Interlude: Meanderings – UFT in the late 1930s and the 1940s

3 Interlude: Meanderings – UFT in the late 1930s and the 1940s

Prior to a discussion of the main research groups concerned with Einstein–Schrödinger theories, some approaches using the ideas of Kaluza and Klein for a unified field theory, or aspiring to bind together quantum theory and gravitation are discussed.

3.1 Projective and conformal relativity theory

Projective relativity theory had been invented expressly in order to avoid the fifth dimension of Kaluza–Klein theory. In Sections 6.3.2 and 7.2.4 of Part I, Pauli & Solomon’s paper was described. Also, in Section 6.3.2 of Part I, we briefly have discussed what O. Veblen & B. Hoffmann called “projective relativity” [671 *], and the relationship to the Einstein–Mayer theory. Veblen & Hoffmann had introduced projective tensors with components $Tασ......τβ= exp(N x0)fσα......βτ (x1,x2,x3,x4 )$ where $x1,...x4$ are coordinates of space-time, $x0$ is an additional parameter (a gauge variable) and $N$ a constant named “index”.⁴⁰ $0 x$ transforms as $′0 0 α x = x + log ρ(x )$ . The auxiliary 5-dimensional space appearing has no physical significance. A projective symmetric metric $G αβ$ of index $2N$ was given by $Gα β = Φ2γαβ$ where $Φ$ is an arbitrary projective scalar of index $N$ . In addition, a third symmetric tensor $gαβ = γαβ − ϕαϕ β$ , the gravitational metric, appeared. Here, $ϕ α := γα0$ is a projective vector. Likewise, the Levi-Civita connections $Π δ,Γ δ αβ αβ$ with $Π δ(G ) = Γ δ(γ ) + N (δδΦ + δ δΦ − γ Φ δ) αβ στ αβ στ α β β α αβ$ with $Φ = 1-∂logΦ- α N ∂xα$ and $δ δσ Φ = γ Φσ$ were used. For arbitrary index $N$ , the field equations were derived from the curvature scalar $P$ calculated from the connection $Π αδβ(G στ)$ . One equation could be written in the form of a wave equation:

where $R$ is the curvature scalar calculated from $gij$ . Veblen & Hoffmann concluded that: “The use of projective tensors and projective geometry in relativity theory therefore seems to make it possible to bring wave mechanics into the relativity scheme” ([671 ], abstract). How Planck’s constant might be brought in, is left in the dark.

During the 1940s, meson physics became fashionable. Of course, the overwhelming amount of this research happened in connection with nuclear and elementary particle theory, outside of UFT, but sometimes also in classical field theory. Cf. the papers by F. J. Belinfante on the meson field, in which he used the undor-formalism⁴¹ [16 , 15 ]. In his doctoral thesis of 1941, “Projective theory of meson fields and electromagnetic properties of atomic nuclei” suggested by L. Rosenfeld, Abraham Pais in Utrecht kept away from UFT and calculated the projective energy momentum tensor of an arbitrary field. Although citing the paper of Veblen and Hoffmann, in projective theory he followed the formalism of Pauli; in his application to the Dirac spinor-field, he used Belinfantes undors [466 ]. After this paper, he examined which of Kemmer’s five types of meson fields were “in accordance with the requirements of projective relativity” ([467 ], p. 268).

It is unsurprising that B. Hoffmann in Princeton also applied the projective formalism to a theory intended to unify the gravitational and vector meson fields [278 *]. The meson field $𝜃α$ was defined by Hoffmann via: $𝜃α = ϕα − Φα$ with $ϕα = γ0α$ and $Φα$ given above. Its space-time components $𝜃j$ form an affine vector from which the vector meson field tensor $𝜃ij = 1(∂𝜃i− ∂𝜃j) 2 ∂xj ∂xi$ follows. The theory again contained three Riemannian curvature tensors (scalars). By skipping all calculations, we arrive at the affine form of Hoffmann’s field equations

1 1 1 1 Rij − -gij(R + 12N 2) + -(grs𝜃ir𝜃js + --gij𝜃sr𝜃rs) − 12N 2(𝜃i𝜃j − -gij𝜃s𝜃s) = 0, 2 2 4 2 𝜃ij,j + 12N 2𝜃i = 0. (100 )

In Hoffmann’s words: “except for the term $2 ij − 6N g$ , these are the classical (i.e. unquantized) field equations for a vector meson and gravitational field in the general theory of relativity” ([278 ], p. 464). We could name them as well “Einstein-meson” equations in analogy to “Einstein–Maxwell” equations: no unification of both field had been reached. Also, no scalar meson field and the electromagnetic field were present in the theory.

Hoffmann then looked for a “broader geometrical base” than projective geometry in order to include the electromagnetic field. He found it in conformal geometry, or rather in a special subcase, similarity geometry [279 ].⁴² It turned out that a 6-dimensional auxiliary space was needed. We shall denote the coordinates in this $R6$ by $A, B = 0,1,2,3,4,5$ . The components of a similarity tensor are $Sασ......τβ= exp((p − q)N x0)fασ......τβ(x α)$ , where $p,q$ are the number of covariant and contravariant coordinate indices while $N$ again is named the index of the tensor. In place of the transformations in projective geometry, now

hold. A symmetric tensor $SAB$ in $R6$ was given the role of metric; the assumptions $S05 = 0,$ and $SS55 00$ independent of $xα$ reduced the number of free functions. The definitions $s := SAB- AB S00$ and $γAB := sAB − sA5sB5 ∕s55,γ05 = 0,γ00 = 1$ led back to the former vector meson field via $γj0$ and to a vector in $R6$ $s ψA := √As555$ with $ψ0 = 0$ , $ψ5$ independent of $xα$ and containing the electromagnetic 4-vector $ψj$ . To abreviate the story, Hoffmann’s final field equations in space-time were:

ij 1 ij rs i j 1 ij s r 1 rs i j 1 ij s r R − --g R + 2(g ψ rψs + -g ψ rψ s) + -(g 𝜃 r𝜃 s + -g 𝜃 r𝜃s) 2 4 2 4 1 − 20N 2(𝜃i𝜃j − -gij𝜃s𝜃s) = 0, 2 𝜃ij,j + 20N 2𝜃i = 0, ir ψ ,r = 0. (102 )

The last equation with $ψ = 1(∂ψi − ∂ψj) ij 2 ∂xj ∂xi$ reproduced Maxwell’s equations. In a sequel to this paper, Hoffmann claimed to have derived “the correct trajectories of charged meson testparticles in a combined gravitational, electromagnetic, and vector meson field” ([280 ], p. 1045).

3.1.1 Geometrical approach

It was Pascual Jordan who in physics re-applied projective geometry (cf. Section 2.1.3.3 of Part I) by showing that the transformation group $G5$ of the 4-potential $Ak$ in electrodynamics, composed of the gauge transformations

and coordinate transformations

in space-time, is isomorphic to the group of homogeneous transformations in five variables $X0, X1, ...,X4$ [316 *]:

(no summation over $α$ on the r.h.s.).⁴³ Equivalently, the new coordinates $′α X$ are homogeneous functions of degree 1 of the old $α X$ and transform like a vector:

For the coordinates $i x$ of space-time, alternatively we may write $i i 0 4 x = f (X ,...,X )$ or $∂xi- ρ ∂X ρX = 0$ . Jordan defined projector-components $P βα11,.,.....,β,αnm$ to transform under (106 *) like tensor-components $T βα11,,.....,.,βαnm$ which are homogeneous functions of degree $(n − m)$ in the $X ρ$ . Thus, $X ρ$ itself is a projector just as the Minkowski (Euclidean) metric $gαβ$ of $V5$ with the invariant:

Jordan introduced also a covariant derivative in $V5$ . The formalism is described in papers and his book [317 , 319 *, 320 *]; a detailed presentation is given by G. Ludwig [384 *]. More generally, if $V5$ is provided with a non-flat metric $grs$ , the curvature scalar plays a prominent role in the derivation of the field equations within projective relativity. Ludwig also introduced arbitrary matter fields. At first, his Lagrangian for a scalar matter field $ψ$ within projective geometry was [383 ]

but then became generalized to

To obtain the Lagrangian for the metrical field, $ψ$ was replaced by $J$ ([384 *], p. 57):

With

we arrive at:

where $U(J ),V(J ),W (J)$ are arbitrary functions. As can be seen from (110 *), the 5-dimensional curvature scalar used by Jordan and by Thiry (cf. the next Section 3.1.2) follows as the subcase $U (J) = J −1∕2,W (J) = V (J) = 0$ of the general expression (111 *). Ludwig, at the time of writing the preface to his book, e.g., in May 1951, seemingly did not know of Thiry’s paper of 1948 [604 *] nor of his PhD thesis published also in 1951: in his bibliography Thiry’s name and paper are missing.
Pauli had browsed in Ludwig’s book and now distanced himself from his own papers on projective relativity of 1933 discussed briefly in Section 7.2.4 of Part I.⁴⁴ He felt deceived:

“The deception consists in the belief that by the projective form, i.e., the homogeneous coordinates, the shortcomings of Kaluza’s formulation have been repaired, and that one has achieved something beyond Kaluza. At the time, in 1933, I did not know explicitly the transition from Kaluza to the projective form (as in [20 *]); it is too simple and banal to the effect that the factual contents of both equivalent formulations could be somehow different.” (letter of W. P. to P. Jordan, [490 *], p. 735):⁴⁵

3.1.2 Physical approach: Scalar-tensor theory

Toward the end of the second world war, Kaluza’s five- dimensional theory and projective relativity emerged once again as vehicles for a new physical theory which, much later, came to be known as “scalar-tensor theory of gravitation.”⁴⁶ Cosmological considerations related to the origin of stars seem to have played the major role for the building of a theory by P. Jordan in which the gravitational constant $κ = 8πG- c2$ is thought to be varying in (cosmological) time and thus replaced by a scalar function [316 ]⁴⁷ . The theory nicely fit with Dirac’s “large number hypothesis” [122 , 123 ]. The fifteenth field variable in Kaluza’s theory was identified by Jordan with this function, or in projective relativity, with the scalar: $J := gαβX αX β$ by setting $J = 2cκ2 (α,β = 0,1,...,4)$ [321 *]. In space-time, the field equations for the gravitational field $gij$ , the electromagnetic 4-potential $Ak = g4k$ , and the $g44$ -variable $κ$ were derived by Jordan and Müller⁴⁸ to be:

κ 1 ( 1 ) Gik + -2FisFks = − --- ∇k ∇i κ − ---∇iκ ∇k κ , (112 ) c 2κ 2κ sj 3- sj κ∇sF = − 2∇s κ F , (113 ) κ 1 1 G = − --2FrsF rs +---grs∇r κ ∇sκ − -grs∇r ∇s κ. (114 ) 2c 2 κ κ

Jordan & Müller denoted the Ricci-tensor in space-time by $Gik$ . P. G. Bergmann, in a paper submitted in August 1946 but published only in January of 1948, reported that work on a theory with a fifteenth field variable had been going on in Princeton:

“Professor Einstein and the present author had worked on that same idea several years earlier, but had finally rejected it and not published the abortive event” ([21 *], p. 255).

It may be that at the time, they just did not have an idea for a physical interpretation like the one suggested by P. Jordan. Although there were reasons for studying the theory further, Bergmann pointed out that there is an “embarras de richesses” in the theory: too many constructive possibilities for a Lagrangian. Nonetheless, in his subsequent paper on “five-dimensional cosmology”, P. Jordan first stuck to the simplest Lagrangian, i.e. to the Ricci scalar in five dimensions [318 *]. In this paper, Jordan also made a general comment on attempts within unitary field theory of the Einstein–Schrödinger-type to embed corpuscular matter into classical field theory (cf. chapter 6 with Section 6.1.1 below):

“The problem of the structure of matter can only be attacked as a problem in quantum mechanics; nevertheless, investigations of the singularities of solutions of the field equations retain considerable importance in this framework. […] the wave functions of matter must be taken into account. Whether this program can be carried through, and to which extent, in the sense of an extension of geometry (to which Schrödinger’s ideas related to the meson field seem to provide an important beginning) is such a widespread question […]”.⁴⁹ ([318 ], p. 205).

Jordan’s theory received wider attention after his and G. Ludwig’s books had been published in the early 1950s [319 *, 384 ]. In a letter to Jordan mentioned, Pauli also questioned Jordan’s taking the five-dimensional curvature scalar as his Lagrangian. Actually, already in the first edition of his book, Jordan had accepted Pauli’s criticism and replaced (110 *) by [compare with (109 *)]:

He thus severed his “extended theory of gravitation” from Kaluza’s theory. He also displayed the Lagrangian ([319 *], p. 139):

but then set one of the two free parameters $η = 1$ . One of those responding to this book was M. Fierz in Basel [195 *]. Before publication, he had corresponded with W. Pauli, sent him first versions of the paper and eventually received Pauli’s placet; cf. the letter of Pauli to Fierz of 2 June 1956 in [492 *], p. 578. In the second edition of his book, Jordan also commented on a difficulty of his theory pointed out by W. Pauli: instead of $gik$ equally well $ϕ(x )gik$ with arbitrary function $ϕ$ could serve as a metric.⁵⁰ This conformal invariance of the theory is preserved in the case that an electromagnetic field forms the matter tensor. A problem for the interpretation of mathematical objects as physical variables results: by a suitable choice of the conformal factor $ϕ$ , a “constant” gravitational coupling function could be reached, again. In his paper, M. Fierz suggested to either couple Jordan’s gravitational theory to point particles or to a (quantum-) Klein–Gordon field in order to remove the difficulty. Fierz also claimed that Jordan had overlooked a physical effect. According to him, a dielectricity “constant” of the vacuum could be introduced $𝜖 = κ1+ 1η = J1+η 0$ from looking at Jordan’s field equations. His Lagrangian corresponded to $κ,rκ,r 𝜖0- rs κ (R − ζ κ2 ) − c2FrsF$ with $η J = κ$ . On the other hand, in the MKSA system of physical units, the fine-structure constant is $2 α = ℏec𝜖 0$ . Thus, the fine-structure constant would depend on $κ$ .

“Assumed that $κ$ be variable in cosmic spaces, then this variability must show up in the redshift of light radiated from distant stars.” ([195 ], p. 134)⁵¹

Because this had not been observed, Fierz concluded that $κ = − 1$ . Both, Pauli and Fierz gave a low rating to Jordan’s theory⁵²

Neither Pauli nor Fierz seem to have known that the mathematician Willy Scherrer at the university of Bern had suggested scalar-tensor theory already in 1941 before P. Jordan, and without alluding to Kaluza or Pauli’s projective formulation.⁵³ Also, in 1949, Scherrer had suggested a more general Lagrangian [532 ]:

$ψ$ is considered to be a scalar matter field. He had advised a student, K. Fink, to work on the Lagrangian $ℒ = (R + 2ωgrs ∂ψr ∂ψs)√ − g ∂x ∂x$ [198 ]. The ensuing field equations correspond to those following from Jordan’s Lagrangian if his parameter $η = 0$ ([319 *] p. 140). Exact solutions in the static, spherically symmetric case and for a homogeneous and isotropic cosmological model were published in 1951 almost simultaneously by Fink ( $η = 0$ ) and Heckmann, Jordan & Fricke ( $η = 1$ ) [243 ].

In 1953, W. Scherrer asked Pauli to support another manuscript on unified field theory entitled “Grundlagen einer linearen Feldtheorie” for publication in Helvetica Physica Acta but apparently sent him only a reprint of a preliminary short note [534 ]. Pauli was loath to get involved and asked the editor of this journal, the very same M. Fierz, what the most appropriate answer to Scherrer could be. He also commented:

“Because according to my opinion all “unified field theories” are based on dubious ideas – in particular it is a typically suspect idea of the great masters Einstein and Schrödinger to add up the symmetric and antisymmetric parts of a tensor – I have to pose the question […].” (W. Pauli to M. Fierz, 15 Dec. 1953) ([491 *], p. 390–391)⁵⁴

Scherrer’s paper eventually was published in Zeitschrift für Physik [535 ]. In fact, he proposed a unified field theory based on linear forms, not on a quadratic form such as it is used in general relativity or Einstein–Schrödinger UFT. His notation for differential forms and tangent vectors living in two reference systems is non-standard. As his most important achievement he regarded “the absolutely invariant and at the same time locally exact conservation laws.” In his correspondence with Fierz, Pauli expressed his lack of understanding: “What he means with this, I do not know, because all generally relativistic field theories abound with energy laws”([491 *], p. 403). H. T. Flint wrote a comment in which he claimed to have shown that Scherrer’s theory is kin to Einstein’s teleparallelism theory [213 ]. For studies of Kaluza’s theory in Paris (Jordan–Thiry theory) cf. Section 11.1.

3.2 Continued studies of Kaluza–Klein theory in Princeton, and elsewhere

As described in Section 6.3 of Part I, since 1927 Einstein and again in 1931 Einstein and Mayer⁵⁵ , within a calculus using 5-component tensorial objects in space-time, had studied Kaluza’s approach to a unification of gravitation and electromagnetism in a formal 5-dimensional space with Lorentz-signature. A decade later, Einstein returned to this topic in collaboration with his assistant Peter Bergmann [167 *, 166 ]. The last two chapters of Bergmann’s book on relativity theory are devoted to Kaluza’s theory and its generalization ([20 *]. Einstein wrote a foreword, in which he did not comment on “Kaluza’s unified field theory” as the theory is listed in the book’s index. He admitted that general relativity “[…] has contributed little to atomic physics and our understanding of quantum phenomena.” He hoped, however, that some of its features as were “general covariance of the laws of nature and their nonlinearity” could contribute to “overcome the difficulties encountered at present in the theory of atomic and nuclear processes” ([20 *] p. V). Two years before Bergmann’s book appeared, Einstein already had made up his mind against the five-dimensional approach:

“The striving for most possible simplicity of the foundations of the theory has prompted several attempts at joining the gravitational field and the electromagnetic field from a unitary, formal point of view. Here, in particular, the five-dimensional theory of Kaluza and Klein must be mentioned. Yet, after careful consideration of this possibility, I think it more proper to accept the mentioned lack of inner unity, because it seems to me that the embodiment of the hypotheses underlying the five-dimensional theory contains no less arbitrariness than the original theory.”⁵⁶

Nonetheless, in their new approach, Einstein and Bergmann claimed to ascribe “physical reality to the fifth dimension whereas in Kaluza’s theory this fifth dimension was introduced only in order to obtain new components of the metric tensor representing the electromagnetic field” ([167 ], p. 683). Using ideas of O. Klein, this five-dimensional space was seen by them essentially as a four-dimensional one with a small periodical strip or a tube in the additional spacelike dimension affixed. The 4-dimensional metric then is periodic in the additional coordinate $x4$ .⁵⁷ With the fifth dimension being compact, this lessened the need for a physical interpretation of its empirical meaning. Now, the authors partially removed Kaluza’s ‘cylinder condition’ $g = 0 αβ,4$ (cf. Section 4.2 of Part I, Eq. (109)): they set $gik,4 ⁄= 0$ , but assumed $gi4,4 = 0$ and $g44 = 1$ : the electrodynamic 4-potential remains independent of $x4$ . Due to the restriction of the covariance group (cf. Section 4.2, Part I, Eq. (112)), in space-time many more possibilities for setting up a variational principle than the curvature scalar of 5-dimensional space exist: besides the 4-dimensional curvature scalar $R$ , Einstein & Bergmann list three further quadratic invariants: $rs rs rs 2 ArsA ,g ,4grs,4,(g grs,4)$ where $Ars := ∂sAr − ∂rAs$ . The ensuing field equations for the fourteen variables $gik$ and $Ak$ contain two new free parameters besides the gravitational and cosmological constants. Scalar-tensor theory is excluded due to the restrictions introduced by the authors. Except for the addition of some new technical concepts (p-tensors, p-differentiation) and the inclusion of projective geometry, Bergmann’s treatment of Kaluza’s idea in his book did not advance the field.

The mathematicians K. Yano and G. Vranceanu showed that Einstein’s and Bergmann’s generalization may be treated as part of the non-holonomic UFT proposed by them [713 , 681 *]. Vranceanu considered space-time to be a “non-holonomic” totally geodesic hypersurface in a 5-dimensional space $V5$ , i.e., the hypersurface cannot be generated by the set of tangent spaces in each point. Besides the metric of space-time $ds2 = gabdxadxb, (a,b = 1,2,3,4)$ , a non-integrable differential form $ds5 = dx5 − ϕadxa$ defining the hypersurface was introduced together with the additional assumption $-∂- ∂x5ϕa = 0$ . The path of a particle with charge $e$ , mass $m0$ and 5-vector $A v ,(A = 1,...,5)$ was chosen to be a geodesic tangent to the non-holonomic hypersurface. Thus $dv5 =! 0$ , and Vranceanu then took $v5 = e-- m0$ . The electromagnetic field was defined as $F = 1(ϕ − ϕ ) ab 2 b,a a,b$ . Both Einstein’s and Maxwell’s equations followed, separately, with the energy-momentum tensor of matter as possible source of the gravitational field equations: “One can also assume that the energy tensor $Tab$ be the sum of two tensors one of which is due to the electromagnetic field […]”. ([681 *], p. 525).⁵⁸ His interpretation of the null geodesics which turn out to be independent of the electromagnetic field is in the spirit of the time: “This amounts to suppose for light, or as well for the photon, that its charge be null and its mass $m0$ be different from zero, a fact which is in accord with the hypothesis of Louis de Broglie (Une nouvelle conception de la lumière; Hermann, Paris 1934).” ([681 ], p. 524)⁵⁹ More than a decade later, K. Yano and M. Ohgane generalized the non-holonomic UFT to arbitray dimension: $n$ -dimensional space is a non-holonomic hypersurface of $(n + 1)$ -dimensional Riemannian space. It is shown that the theory “[…] seems to contain all the geometries appearing in the five-dimensional unified field theories proposed in the past and to suggest a natural generalization of the six-dimensional unified field theories proposed by B. Hoffmann, J. Podolanski, and one of the present authors” ([714 ], pp. 318, 325–326). They listed the theories by Kaluza–Klein, Veblen–Hoffmann, Einstein–Mayer, Schouten–Dantzig, Vranceanu and Yano; cf. also Sections 3.1 and 11.2.1.

B. Hoffmann derived the geodesic equations of a magnetic monopole in the framework of a 6-dimensional theory [277 *]; cf. Section 11.2.1. The one who really made progress, although unintentionally and unnoticed at the time, was O. Klein who extended Abelian gauge theory for a particular non-Abelian group, which almost corresponds to SU(2) gauge theory [333 ]. For a detailed discussion of Klein’s contribution cf. [237 ].

Einstein unceasingly continued his work on the “total field” but was aware of inherent difficulties. In a letter to his friend H. Zangger in Zurich of 27 February 1938, he wrote:

“I still work as passionately even though most of my intellectual children, in a very young age, end in the graveyard of disappointed hopes”. ([560 *], p. 552)⁶⁰

At first he was very much fascinated by the renewed approach to unified field theory by way of Kaluza’s idea. We learn this from the letter of 8 August 1938 to his friend Besso:

“After twenty years of vain searching, this year now I have found a promising field theory which is a quite natural sequel to the relativistic gravitational theory. It is in line with Kaluza’s idea about the essence of the electromagnetic field.” ([163 *], p. 321)⁶¹

3.3 Non-local fields

3.3.1 Bi-vectors; generalized teleparallel geometry

In 1943, Einstein had come to the conclusion that the failure of “finding a unified theory of the physical field by some generalization of the relativistic theory of gravitation” seemed to require “a decisive modification of the fundamental concepts” ([165 *], p. 1). He wanted to keep the four-dimensional space-time continuum and the diffeomorphism group as the covariance group, but wished to replace the Riemannian metric by a generalized concept. Together with the assistant at the Institute for Advanced Studies, Valentine Bargmann , he set out to develop a new scheme involving “bi-vector fields”. Unlike the concept of a bi-vector used by Schouten in 1924 ([537 ], p. 17) and ever since in the literature, i.e., for the name of a special antisymmetric tensor, in the definition by Einstein and Bargmann the concept meant a tensor depending on the coordinates of two points in space-time, an object which would be called “bi-local” or “non-local”, nowadays. The two points, alternatively, could be imagined to lie in the same manifold (“single space”), or in two different spaces (“double space”). In the latter case, the coordinate transformations for each point are independent. In place of the Riemannian metric, a contravariant bi-vector $k21l g$ is defined via

and, similarly,

Here, the numbers refer to the two points, while the Latin indices denote the usual tensor indices. The coordinate transformation $xi → ∗xi$ for a simple “bi-vector” $Ti2k1$ is given by:

Already here a problem was mentioned in the paper: there exist too many covariant geometric objects available for deriving field equations. This is due to the possibility to form covariant quantities containing only first order derivatives like the tensorial quantity: $γ := g − g i1j2k2 i1j2,k2 i1k2,j2$ . In order to cut down on this wealth, a new operation called “rimming” was introduced which correlated a new “bi-vector” $∗ giαkβ$ with $giαkβ$ by multiplying it from left and right by tensors of full rank $kα ′ kβ ω iα , ωβi$ where each is taken from one of the two manifolds (now Greek indices refer to the two different points)⁶² :

All tensors $∗ g$ obtained by rimming $g$ were considered as different representations of the same field. The rimming of a contravariant “bi-vector” was done similarly by multiplying from the left with $k ′σ β iβ$ and from the right with tensors $kα σαi$ . In order that (118 *), (119 *) remain invariant under rimming, the relations $σ = ω−1, ′σ = (′ω )−1 α α α α$ , must hold. A comparison of (120 *) and (121 *) shows that a coordinate transformation can be combined with a rimming operation “in such a way that the bi-vector components remain invariant – i.e., behave like scalars under the resultant transformation” ([165 *], p. 4).

As a possible field equation, the authors now introduced “tensorial four-point equations”:

If $giαkβ$ is treated as a matrix, it is easy to see that all higher-order tensorial equations (6, 8, …-point) are dependent on the four-point equation. Next, the authors showed that by a special rimming operation $gik αβ$ can be transformed into $δik$ such that the corresponding space-time is flat. The trace on (i,j) in (122 *) leads to a scalar equation:

This equations apparently allows for non-flat solutions (cf. Eq. (13), p. 6 in [165 ]).

In another paragraph, the authors returned to the “single space”-version. Here, a symmetry condition is demanded: $gik= gki αβ βα$ . Now, the tensorial p-point equations admit the special case that the two points (coordinates $α,β$ ) coincide. By a suitable rimming operation $giαkα = ηik$ was reached where $ηik$ is the matrix diag $(1,− 1,− 1,− 1)$ . The rimming operations were performed with representations of the Lorentz group. Also, mixed bi-vectors $i gαk = ηilg lαk β β$ were introduced and a tensorial three-point equation had to be satisfied:

It characterizes flat space as well. After a discussion of complex rimming transformations, the theory was put into spinor form. Solutions were obtained of the relevant matrix equations, some of them due to the mathematician Carl L. Siegel (1896 – 1981), who stayed in Princeton at the time. Neither a link to physics nor a new UFT followed from this paper. The truly new feature of its approach was that the “metric” can join arbitrarily distant points, not just infinitesimally neighbouring ones. At this time, as in many other cases, Einstein expected the solution to physical problems from a solution to still unanswered mathematical questions.

In a continuation of this paper, Einstein explicitly introduced the concept of connection: “I show that just as in the case of the infinitesimal theory this theory can be made very simple by separating the concepts and relations into those based exclusively on the affine connection and those where the affine connection is specialized by hypotheses on the structure of the field” ([146 *], p. 15). The mixed bi-vector $i gαk β$ is interpreted as the (non-infinitesimal) affine connection because the relation

connects the points with coordinates $α$ and $β$ .⁶³ Two conditions are to be fulfilled: the displacement of a vector from $α$ to $β$ and back does not change it, and “the scalar product of a covariant and a contravariant vector is invariant with respect to the affine connection”. This led to

Taking into account the rimming operation, it is seen that (125 *) and (126 *) are invariant with respect to rimming. Again a tensorial three-point equation was written down which, in matrix form, reads as:

The consequences of (127 *) were the same as for the 4-point equation before: space-time is flat. To escape this conclusion, the trace of (127 *) could also be used “as a possible choice of a field law of a bi-vector field” ([146 *], p. 17). Furthermore, the symmetry of the metric could be replaced by the more general:

In the remaining part of the paper, various possible cases were discussed and a new concept introduced: “volume invariance”. For this, the rimming matrices were restricted to have determinant $±1$ . Another field law then was proposed:

In a note in proof Einstein remarked that “W. Pauli and V. Bargmann have meanwhile succeeded in proving that (129 *), too, admits only “flat space” solutions”([146 ], p. 23). He slightly changed the equation into:

and ended the paper (and his publications on “bi-vectors”) by stating: “At present, the author, in collaboration with W. Pauli, is trying to find out whether this equation has non-trivial solutions”. Thus, besides a new mathematical scheme, and Einstein’s method of “trial and error” for finding field equations, no progress in terms of unified field theory had been achieved.

The only physicist outside of Princeton who expressed an interest in this discovery of “a new form of geometrical connection of a continuum, the distant affine connection” was Schrödinger in Dublin. In his paper, he set out to

“[…] show how the new geometrical structure emerges, by generalization, from the one that was at the basis of Einstein’s ‘Distant Parallelism’ (Fernparallelismus), and consisted in the natural union of an integrable (but in general non-symmetric) infinitesimal affine connection and a (in general not flat) Riemannian metric” ([550 ], p. 143).

He rewrote Einstein’s mixed bi-vector with the help of the tetrads used in teleparallel geometry:

Note that Schrödinger denoted the tetrad by $hνa$ with Latin indices $a, b,...$ for the number of the leg, and Greek space-time indices such that $h νa h νb = δab$ . By comparing the parallel displacement of a 4-vector $δA ν = − Δ ρσνAρdx σ$ with $i iβ k Aβ = g kA α α$ he arrived at his interpretation of $βi g k α$ as a connection. As the relation between the affine connection – which he called “infinitesimal connection” – and the bi-vector connection, named “distant connection”, he took:

( ( ) ) k { ∂ kβ } Δ ij = ---- g i . (132 ) ( ∂xjα α ) α=β

In the following paper with Friedrich Mautner, scholar at the Dublin Institute for Advanced Studies, who already had been acknowledged as coworker in the first publication, Schrödinger likened Einstein’s three-point tensor in (127 *) to the curvature tensor: the three-point matrices of a distant affinity perform a transfer from the point $α$ via $→ β → γ$ back to the starting point $α$ . This is analogous to parallel transfer around a triangle formed by the points $α, β,γ$ ; the change of a vector thus transported along the closed circuit is proportional to the curvature tensor, if it is skew symmetric also in the first pair of indices ([399 ], pp. 224–225). This analogy was carried on further; a formula corresponding to (59 *) of Section 2.3.1 was derived, with the symmetric affine connection replaced by the Christoffel symbols and torsion by contorsion, i.e., by $Skli + Slik + Skil$ . Both cases in (128 *) were treated and tentative field equations including a cosmological constant written down. The paper ended with the sentence: “Are these equations likely to give an appropriate description of physical fields?” As is known now, the answer to this question should have been “no”. Thus, in this context, Schrödinger’s papers also did not bring progress for UFT; nevertheless, they helped to make Einstein’s papers more readable. The subject was also taken up by a Romanian mathematician, M. Haimovici (1906 – 1973), who instead introduced a space of point-couples, introduced axioms for defining a connection there, and established a relation to work by E. Cartan [239 ].

3.3.2 From Born’s principle of reciprocity to Yukawa’s non-local field theory

Much earlier, Max Born had followed a different if not entirely unrelated conceptual course: in 1938, he had introduced a “principle of reciprocity”: “ […] each general law on the $x$ -space has an ’inverse image’ in the $p$ -space, in the first instance the laws of relativity” ([38 ], p. 327). In this note in Nature, Born added a Lorentzian metric $g (p)dpadpb ab$ in momentum space satisfying as well the corresponding Einstein field equations (cf. Section 4.2). Infeld in Princeton wanted to get some further information about this principle of reciprocity from Born, who was afraid that his idea be seized by the “terribly clever people over there”. However, in his letter to Einstein of 11 April 1938, he described his joint work with Klaus Fuchs: to derive a “super-mechanics” with an 8-dimensional metric in phase space. A new fundamental (“natural”) constant appeared leading to both an absolute length and an absolute momentum ([168 *], pp. 182–184). In a way, Born’s formalism came near to Einstein’s “double space” in his bi-vector theory.

About a decade later, in 1947, H. Yukawa in an attempt to arrive at a theory of elementary particles, used exactly this “double space”-approach: “[…] the field in our case is not necessarily a function of $x μ$ alone, but may depend on $pμ$ also. […] The generalized field can include not only the electromagnetic field and various types of meson fields but also the so-called pair-fields such as the meson pair field and the electron-neutrino field” ([716 ], p. 211–212). It seems that he saw his non-local field theory as introducing new degrees of freedom and leading a step away from the point-particle concept: a possibility for avoiding infinite self-energies. During his stay at the Princeton Institute for Advanced Study (PIAS) in 1948/49, Yukawa must have learned about Born’s reciprocity principle and work around it [45 , 46 , 47 ], refered to it (cf. [718 ]) and embarked on his quantum-theory of non-local fields. In general, his theory now corresponded to the “single space” of Einstein and Bargmann: operators $U (Xμ,rμ$ ) depended on “two sets of real variables” (in the same space) [719 , 720 , 721 ]. According to Yukawa: “ $X μ$ coincide with the ordinary coordinates $xμ$ in the limit of local field, so that the dependence on $X μ$ describes the asymptotic behavior and the dependence on $rμ$ characterizes the internal structure or inertial motion” ([722 ], p. 3 ). At this time, both Yukawa and Einstein lived in Princeton and got to know each other. It can reasonably be doubted, though, that they took note of each other’s scientific work except superficially. In 1947, Yukawa possibly had not the least inkling of the Einstein–Bargmann paper of 1944. Of course, Bargmann in Princeton then could have told him. In the special issue of Review of Modern Physics for Einstein’s 70th birthday in 1949, Max Born in Edinburgh dedicated an article about his reciprocity principle to Einstein:

“The theory of elementary particles which I propose in the following pages is based on the current conceptions of quantum mechanics and differs widely from the ideas which Einstein himself has developed in regard to this problem. […] It can be interpreted as a rational generalization of his (“special”) theory of relativity.” ([40 ], p. 463)

Right next to Born’s article, one by Yukawa was placed which dealt with meson theory. He made only a cryptic remark about his non-local field theory when expressing the then prevailing ignorance about elementary particle theory:

“Probably we need a broader background (such as the five-dimensional space or the quantized phase space) for field theory in order to cope with these problems, although it is premature to say anything definite in this connection.” ([717 ], p.479)

He then gave references to his papers on non-local field theory. In a letter of 23 January 1949, Born informed Einstein also privately about his theory of elementary particles:

The laws of nature are invariant not only with regard to the relativistic transformations but also with regard to the substitutions $xα → pα,pα → − xα$ […]. All amounts to replace your fundamental invariant $xαxα = R$ by the symmetrical quantity $S = R + P$ where $P = pαpα$ . $S$ is an operator, the integer eigenvalues of which are the distances […].” ([168 *], p. 242.)⁶⁴

This story shows that a very loose kinship existed between the Einstein–Bargmann “bi-vector” method and Yukawa’s non-local field theory with Born’s reciprocity theory in the middle. Although some of those involved were in direct personal contact, no concrete evidence for a conscious transfer of ideas could be established.

	Abstract
1	Introduction
2	Mathematical Preliminaries
	2.1	Metrical structure
	2.2	Symmetries
	2.3	Affine geometry
	2.4	Differential forms
	2.5	Classification of geometries
	2.6	Number fields
3	Interlude: Meanderings – UFT in the late 1930s and the 1940s
	3.1	Projective and conformal relativity theory
	3.2	Continued studies of Kaluza–Klein theory in Princeton, and elsewhere
	3.3	Non-local fields
4	Unified Field Theory and Quantum Mechanics
	4.1	The impact of Schrödinger’s and Dirac’s equations
	4.2	Other approaches
	4.3	Wave geometry
5	Born–Infeld Theory
6	Affine Geometry: Schrödinger as an Ardent Player
	6.1	A unitary theory of physical fields
	6.2	Semi-symmetric connection
7	Mixed Geometry: Einstein’s New Attempt
	7.1	Formal and physical motivation
	7.2	Einstein 1945
	7.3	Einstein–Straus 1946 and the weak field equations
8	Schrödinger II: Arbitrary Affine Connection
	8.1	Schrödinger’s debacle
	8.2	Recovery
	8.3	First exact solutions
9	Einstein II: From 1948 on
	9.1	A period of undecidedness (1949/50)
	9.2	Einstein 1950
	9.3	Einstein 1953
	9.4	Einstein 1954/55
	9.5	Reactions to Einstein–Kaufman
	9.6	More exact solutions
	9.7	Interpretative problems
	9.8	The role of additional symmetries
10	Einstein–Schrödinger Theory in Paris
	10.1	Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory
	10.2	Tonnelat’s research on UFT in 1946 – 1952
	10.3	Some further developments
	10.4	Further work on unified field theory around M.-A. Tonnelat
	10.5	Research by and around André Lichnerowicz
11	Higher-Dimensional Theories Generalizing Kaluza’s
	11.1	5-dimensional theories: Jordan–Thiry theory
	11.2	6- and 8-dimensional theories
12	Further Contributions from the United States
	12.1	Eisenhart in Princeton
	12.2	Hlavatý at Indiana University
	12.3	Other contributions
13	Research in other English Speaking Countries
	13.1	England and elsewhere
	13.2	Australia
	13.3	India
14	Additional Contributions from Japan
15	Research in Italy
	15.1	Introduction
	15.2	Approximative study of field equations
	15.3	Equations of motion for point particles
16	The Move Away from Einstein–Schrödinger Theory and UFT
	16.1	Theories of gravitation and electricity in Minkowski space
	16.2	Linear theory and quantization
	16.3	Linear theory and spin-1/2-particles
	16.4	Quantization of Einstein–Schrödinger theory?
17	Alternative Geometries
	17.1	Lyra geometry
	17.2	Finsler geometry and unified field theory
18	Mutual Influence and Interaction of Research Groups
	18.1	Sociology of science
	18.2	After 1945: an international research effort
19	On the Conceptual and Methodic Structure of Unified Field Theory
	19.1	General issues
	19.2	Observations on psychological and philosophical positions
20	Concluding Comment
	Acknowledgements
	References
	Footnotes
	Biographies