged in a discussion with the philosophical background of Lanczos and criticised his support for Mie’s theory of matter of 1913 according to which

“the atomism of electricity and matter, fully separated from the existence of the quantum of action, is to be reduced to the properties of (singularity-free) eigen-solutions of still-to-be-found nonlinear differential equations for the field variables.”²³⁸

Thus, Pauli lightly pushed aside as untenable one of Einstein’s repeated motivations and hoped-for tests for his unified field theories.

The first reactions to Einstein’s papers came quickly. On 29 October 1928, de Donder suggested a generalisation by using two metric tensors, a space-time metric g_ik, and a bein-metric g_ik^⋆, connected to the 4-bein components \({h_{l\hat \jmath }}\) by In place of Einstein’s connection (161), defined through the 4-bein only, he took: where the dot-symbol denotes covariant derivation by help of the Levi-Civita connection derived from g_ik^⋆. If the Minkowski metric is used as a bein metric g^⋆, then the dot derivative reduces to partial derivation, and Einstein’s original connection is obtained [48].

Another application of Einstein’s new theory came from Eugen Wigner in Berlin whose paper showing that the tetrads in distant-parallelism-theory permitted a generally covariant formulation of “Diracs equation for the spinning electron”, was received by Zeitschrift für Physik on 29 December 1928 [419]. He did point out that “up to now, grave difficulties stood in the way of a general relativistic generalisation of Dirac’s theory” and referred to a paper of Tetrode [344]. Tetrode, about a week after Einstein’s first paper on distant parallelism had appeared on 14 June 1928, had given just such a generally relativistic formulation of Dirac’s equation through coordinate dependent Gamma-matrices²³⁹; he also wrote down a (symmetric) energy-momentum 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 matrix-valued 4-vector γⁱ he prescribed the condition of vanishing divergence. Wigner did not fully accept Tetrode’s derivations because there, implicitly and erroneously, it had been assumed that the two-dimensional representation of the Lorentz group (2-spinors) could be extended to a representation of the affine group. Wigner stated that such difficulties would disappear if Einstein’s teleparallelism theory were used. Nevertheless, nowhere did he claim that the Dirac equation could only be formulated covariantly with the help of Einstein’s new theory.

Zaycoff of the Physics Institute of the University in Sofia also followed Einstein’s work closely. Half a year after Einstein’s 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 Levi-Civita connections. His second and third papers came quickly after Einstein’s 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 Einstein’s 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 \({D_l}: = {\nabla _l} - {S_l}\), ∇_lbeing 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 “for not having shown, in his most recent publication, whether his constraints on the world metric be permissible.” He then derived additional exact compatibility conditions for Einstein’s 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 Einstein’s Hamiltonian principle and rederived for himself Einstein’s 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 semi-symmetric connection. Zaycoff praised Einstein’s teleparallelism theory in words reminding me of the creation of the world as described in Genesis:

“We may say that A. Einstein built a plane world which is no longer waste like the Euclidean space-time-world of H. Minkowski, but, on the contrary, contains in it all that we usually call physical reality.”²⁴⁰ ([428], p. 724)

A conference on theoretical physics at the Ukrainian Physical-Technical 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. Einstein’s former calculational assistant Grommer, now on his own in Minsk, in a brief contribution stressed Einstein’s 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 Einstein’s first note) although Einstein never used \(\Gamma {(g)_{lm}}^k\). According to Grommer the anti-symmetry of P is needed, because its contraction leads to the electromagnetic 4-potential 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 Einstein’s program of deriving the equations of motion from the overdetermined system:

“If 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.”²⁴¹ ([153], p. 646)

Levi-Civita also had sent a paper on distant parallelism to Einstein, who had it appear in the reports of the Berlin Academy [207]. Levi-Civita 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 “bein”-indices; the Greek letters denote coordinates. They obey The electromagnetic field tensor F_ik was entered via Levi-Civita chose as his field equations the Einstein-Maxwell equations projected on a rigidly fixed “world-lattice” of 4-beins. He used the time until the printing was done to give a short preview of his paper in Nature [206]. About a month before Levi-Civita’s paper was issued by the Berlin Academy, Fock and Ivanenko [135] had had the same idea and compared Einstein’s notation and the one used by Levi-Civita in his monograph on the absolute differential calculus [205]:

“Einstein’s 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 […] with those of Einstein.”²⁴²

One of those hunting for exact solutions was G. C. McVittie who referred to Einstein’s paper [88]:

“[…] 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.”

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 4-potential \({\phi _4} = \tfrac{1}{{2\sqrt \pi }}{e^{\tfrac{1}{2}a{x_1}}}\) [225]. He also wrote a paper on exact axially symmetric solutions of Einstein’s teleparallelism theory [226].

Wiener²⁴⁵ and Vallarta²⁴⁶ were after particular exact solutions of Einstein’s field equations in the teleparallelism theory. By referring to Einstein’s first two papers concerning distant parallelism, they set out to show that the

“[…] electromagnetic field is incompatible in the new Einstein theory with the assumption of static spherical symmetry and symmetry of the past and the future. […] the new Einstein theory lacks at present all experimental confirmation.”

In footnote 4, they added:

“Since writing this paper the authors have learned from Dr. H. Müntz 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” ([416], p. 356)

Müntz is mentioned in [88, 85].

In his paper “On unified field theory” of January 1929, Einstein acknowledges work of a Mr. Müntz:

“I am pleased to dutifully thank Mr. Dr. H. Müntz for the laborious exact calculation of the centrally-symmetric problem based on the Hamiltonian principle; by the results of this investigation I was led to the discovery of the road following here.”²⁴⁷

Again, two months later in his next paper, “Unified field theory and Hamiltonian principle”, Einstein remarks:

“Mr. Lanczos and Müntz have raised doubt about the compatibility of the field equations obtained in the previous paper […].”

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:

“Solutions 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 Müntz.”

In the same paper, he states: “I did not see the papers of Lanczos and Müntz.” Even before this, in the same year, in a footnote to the paper of Wiener and Vallarta, we read:

“Since writing this paper the authors have learned from Dr. H. Müntz 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.”

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 relativity²⁴⁸.

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 Einstein’s 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.

“In 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 Einstein’s 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 Einstein’s second 1928 paper, the electromagnetic potential vanishes and the gravitational field also vanishes.”

And they hasten to reassure the reader:

“None of the conclusions of the previous paper are vitiated by this investigation, although some of the final formulas are supplemented by an additional term.” ([417], p. 802)

Vallarta also wrote a paper by himself ([358], p. 784) whose abstract reads:

“In recent papers Wiener and the author have determined the tensors \(^s{h_\lambda }\) of Einstein’s 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 Einstein’s 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].”

He also claims

“that 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]).”

Finally, Rosen and Vallarta [283] got together for a systematic investigation of the spherically symmetric, static field in Einstein’s unified field theory of the electromagnetic and gravitational fields [93].

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 Einstein’s 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 Cauchy-Kowlewsky type of equations in unified field theory, by studying the characteristics and bi-characteristic, the characteristic Cauchy problem, and Huygen’s principle. T. Y. Thomas described the contents of his first paper as follows:

“In 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. […] 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_i^k̂ 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_i^k̂ 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.” [350]

This looks as if he had introduced four vector potentials for the electromagnetic field, and this, in fact, T. Y. Thomas does: “the components h_i^k̂ will play the role of electromagnetic potentials in the present theory.” 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 k̂, with k̂= 1,…4, 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.

“This latter point of view is made the basis for the construction of a system of field equations in the present note — 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_i^k̂ as electromagnetic potentials; doing this, we can say that Maxwell’s equations hold approximately in the local coordinate system in the presence of weak electromagnetic fields.” [351]

The third paper contains a remark as to the content of the concept “unified field theory”:

“It is the objective of the present note to deduce the general existence theorem of the Cauchy-Kowalewsky type for the system of field equations of the unified field theory. […] Einstein (Sitzber. 1930, 18–23) has pointed out that the vanishing of the invariant \(h_{j,k}^{\hat \imath }\) is the condition for the four-dimensional world to be Euclidean, or more properly, pseudo-Euclidean. From the point of view of our previous notes this fact has its interpretation in the statement that the world will be pseudo-Euclidean 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 — provided, of course, that the theory shows itself to be tenable as a theory in agreement with experience.” [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 “at a distance” of directions but not of lengths, i.e., only of the quotient of vector components, Aⁱ/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 Einstein’s first note on distant parallelism [84, 89], introduced a D-bein h_i^k, and determined his connection such that the quotients Aⁱ/A^k of vector components with regard to the D-bein remained invariant under parallel transport. The resulting connection is given by where ψ_m corresponds to \({\Gamma _{lm}}^k\).

Schouten and van Dantzig also used a geometry built on complex numbers, and on Hermitian forms:

“[…] we were able to show that the metric geometry used by Einstein in his most recent approach to relativity theory [84, 83] coincides with the geometry of a Hermitian tensor of highest rank, which is real on the real axis and satisfies certain differential equations.” ([313], p. 319)

The Hermitian tensor referred to leads to a linear integrable connection that, in the special case that it “is real in the real”, coincides with Einstein’s teleparallel connection.

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, “overdetermination” was brought into the game by Einstein:

“The demand for the existence of an ‘overdetermined’ system of equations does provide us with the means for the discovery of the field equations”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 “in involution,” 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 “degree of generality” and “generality index” not familiar to Einstein ²⁵⁰ . Cartan admitted²⁵¹:

“I 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 […] 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 […]. 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 […].” ([50], pp. 25–26)

Einstein’s rapid answer of 9 December 1929 referred to the letter only; he had not been able to study Cartan’s note. As the further correspondence shows, he had difficulties in following Cartan:

“For you have exactly that which I lack: an enviable facility in mathematics. Your explanation of the indice de généralité 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.” ([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 note²⁵²

“[…] 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 […]”²⁵³

which was better suited to physicists. Through this note, Einstein came to understand Cartan’s theory of systems in involution:

“I have read your manuscript, and this enthusiastically. Now, everything is clear to me. Previously, my assistant Prof. Müntz and I had sought something similar — but we were unsuccessful.”²⁵⁴ ([50], pp. 87, 94)

In the correspondence, Einstein made it very clear that he considered Maxwell’s equations only as an approximation for weak fields, because they did not allow for non-singular exact solutions approaching zero at spacelike infinity.

“It 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 far-reaching way — more strongly than the restrictions corresponding to your degrees of determination.”²⁵⁵ ([50], p. 92)

Although Einstein was grateful for Cartan’s help, he abandoned the geometry with distant parallelism.

For some time, the new concept of spinorial wave function stayed unfamiliar to many physicists deeply entrenched in the customary tensorial formulation of their equations²⁵⁹. For example, J. M. Whittaker was convinced that Dirac’s theory for the electron

“has been brilliantly successful in accounting for the ‘duplexity’ phenomena of the atom, but has the defect that the wave equations are unsymmetrical and have not the tensor form.” ([415], p. 543)

Some early nomenclature reflects this unfamiliarity with spinors. For the 4-component spinors or Dirac-spinors (cf. Section 2.1.5) the name “half-vectors” coined by Landau was in use²⁶⁰. Podolsky even purported to show that it was unnecessary to employ this concept of “half-vector” if general curvilinear coordinates are used [259]. Although van der Waerden had written on spinor analysis as early as 1929 [368] and Weyl’s [407, 408], Fock’s [133, 131], and Schouten’s [306] treatments in the context of the general relativistic Dirac equation were available, it seems that only with van der Waerden’s book [369], Schrödinger’s and Bargmann’s 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 complained²⁶¹:

“Yet still a thin booklet is missing from which one could leasurely learn spinor- and tensor-calculus combined.”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 2-spinors “a sort of geometric commentary on the paper of Weyl” [378]. Veblen had studied Weyl’s, Fock’s, and Schouten’s papers, and now introduced a “spinor connection of the first kind” \(\Lambda _{\beta \alpha }^A\), α=1,…,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_D^B is the inverse matrix T^-1. T_D^A corresponds to A_B^A of Equation (75); however, the transformation need not be unimodular. Thus, Veblen took up Schouten’s concept of “spin density” [306] by considering quantities transforming like \({\bar \psi ^A} = {t^N}T_D^A{\psi ^D}\) (A, D=1, 2), with t:= det t_D^B. Then, the covariant derivative of a spinor of weight N, ψ^A is considered; the expression²⁶³ gives “the 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 α”. In his first paper [378], a further generalisation is introduced including “gauge-transformations” in an additional variable x⁰: \({\bar \psi ^A} = {e^{k{x^0}}}{f^A}\) (A, D= 1, 2), with the gauge transformations k is called the “index” of the spinor. In order to deal with 4-spinors Veblen considered a complex projective 3-space and defined 6 real homogeneous coordinates \({X^\sigma }\), with σ=0, …, 5, through Hermitian forms of the 4-spinor components. The subspace X⁰=0, X⁵=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]:

“[…] 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. […] 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.” [380]

The transformation law for spinors is the same as before²⁶⁴: In part, he also takes over van der Waerden’s notation (dotted indices.) As to Veblen’s papers on 2- and 4-spinors, 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 Weyl’s, Fock’s, van der Waerden’s, and Pauli’s 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 “to make things more complicated” (cf. the paragraphs on projective geometry in Section 2.1.3). Veblen’s inhomogeneous coordinates xⁱ, (i = 1, 2, 3, 4) and the homogeneous coordinates \({X^\mu }\), with μ=0, …, 4, are connected by According to Veblen,

“In a five-dimensional representation the use of the homogeneous coordinates (X⁰, … ,X⁴) amounts to representing the points of space-time by the straight lines through the origin, whereas the use of x¹, … , x⁴, and the gauge variable amounts to using the system of straight lines parallel to the x⁰-axis for the same purpose. The transformation (192) given above carries the system of lines into the other.“ [382]

After Tetrode and Wigner, whose contributions were mentioned in Section 6.4.5, Weyl also gave a general relativistic formulation of Dirac’s 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 Dirac’s 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 gauge-symmetry in modern physics ([424], pp. 13–19), already in 1928 in his book on group theory and quantum mechanics ([406], pp. 87–88). We have noted before his refutation of distant parallelism (cf. Section 6.4.4). In his papers, Weyl used a 2-spinor formalism and a tetrad notation different from Einstein’s and Levi-Civita’s: He wrote e^p(k̂) in place of h_k̂^p, and o(l; kj) for the Ricci rotation coefficients γ_jkl; this did not ease the reading of his paper. He partly agreed with what Einstein imagined:

“It is natural to expect that one of the two pairs of components of D i r a c’s quantity belongs to the electron, the other to the proton.”²⁶⁵

In contrast to Einstein, Weyl did not expect to find the electron as a solution of “classical” spinorial equations:

“For every attempt at establishing the quantum-theoretical 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.”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 Born-Jordan-Heisenberg statistical interpretation of the quantum state. For Weyl and statistics, cf. Section V of Sigurdsson’s dissertation ([326], pp. 180–192).

At about the same time, Fock in May 1929 and later in the year wrote several papers on the subject of “geometrizing” Dirac’s equation:

“In 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 n’s 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.”²⁶⁷ ([135], p. 798)

In a subsequent note in the Reports of the Parisian Academy, Fock and Ivanenko introduced Dirac’s 4-spinors under Landau’s name “half vector” and defined their parallel transport with the help of Ricci’s coefficients. In modern parlance, by introducing a covariant derivative for the spinors, they in fact already obtained the “gauge-covariant” 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:

“Thus, it is in the law for the transport of a half-vector that Weyl’s differential linear form must appear.”²⁶⁸ ([134], p. 1469)

In order that gauge-invariance results, ψ 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 matrix-valued spin connection in the expression for the parallel transport of a half-vector ψ: The covariant derivative then is \({D_k} = \tfrac{\partial }{{\partial {x^k}}} - {\Gamma _k}\). Fock made clear that the covariant formulation of Dirac’s equation did not need the special geometry of Einstein’s theory of distant parallelism:

“By help of the concept of parallel transport of a half-vector, Dirac’s equations will be written in a generally invariant form. […] The appearance of the 4-potential φ_l besides the Ricci-coefficients γ_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 φ_l have a place of their own in the geometrical world-view, contrary to Einstein’s opinion; they need not be functions of the γ_ikl.”²⁶⁹ ([131], p. 261, Abstract)

For his calculations, Fock used Eisenhart’s book [119] and “the excellent collection of the most important formulas and facts in the paper of Levi-Civita” [207]. Again, Weyl’s “principle of gauge invariance” as formulated in Weyl’s book of 1928 [406] is mentioned, and Fock stressed that he had found this principle independently and earlier²⁷⁰:

“The appearance of Weyl’s 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 4-potential corresponds to multiplication of the ψ- function with a factor of modulus 1.”²⁷¹ ([130], p. 266)

The divergence of the complex energy-momentum 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 “completed presentation”; in it Fock had noted:

“The 4-potential 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 ‘gravitation and the electron’.”²⁷² ([132], p. 405)

In both of his papers, Fock thus stressed that Einstein’s teleparallel theory was not needed for the general covariant formulation of Dirac’s equation. In this regard he found himself in accord with Weyl, whose approach to the Dirac equation he nevertheless criticised:

“The main subject of this paper is ‘Dirac’s difficulty’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.”²⁷⁴

Weyl’s paper is seminal for the further development of the gauge idea [407].

In another paper, Zaycoff wanted to build a theory explaining the “equilibrium of the electron”. This means that he considered the electron as extended. At this occasion, he fought with himself about the admissibility of the Kaluza-Klein approach:

“Recently, 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 Poincaré’s pressure of the electron within the normal spacetime scheme. However, the introduction of such metaphysical elements is in gross contradiction with space-time causality, although we may doubt in causality in the usual sense due to Heisenberg’s uncertainty relations. A multi-dimensional causality cannot be understood as long as we are unable to give the extra dimensions an intuitive meaning.”²⁷⁶ [433]

Rumer’s paper is [285] (cf. Section 8). In the paper, Zaycoff introduced a six-dimensional manifold with local coordinates x₀, … , x₅ where x₀, x₅ belong to the additional dimensions. His local 6-bein comprises, besides the 4-bein, four electromagnetic potentials and a further one called “eigenpotential” of the electromagnetic field. As he used a “sharpened cylinder condition, ” no further scalar field is taken into account. For x⁰ to x⁴ he used the subgroup of coordinate transformations given in Klein’s approach, augmented by x^5′=x⁵.

Schouten seemingly became interested in Dirac’s equation through Weyl’s publications. He wrote two papers, one concerned with the four-dimensional and a second one with the five-dimensional 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; Weyl’s paper referred to is in Zeitschrift für Physik [407]. Schouten relied on his particular representation of the Lorentz group in a complex space, which later attracted Schrödinger’s criticism. [305]. His comment on Fock’s paper [131] is²⁷⁷:

“Fock has tried to make use of the indetermination of the displacement of spin-vectors to introduce the electromagnetic vector potential. However the displacement of contravariant tensor-densities of weight +1/2 being wholly determined and only these vector-densities playing a role, the idea of Weyl of replacing the potential vector by pseudovectors of class +1 and −1 seems much better.” ([306], p. 261, footnote 19)

Schouten wrote down Dirac’s equation in a space with torsion; his iterated wave equation, besides the mass term, contains a contribution ∼−1/4R if torsion is set equal to zero. Whether Schouten could fully appreciate the importance of Weyl’s new idea of gauging remains open. For him an important conclusion is that

“by the influence of a gravitational field the components of the potential vector change from ordinary numbers into Dirac-numbers.” ([306], p. 265)

Two years later, Schrödinger as well became interested in Dirac’s equation. We reproduce a remark from his publication [319]:

“The joining of Dirac’s 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 Dirac-matrices. This procedure makes it a little difficult to find out whether Einstein’s 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 ψ-function.”²⁷⁸ ([319], p. 105)

In the course of his calculations, Schrödinger obtained the wave equation where μ=2πmc/h, f_kl is the electromagnetic field tensor, and \({s^{kl}}: = \tfrac{1}{2}{\gamma ^{[k}}{\gamma ^{l]}}\) with the γ-matrices γ^k, i.e., the spin tensor. As to the term with the curvature scalar R, Schrödinger was startled:

“To 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 μ is the reciprocal Compton length, about 10¹¹ 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.”²⁷⁹ ([319], p. 128)

The coefficient -1/4 in front of the Ricci scalar in Schrödinger’s (Klein-Gordon) wave equation differs from the 1/6 needed for a conformally invariant version of the scalar wave equation²⁸⁰ (cf. [257], p. 395).

Levi-Civita wrote a letter to Schrödinger in the form of a scientific paper, excerpts of which became published by the Berlin Academy:

“Your fundamental memoir induced me to develop the calculational details for obtaining, from Dirac’s 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 space-time manifold: It seems that without such a tetrad one cannot obtain Dirac’s equation.”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 “Please, copy and distribute!”):

“The heap of corpses, behind which quite a lot of bums look for cover, has got an increment. Beware of the paper by Levi-Civita: Dirac- and Schrödinger-type 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.”²⁸² ([252], p. 170)

Pauli really must have been enraged: Among the publications banned by him is also Weyl’s wellknown article on the electron and gravitation of 1929 [407].

Schrödinger’s 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 Schouten’s [306]:

“In the end, Schouten arrives at almost the same formalism developed in this paper; only that he uses without need n-bein components and theorems on sedenions²⁸³, while afterwards the formalism is still burdened with auxiliary variables and pseudo-quantities. We have taken over the introduction of ‘spin densities’ by Schouten.”²⁸⁴ ([168], p. 4)

Unlike Schrödinger’s, the wave equation derived from Dirac’s equation by Infeld and Waerden contained a term +1/4R, with R the Ricci scalar.

Einstein’s papers on distant parallelism had a strong but shortlived impact on theoretical physicists, in particular in connection with the discussion of Dirac’s equation for the electron, where the 4-spinor ψ and the γ-matrices are used. At the time, there existed some hope that a unified field theory for gravitation, electromagnetism, and the “electron field” was in reach. This may have been caused by a poor understanding of the new quantum theory in Schrödinger’s version: The new complex wave function obeying Schrödinger’s, and, more interestingly for relativists, Dirac’s equation or the ensuing Klein-Gordon wave equation, was interpreted in the spirit of de Broglie’s “onde pilote”, i.e., as a classical matter wave, not — as it should have been — as a probability amplitude for an ensemble of indistinguishable electrons. One of the essential features of quantum mechanics, the non-commutativity 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 space-time. Although he knew well that already for two particles the wave function no longer “lived” in space-time but in 7-dimensional 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 “Does Schrödinger’s wave mechanics determine the motion of a system completely or only in the statistical sense?”. It should have become a 4-page publication in the Sitzungsberichte. As he wrote to Max Born:

“Last week I presented a short paper to the Academy in which I showed that one can ascribe fully determined motions to Schrödinger’s wave mechanics without any statistical interpretation. Will appear soon in Sitz.-Ber. [Reports of the Berlin Academy].”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 Academy’s 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. 134–135).

Each month during 1929, papers appeared in which a link between Einstein’s teleparallelism theory and quantum physics was foreseen. Thus, in February 1929, Wiener and Vallarta stressed that

“the quantities ^sh_λ²⁸⁶ of Einstein seem to have one foot in the macro-mechanical world formally described by Einstein’s gravitational potentials and characterised by the index λ, and the other foot in a Minkowskian world of micro-mechanics characterised by the index s. That the micro-mechanical 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 Einstein’s recent work, and by far the most hopeful portent for a unification of the divergent theories of quanta and gravitational relativity.” [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

“that for the new field theory of Einstein [84, 88] certain quantum-mechanical features are characteristic, and that we may hope that the theory will enable one to seize the quantum laws of the microcosm.”²⁸⁷ ([341], p. 288)

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 (Schrödinger) wave equation in an external electromagnetic field with a contribution from the spin tensor coupled to a torsion term²⁸⁸\({\alpha ^{[i}}{\alpha ^{k]}}{S_{ik}}^l\). 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 4-potential φ_k. This induced him to see another link to quantum physics; by returning to the first of Einstein’s field equations (170) and replacing ∊ in Equation (169) by \(i\tfrac{e}{c}\hbar\) in the limit ħ→0, he obtained the laws of electricity and gravitation, separately. From this he conjectured that, for finite h, Einstein’s field equations might correctly reproduce the quantum features of “the microcosm” ([341], p. 291); cf. also [340].

For some, Kaluza’s 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 Kaluza’s approach, Reichenbächer 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 Schrödinger’s matter wave function with the new metrical component g₅₅ was put to work. Gonseth²⁸⁹ and Juvet, in the first of four consecutive notes submitted in August 1927 [150, 148, 149, 147] stated:

“The objective of this note is to formulate a five-dimensional 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. Schrödinger. 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.”²⁹⁰ ([150], p. 543)

It turned out that from the R₅₅-component of the Einstein vacuum equations \({R_{\alpha \beta }} = 0\), α, β=1, … , 5, with the identification g₅₅=ψ made, and the assumption that ψ, \(\tfrac{{\partial \psi }}{{\partial {x^i}}}\) be “very small”, while ψ, \(\tfrac{{\partial \psi }}{{\partial {x^5}}}\) be “even smaller”, the covariant d‘Alembert equation followed, an equation that was identified by the authors with Schrödinger’s equation. Their further comment is:

“We thus can see that the fiction of a five-dimensional universe provides a deep reason for Schrödinger’s equation. Obviously, this artifice will be needed if some phenomenon would force the physicists to believe in a variability of the [electric] charge.”²⁹¹ ([149], p. 450)

In the last note, with the changed identification g₅₅=ψ² and slightly altered weakness assumptions, Gonseth and Juvet gained the relativistic wave equation with a non-linear mass term.

Interestingly, a couple of months later, O. Klein had the same idea about a link between the g₅₅-component of the metric and the wave function for matter in the sense of de Broglie and Schrödinger. However, as he remarked, his hopes had been shattered [189]. Klein’s 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 Heisenberg’s 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, Klein’s argument is fallacious: He had compactified the fifth dimension. Consequently, the variance of position could not become larger than the compactification length l ∼ 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 space-time might be implied by equations in five-dimensional 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 Kaluza’s idea, in which he had hoped to gain some hold on quantum mechanics, Klein wrote:

“Particularly, 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.” ([189], p. 191, footnote)

At about the same time, W. Wilson of the University of London rederived the Schrödinger equation in the spirit of O. Klein and noted:

“Dr. 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 Klein’s work […].” ([420], p. 441)

Even Eddington ventured into the fifth dimension in an attempt to reformulate Dirac’s equation for more than one electron; he used matrix algebra extensively:

“The matrix theory leads to a very simple derivation of the first order wave equation, equivalent to Dirac’s 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. […] It is interesting to note the way in which the existence of electrons with opposite spins locks the ‘fifth dimension,’ 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.” ([61], pp. 524, 542)

Eddington’s “pentads” built up from sedenions later were generalised by Schouten [307].

Mandel of Petersburg/Leningrad believed that

“a consideration in five dimensions has proven to be well suited for the geometrical interpretation of macroscopic electrodynamics.” ([221], p. 567)

He now posed the question whether this would be the same for Dirac’s theory. Seemingly, he also believed that a tensorial formulation of Dirac’s equation was handy for answering this question and availed himself of “the tensorial form given by W. Gordon [151], and by J. Frenkel”²⁹² [140]. Mandel used, in five-dimensional space, the complex-valued tensorial wave function \({\Psi ^{ik}} = \psi {\gamma ^{ik}} + {\Psi ^{[ik]}}\) with a 5-scalar ψ. Here, he had taken up a suggestion J. Frenkel had developed during his attempt to describe the “rotating electron,” i.e., Frenkel’s introduction of a skew-symmetric wave function proportional to the “tensor of magneto-electric moment” m_ik of the electron by m_ikψ=m₀ψ_ik [141, 140]. Ψ^ik may depend on x⁵; by taking Ψ periodic in x⁵, Mandel derived a wave equation “which can be understood as a generalisation of the Klein-Fock five-dimensional wave equation […].” He also claimed that the vanishing of ψ made M₅ 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 Ψ-field might then represent polarised electrons as its quanta. However, he restricted himself to the consideration of classical one-particle wave equations because

“in some cases one can properly speak of a quasi-macroscopical one-body problem — think of a beam of monochromatic cathod-rays in an arbitrary external force-field.”²⁹³

In a later paper, Mandel came back to his wave equation with a skew-symmetric 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 “Dirac’s spin variable” ζ and the spatial coordinate x⁵ as a pair of canonically-conjugate operator-valued variables; ζ is linked to positive and negative elementary charge (of proton and electron) as its eigenvalues. In Mandel’s five-dimensional space, the fifth coordinate, as a “charge” coordinate, thus assumed only 2 discrete values αe.

“This completely corresponds to the procedure of the Dirac theory, with the only difference that for Dirac the coordinate ζ could assume not 2 but 4 values; from our point of view this remains unintelligible.”²⁹⁴ ([222], p. 785)

In following Klein, Mandel concluded from the Heisenberg uncertainty relations that

“[…] all possible values of this quantity [x⁵] still remain completely undetermined such that all its possible values from -inf to +inf are of equal probability.”

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 “save” his argument [222].

Another one of the many versions of “Dirac’s equation” was presented, in December 1930, by Zaycoff who worked both in the framework of Einstein’s teleparallel theory and of Kaluza’s five-dimensional space. His Lagrangian is complicated²⁹⁵, 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 4-potential. The remaining variables in (198) are \({J_m}: = i\tilde \psi {\gamma _m}{\gamma _l}^\dag {\gamma _0}\psi\), with k≠l, and \({J_{klm}}: = i\tilde \psi {\gamma _k}{\gamma _l}^\dag {\gamma _m}\psi\), with k≠l≠m [434].

While Zaycoff submitted his paper, Schouten lectured at the MIT. and, among other things, showed “how the mass-term in the Dirac equations comes in automatically if we start with a five-dimensional instead of a four-dimensional Riemannian manifold” ([306], p. 272). He proved a theorem:

The Dirac equations for Riemannian space-time 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 R₅.

Here α^b is the set of Dirac numbers defined by α_(aα^b)=g^ab, (α)^aα^b)α^c=α^a(α^bα^c) with a, b, c = 0, … , 4, and ∇_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 Dirac’s equation as a prime motivation:

“The following deductions are intended to show […] 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 field-physics 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 Dirac’s theory.”²⁹⁶ ([250], pp. 837–838)

Pauli started with the observation that the group of orthogonal transformations in five-dimensional space had an irreducible, four-dimensional matrix representation satisfying where \({\alpha _\mu }\) are 4×4 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 μ = 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 bein-quantities. He followed the methods of Schrödinger 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 4-spinors Ψ 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 vector²⁹⁷.

Pauli criticised an analogous attempt at formulating Dirac’s equation with the help of five homogeneous coordinates by Schouten and van Dantzig [317, 308, 309] as being “difficult to understand and less than transparent”²⁹⁸. A projective spinor is defined via where ψ is a normal (“affine”) 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) spin-connections Λ_k for projective spinors Ψ and \({\mathop \Lambda \limits^{\rm{R}} _k}\) for spinors ψ.

Pauli’s 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 Λ_k. An involved calculation leads to Dirac’s equation in four dimensions: with the numerical factor r, the electrical 4-potential Φ_k, and the electromagnetic field tensor F_kl; furthermore, \({\alpha ^{\mu \nu }} = {\alpha ^{[\mu }}{\alpha ^{\nu ]}}\).

Pauli succeeded also in formulating a five-dimensional energy-momentum tensor containing, besides the four-dimensional energy-momentum tensor, the four-dimensional Dirac current vector. At the end of his paper Pauli stressed the

“more provisional character of his 5-dimensional-projective form of Dirac’s theory. […] In contrast to the joinder of the gravitational and electromagnetic fields, a direct logical coupling of the matter-wave-field with these has not been achieved in the form of the theory developed here.”