On the History of Unified Field Theories. Part II. (ca. 1930–ca. 1965)
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Abstract
The present review intends to provide an overall picture of the research concerning classical unified field theory, worldwide, in the decades between the mid1930 and mid1960. Main themes are the conceptual and methodical development of the field, the interaction among the scientists working in it, their opinions and interpretations. Next to the most prominent players, A. Einstein and E. Schrödinger, V. Hlavatý and the French groups around A. Lichnerowicz, M.A. Tonnelat, and Y. Thiry are presented. It is shown that they have given contributions of comparable importance. The review also includes a few sections on the fringes of the central topic like BornInfeld electromagnetic theory or scalartensor theory. Some comments on the structure and organization of researchgroups are also made.
Keywords
Unified field theory Differential geometry History of science1 Introduction
The dream of unifying all fundamental interactions in a single theory by one common representation still captures the mind of many a theoretical physicist. In the following, I will focus on the development of classical unified field theory (UFT) in the period from the mid1930s to the mid1960s. One of the intentions then was to join the gravitational to the electromagnetic field, and, hopefully, to other fields (mesonic, …) in “a single hyperfield, whose basis would be equivalent to that of the geometrical structure for the universe” ([376], p. 3). Einstein referred to his corresponding theories alternatively as the “generalized theory of gravitation”, “(relativistic) theory of the nonsymmetric (or asymmetric) field”, and of “the theory of the total field”. Schrödinger spoke of “unitary field theory”; this name was taken up later by Bergmann [24] or Takasu [598]. In Mme. Tonnelat’s group, the name “théorie du champ unifié d’Einstein” (or d’EinsteinSchrödinger), or just “théorie unitaire (du champ)(d’Einstein)” was in use; Hlavatý called it “(Einstein) Unified (Field) Theory of Relativity”. In other papers we read of “Einstein’s Generalized Theory of Gravitation”, “Einstein’s equations of unified field”, “theory of the nonsymmetric field”, “einheitliche Feldtheorie” etc. However, we should not forget that other types of unitary field theory were investigated during the period studied, among them KaluzaKlein theory and its generalizations. In France, one of these ran under the name of “JordanThiry” theory, cf. Sections 3.1.2 and 11.1.
Most important centers for research on unified field theory in the 1930s until the early 1950s were those around Albert Einstein in Princeton and Erwin Schrödinger in Dublin. Paris became a focus of UFT in the late 1940s up to the late 1960s, with a large group of students around both Mme. M.A. Tonnelat in theoretical physics, and the mathematician A. Lichnerowicz. In comparison with the work of Einstein and Schrödinger, the contributions to UFT of the Paris groups have been neglected up to now by historians of physics although they helped to clarify consequences of the theory. These groups had a share both in the derivation of exact mathematical results and in contributing arguments for the eventual demise of UFT. The mathematician V. Hlavatý from Indiana University, Bloomington (USA), with one or two students, enriched the mathematicallyoriented part of the UFTcommunity with his systematical studies in the 1950s. We will encounter many further researchers worldwide, especially sizeable groups in Italy, and in countries like Canada, England, India, and Japan. The time period is chosen such that Einstein’s move from Berlin to Princeton approximately defines its beginning while its end falls into the 1960s which saw a revival of interest in general relativity theory [192], and the dying off of some still existing interest into classical unified field theory. Up to the 1940s, some hope was justified that the gravitational interaction might play an important role in the unification of the fundamental fields. With the growth of quantum field theory and developments in elementary particle physics, gravity became crowded out, however.
“to relate the physical phenomena in the submicroscopic world of the atom to those in the macroscopic world of universal spacetime, to find a common principle explaining both electromagnetic forces and gravitational force […]. In this inquiry Einstein has pursued a lonely course; most physicists have taken the apparently more promising road of quantum theory.” ([564], p. 26)
In fact, the majority of the theoretical physicists working in field theory considered UFT of the EinsteinSchrödinger type as inadequate. Due to Einstein’s earlier achievements, his fame and, possibly, due to his, Schrödinger’s and de Broglie’s reserve toward the statistical interpretation of quantum mechanics, classical or semiclassical approaches to field theory were favoured in their scientific research environments in theoretical physics. Convinced by the stature of these men, a rather small number of theoretical physicists devoted their scientific careers to classical unified field theory. Others wrote their PhD theses in the field and then quickly left it. A few mathematicians became attracted by the geometrical structures underlying the field (cf. [677], p. 30).
“[…] the essential achievement of general relativity, namely to overcome ‘rigid’ space (i.e., the inertial frame), is only indirectly connected with the introduction of a Riemannian metric. The direct relevant conceptual element is the ‘displacement field’ (\(\Gamma _{ik}^l\)), which expresses the infinitesimal displacement of vectors. It is this which replaces the parallelism of spatially arbitrarily separated vectors fixed by the inertial frame (i.e., the equality of corresponding components) by an infinitesimal operation. This makes it possible to construct tensors by differentiation and hence to dispense with the introduction of ‘rigid’ space (the inertial frame). In the face of this, it seems to be of secondary importance in some sense that some particular Γfield can be deduced from a Riemannian metric […].”^{3} (A. Einstein, 4 April 1955, letter to M. Pantaleo, in ([473], pp. XV–XVI); English translation taken from Hehl and Obuchov 2007 [244].)
“The quantization of fields turns out more and more to be a problem with thorns and horns, and by and by I get used to think that I will not live to see substantial progress for all these problems.” ([493], p. 519)^{6}
In fact, for elementary particle theory, the 1950s and 1960s could be seen as “a time of frustration and confusion” ([686], p. 99). For weak interactions (fourfermion theory) renormalization did not work; for strong interactions no calculations at all were possible. W. Pauli was very skeptical toward the renormalization schemes developed: “[…] from my point of view, renormalization is a not yet understood palliative.” (Letter to Heisenberg 29 September 1953 [491], p. 268.)
“Einstein just has disappeared leaving us, in addition to many completed works, an enigmatic theory. The scientists look at it — like he himself did — with a mixture of distrust and hope, a theory which carries the imprint of a fundamental ambition of its creator.” (cf. Lichnerowicz, preface of [632], p. VII.)^{7}
“In the literature there are many approaches to the problem of the unified field theory. Some of them strongly influenced the development of geometry, although none has received general recognition as a physical theory.” ([269], preface, p. X.)
The work done in the major “groups” lead by Einstein, Schrödinger, Lichnerowicz, Tonnelat, and Hlavatý was published, at least partially, in monographs (Einstein: [150], Appendix II; [156], Appendix II); (Schrödinger: [557], Chapter XII); (Lichnerowicz: [371]); (Hlavatý: [269]), and, particularly, (Tonnelat: [632, 641, 642]). To my knowledge, the only textbook including the EinsteinSchrödinger nonsymmetric theory has been written in the late 1960s by D. K. Sen [572]. The last monograph on the subject seems to have been published in 1982 by A. H. Klotz [334]. There exist a number of helpful review articles covering various stages of UFT like Bertotti [26], Bergia [19], Borzeszkowsi & Treder [679], Cap [71], Hittmair [256], Kilmister and Stephenson [330, 331], Narlikar [453], Pinl [497], Rao [504], Sauer [528, 529], and Tonnelat ([645], Chapter 11), but no attempt at giving an overall picture beyond Goenner [228] seems to have surfaced. Vizgin’s book ends with Einstein’s research in the 1930s [678]. In 1957, V. Bargmann has given a clear fourpage résumé of both the EinsteinSchrödinger and the KaluzaKlein approaches to unified field theory [12].^{8} In van Dongen’s recent book, the epistemological and methodological positions of Einstein during his work on unified field theory are discussed [667].
The present review intends to provide a feeling for what went on in research concerning UFT at the time, worldwide. Its main themes are the conceptual and methodical development of the field, the interaction among the scientists working in it, and their opinions and interpretations. The review also includes a few sections on the fringes of the general approach. A weighty problem has been to embed the numerous technical details in a narrative readable to those historians of science lacking the mathematical tools which are required in many sections. In order to ease reading of chapters, separately, a minor number of repetitions was deemed helpful. Some sociological and philosophical questions coming up in connection with this review will be touched in Sections 18 and 19. These two chapters can be read also by those without any knowledge of the mathematical and physical background. Up to now, philosophers of science apparently have not written much on Einstein’s unified field theory, with the exception of remarks following from a nontechnical comparison of the field with general relativity. Speculation about the motivation of the central figures are omitted here if they cannot be extracted from some source.
The main groups involved in research on classical unified field theory will be presented here more or less in chronological order. The longest account is given of EinsteinSchrödinger theory. In the presentation of researchers we also follow geographical and language aspects due to publications in France being mostly in French, in Italy mostly in Italian, in Japan and India in English.^{9}
We cannot embed the history of unified field theory into the external (political) history of the period considered; progress in UFT was both hindered by the second world war, Nazi and communist regimes, and helped, after 1945, by an increasing cooperation among countries and the beginning globalization of communications.^{10}
Part II of the “History of Unified Field Theory” is written such that it can be read independently from Part I. Some links to the earlier part [229] in Living Reviews in Relativity are provided.
2 Mathematical Preliminaries
For the convenience of the reader, some of the mathematical formalism given in the first part of this review is repeated in a slightly extended form: It is complemented by further special material needed for an understanding of papers to be described.
2.1 Metrical structure
First, a definition of the distance ds between two infinitesimally close points on a Ddimensional differential manifold M_{ D } is to be given, eventually corresponding to temporal and spatial distances in the external world. For ds, positivity, symmetry in the two points, and the validity of the triangle equation are needed. We assume ds to be homogeneous of degree one in the coordinate differentials dx^{ i } connecting neighboring points. This condition is not very restrictive; it includes Finsler geometry [510, 199, 394, 4] to be briefly discussed in Section 17.2.
2.1.1 Affine structure
Both derivatives are used in versions of unified field theory by Einstein and others.^{18}
A manifold provided with only a linear (affine) connection L is called affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special rôle: with regard to it the connection transforms as a tensor; cf. Section 2.1.5 of Part I.
In Section 2.2.3, we shall find the set of transformations \({L_{jk}}^j \to {L_{ik}}^j + \delta _i^j{{\partial \omega} \over {\partial {x^k}}}\) playing a role in versions of Einstein’s unified field theory.
2.1.2 Metric compatibility, nonmetricity
From (30) or (33), the connection L may in principle be determined as a functional of the metric tensor, its first derivatives, and of torsion.^{28} After multiplication with ν_{ s }, (33) can be rewritten as \(\overset{}{\nabla}_{i}{\overset{\rightarrow}{\nu}}^{k}= g^{ks}\underset{}{\tilde{\nabla}_{i}}\nu_{s}\), where \(\tilde \nabla\) is formed with the Hermitian conjugate connection (cf. Section 2.2.2) [396].^{29}
Remark:
Although used often in research on UFT, the ±notation is clumsy and ambiguous. We apply the ±differentiation to (2), and obtain: \({({g_{mi}}\,{g^{mj}})_{\underset {\pm} {\vert\vert}l}} = {g_{\underset + m \underset  i}}_{\vert\vert l}\,{g^{mj}} + {g_{mi\,}}\,{g^{\overset m + \overset j }}_{{\vert\vert}l} = {(\delta _i^j)_{\underset \pm {\vert\vert}l}}\). While the l.h.s. of the last equation is well defined and must vanish by definition, the r.h.s. is ambiguous and does not vanish: in both cases \(\delta _{\underset  i}^{\overset j +} =  {S_{ij}}^j \ne 0,\,\,\delta _{\underset {+}{i}}^{\overset{j}{}}{}_{\Vert l} =S_{il}^j \ne 0\). Einstein had noted this when pointing out that only \(\delta _{\underset + i}^{\overset j +}{}_{\Vert l}=0=\delta _{\underset  i}^{\overset j }{}_{\Vert l}\) but \(\delta _{\underset + i}^{\overset j +}{}_{\Vert l}\neq 0,\, \, \delta _{\underset  i}^{\overset j }{}_{\Vert l}\neq 0\) ([147], p. 580). Already in 1926, J. M. Thomas had seen the ambiguity of \({({A_{i{B_i}}})_{\Vert l}}\) and defined a procedure for keeping valid the product rule for derivatives [607]. Obviously, \({\overset 0 \nabla _k}\delta _i^j = 0\).
2.2 Symmetries
2.2.1 Transformation with regard to a Lie group
2.2.2 Hermitian symmetry
2.2.3 λtransformation
The compatibility equation (30) is not conserved under λtransformations because of \({g_{\underset + i \,\underset  k \Vert l}} \to {g_{\underset + i \,\underset  k \Vert l}}  2{g_{i(k}}{\lambda _{l)}}\). The same holds for the projective transformations (51), cf. ([430], p. 84). No generally accepted physical interpretation of the λtransformations is known.
2.3 Affine geometry
We will speak of affine geometry in particular if only an affine connection exists on the 4manifold, not a metric. Thus the concept of curvature is defined.
2.3.1 Curvature
For a complete decomposition of the curvature tensor (53) into irreducible parts with regard to the permutation group further objects are needed, as e.g., \({\epsilon ^{ajkl}}\underset + K _{jkl}^b = 2{\epsilon ^{ajkl}}{\partial _{\left[ k \right.}}{S_{\left. l \right]}}_j^b\); cf. [348].
2.3.2 A list of “Ricci”tensors
One of the puzzles remaining in Einstein’s research on UFT is his optimism in the search for a preferred Riccitensor although he had known, already in 1931, that presence of torsion makes the problem ambiguous, at best. At that time, he had found a totality of four possible field equations within his teleparallelism theory [176]. As the preceding list shows, now a 6parameter object could be formed. The additional symmetries without physical support suggested by Einstein did not help. Possibly, he was too much influenced by the quasiuniqueness of his field equations for the gravitational field.
2.3.3 Curvature and scalar densities
From the expressions (73) to (81) we can form scalar densities of the type: \({\hat g^{ik}}\overset {\rm{Her}} {{{\underset  K}_{ik}}}\) to \({\hat g^{ik}}{U_{ik}}\) etc. As the preceding formulas show, it would be sufficient to just pick \({\hat g^{ik}}{\underset 0 K_{ik}}\) and add scalar densities built from homothetic curvature, torsion and its first derivatives in order to form a most general Lagrangian. As will be discussed in Section 19.1.1, this would draw criticism to the extent that such a theory does not qualify as a unified field theory in a stronger sense.
2.3.4 Curvature and λtransformation
2.4 Differential forms
2.5 Classification of geometries
A differentiable manifold with an affine structure is called affine geometry. If both, a (possibly nonsymmetric) “metric” and an affine structure, are present we name the geometry “mixed”. A subcase, i.e., metricaffine geometry demands for a symmetric metric. When interpreted just as a gravitational theory, it sometimes is called MAG. A further subdivision derives from the nonmetricity tensor being zero or ≠ 0. RiemannCartan geometry is the special case of metricaffine geometry with vanishing nonmetricity tensor and nonvanishing torsion. Weyl’s geometry had nonvanishing nonmetricity tensor but vanishing torsion. In Sections 2.1.3 and 4.1.1 of Part I, these geometries were described in greater detail.
2.5.1 Generalized RiemannCartan geometry
2.5.2 Mixed geometry
Whenever a symmetric tensor appears which is independent of the connection and of full rank, it can play the role of a metric. The geometry then may be considered to be a Riemannian geometry with additional geometric objects: torsion tensor, nonmetricity tensor, skewsymmetric part of the “metric” etc. These might be related to physical observables. Therefore, it is moot to believe that two theories are different solely on the basis of the criterion that they can be interpreted either in a background of Riemannian or mixed geometry. However, by a reduction of the more general geometries to a mere Riemannian one plus some additional geometric objects the very spirit of UFT as understood by Einstein would become deformed; UFT explicitly looks for fundamental geometric objects representing the various physical fields to be described.
2.5.3 Conformal geometry
If ξ^{ k } is a Killing vector field for g_{ ij }, then ξ^{ k } is a conformal Killing vector field for ḡ_{ ij }; cf. Eq. (45) in Section 2.1.2.
A particular subcase of conformal geometry is “similarity geometry”, for which the restricted group of transformations acts g_{ ij } (x^{ m }) → γ_{ ij } (x^{ l }) = k^{2}g_{ ij } (x^{ m }), with a constant k, cf. Section 3.1.
2.6 Number fields
In Section 2.3 of Part I, the possibility of choosing number fields different from the real numbers for the field variables was stated. Such field variables then would act in a manifold with real coordinates. A more deeply going change is the move to an underlying manifold with coordinates taken from another number field, e.g., complex spaces. The complex number field was most often used in connection with unified field theory in both roles. cf. A. Einstein, (complex space, Section 7.2), J. Moffat, (complex field on real space, Section 13) and A. Crumeyrolle, (hypercomplex manifold, Section 11.2.2).
3 Interlude: Meanderings — UFT in the late 1930s and the 1940s
Prior to a discussion of the main research groups concerned with EinsteinSchrödinger 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
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 undorformalism^{41} [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 spinorfield, 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).
3.1.1 Geometrical approach
“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):^{46}
3.1.2 Physical approach: Scalartensor theory
“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).
“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 […]”.^{50} ([318], p. 205).
“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)^{52}
Because this had not been observed, Fierz concluded that κ = −1. Both, Pauli and Fierz gave a low rating to Jordan’s theory^{53}
“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)^{55}
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 EinsteinSchrödinger UFT. His notation for differential forms and tangent vectors living in two reference systems is nonstandard. 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 (JordanThiry theory) cf. Section 11.1.
3.2 Continued studies of KaluzaKlein theory in Princeton, and elsewhere
“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 fivedimensional 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 fivedimensional theory contains no less arbitrariness than the original theory.”^{57}
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 fivedimensional space was seen by them essentially as a fourdimensional one with a small periodical strip or a tube in the additional spacelike dimension affixed. The 4dimensional metric then is periodic in the additional coordinate x^{4}.^{58} 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_{αβ,4} = 0 (cf. Section 4.2 of Part I, Eq. (109)): they set g_{ik,4} = 0, but assumed (g_{i4,4} = 0 and g_{44} = 1: the electrodynamic 4potential remains independent of x^{4}. Due to the restriction of the covariance group (cf. Section 4.2, Part I, Eq. (112)), in spacetime many more possibilities for setting up a variational principle than the curvature scalar of 5dimensional space exist: besides the 4dimensional curvature scalar R, Einstein & Bergmann list three further quadratic invariants: A_{ rs }A^{ rs }, g^{ rs } _{,4}g_{rs,4}, (g^{ rs }g_{rs,4})^{2} where A_{ rs }:= ∂_{ s }A_{ r } − ∂_{ r }A_{ s }. The ensuing field equations for the fourteen variables g_{ ik } and A_{ k } contain two new free parameters besides the gravitational and cosmological constants. Scalartensor theory is excluded due to the restrictions introduced by the authors. Except for the addition of some new technical concepts (ptensors, pdifferentiation) 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 nonholonomic UFT proposed by them [713, 681]. Vranceanu considered spacetime to be a “nonholonomic” totally geodesic hypersurface in a 5dimensional space V^{5}, i.e., the hypersurface cannot be generated by the set of tangent spaces in each point. Besides the metric of spacetime ds^{2} = g_{ ab } dx^{ a } dx^{ b }, (a, b = 1, 2, 3, 4), a nonintegrable differential form ds^{5} = dx^{5} − ϕ_{ a } dx^{ a } defining the hypersurface was introduced together with the additional assumption \({\partial \over {\partial {x^5}}}{\phi _a} = 0\). The path of a particle with charge e, mass m_{0} and 5vector v^{ A }, (A = 1, …, 5) was chosen to be a geodesic tangent to the nonholonomic hypersurface. Thus \(d{\upsilon ^5}\overset i = 0\), and Vranceanu then took \({\upsilon ^5} = {e \over {{m_0}}}\). The electromagnetic field was defined as F_{ ab } = ½(ϕ_{ b,a } − ϕ_{ a,b }). Both Einstein’s and Maxwell’s equations followed, separately, with the energymomentum tensor of matter as possible source of the gravitational field equations: “One can also assume that the energy tensor T_{ ab } be the sum of two tensors one of which is due to the electromagnetic field […]”. ([681], p. 525).^{59} 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 m_{0} 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)^{60} More than a decade later, K. Yano and M. Ohgane generalized the nonholonomic UFT to arbitray dimension: ndimensional space is a nonholonomic hypersurface of (n + 1)dimensional Riemannian space. It is shown that the theory “[…] seems to contain all the geometries appearing in the fivedimensional unified field theories proposed in the past and to suggest a natural generalization of the sixdimensional 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 KaluzaKlein, VeblenHoffmann, EinsteinMayer, SchoutenDantzig, 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 6dimensional 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 nonAbelian group, which almost corresponds to SU(2) gauge theory [333]. For a detailed discussion of Klein’s contribution cf. [237].
“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)^{61}
“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)^{62}
3.3 Nonlocal fields
3.3.1 Bivectors; 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 fourdimensional spacetime 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^{63}, he set out to develop a new scheme involving “bivector fields”. Unlike the concept of a bivector 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 spacetime, an object which would be called “bilocal” or “nonlocal”, 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.
“[…] 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 nonsymmetric) infinitesimal affine connection and a (in general not flat) Riemannian metric” ([550], p. 143).
3.3.2 From Born’s principle of reciprocity to Yukawa’s nonlocal 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 xspace has an ‘inverse image’ in the pspace, in the first instance the laws of relativity” ([38], p. 327). In this note in Nature, Born added a Lorentzian metric g_{ ab } (p) dp^{ a } dp^{ b } 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 “supermechanics” with an 8dimensional 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 bivector theory.
“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)
“Probably we need a broader background (such as the fivedimensional 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)
The laws of nature are invariant not only with regard to the relativistic transformations but also with regard to the substitutions x^{ α } → p_{ α }, p_{ α } → −x^{ α } […]. All amounts to replace your fundamental invariant x^{ α }x_{ α } = R by the symmetrical quantity S = R + P where P = p^{ α }p_{ α }. S is an operator, the integer eigenvalues of which are the distances […].” ([168], p. 242.)^{66}
This story shows that a very loose kinship existed between the EinsteinBargmann “bivector” method and Yukawa’s nonlocal 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.
4 Unified Field Theory and Quantum Mechanics
“The classical field theory of Einstein — electromagnetic and gravitational together — give us a satisfactory explanation of all largescale physical phenomena. […] But they fail completely to describe the behavior of individual atoms and particles. To understand the smallscale side of physics, physicists had to invent quantum mechanics and the idea of the quantum field.” ([137], p. 60)
Nevertheless, there were other physicists like Einstein for whom no divide between classical and quantum field existed, in principle.
4.1 The impact of Schrödinger’s and Dirac’s equations
“In connection with the equation of the electron path we have the suggestion that ψ*ψ respond to the certainty of finding the electron on the track” ([206], same page).
His conclusion, i.e., that quantum phenomena correspond to geometrical conceptions, and that the complete geometrical scheme includes quantum theory, gravitation, and electromagnetism could not hide that all he had achieved was to build a set of classical relativistic wave equations decorated with an ħ. In a further paper of 1938, in the same spirit, Flint arrived at a geometrical “quantum law” built after the vanishing of the curvature scalar from which he obtained the Dirac equation in an external electrical field [207].
During the second world war, Flint refined his research without changing his basic assumption [208, 209, 210], i.e., “that the fundamental equation of the quantum theory, which is the quantum equation for an electron in a gravitational and electromagnetic field, can be developed by an appeal to simple geometric ideas.” His applications to “field theories of the electron, positron and meson” [211] and to “nuclear field theories” [212] follow the same line. No progress, either for the understanding of quantum mechanics nor for the construction of a unified field theory, can be discovered. Flint’s work was not helped by contributions of others [6, 3]. After World War II, Flint continued his ideas with a collaborator [214]; in the meantime he had observed that Mimura also had introduced matrix length in 1935. As in a previous paper, he used the method by which Weyl had derived his first gauge theory combining gravitation and electromagnetism. Strangely enough, Weyl’s later main success, the redirection of his idea of gauging to quantum mechanics was not mentioned by Flint although he was up to show that “equations of the form of Dirac’s equation can be regarded as gaugeequations” ([214], p. 260). Under parallel transport, the matrix length L of a vector A_{ k } is assumed to change by dL = ΘR_{ r }α^{ s }A_{ s }ψ dx^{ r }, where R_{ r } is an operator (a matrix) corresponding to the 5vector ϕ_{ k }. Flint still was deeply entrenched in classical notions when approaching the explanation of the electron’s rest mass: it should contain contributions from the electromagnetic and mesonic fields. The mathematician _{J}. A. Schouten conjectured that “[t]he investigations of H. T. Flint are perhaps in some way connected with conformal meson theory […]” ([539], p. 424).
That Flint was isolated from the physics mainstream may be concluded also from the fact that his papers are not cited in a standard presentation of relativistic waveequations [84]. We dwelled on his research in order to illuminate the time lag in the absorption of new physics results among groups doing research, simultaneously. In this theme, we could have included the “tensor rear guard” (Fisher, Temple, etc.) who believed to be able to get around spinors.
4.2 Other approaches
4.3 Wave geometry
A group of theoreticians at the Physical Institute of Hiroshima University in Japan in the second half of the 1930s intensively developed a program for a unified field theory of a new type with the intention of combining gravitation and quantum theory. Members of the group were Yositaka Mimura, Tôyomon Hosokawa, Kakutarô Morinaga, Takasi Sibata, Toranosuke Iwatsuki, Hyôitirô Takeno, and also Kyosi Sakuma, M. Urabe, K. Itimaru. The research came to a deadly halt when the first atom bomb detonated over Hiroshima, with the hypocenter of the explosion lying 1.5 km away from the Research Institute for Theoretical Physics.^{69} After the second world war, some progress was made by the survivors. The theory became simplified and was summarized in two reports of the 1960s [427, 428].
In an introductory paper by Mimura, the new approach was termed “wave geometry” [425]. His intention was to abandon the then accepted assumption that spacegeometry underlying microscopic phenomena (like in elementary particle physics), be the same as used for macroscopic physics. Schrödinger had argued in this sense and was cited by Mimura [541]. Einstein’s original hope that spacetime must not exist in the absence of matter, unfulfilled by general relativity, became revived on the level of “microscopic physics”: “[…] the microscopic space exists only when an elementary particle exists. In this sense, where there is no elementary particle, no ‘geometry’ exists” ([425], p. 101). Also “[…] according to our new theory, geometry in microscopic space differs radically from that of macroscopic […]” ([425], p. 106).^{70} “wave geometry” must not be considered as one specific theory but rather as the attempt for a theory expressing the claimed equivalence of geometry and physics.
In 1938, T. Hosokawa even had extended wave geometry to Finsler geometry and applied to Milne’s cosmological principle [287].
With its results obtained until 1945, wave geometry could not compete with quantum field theory. After the war, the vague hope was expressed that in a “supermicroscopic” spacetime, elementary particle theory could be developed and that “the problem of internal space’ of elementary particles may be interwoven with some ‘hidden’ relations to the structure of spacetime.” ([428], p. 41.) Clearly, the algebra of γmatrices which is all what is behind the distance operator, was an insufficient substitute for the algebra of noncommuting observables in quantum field theory.
5 BornInfeld Theory
“Possibly, you will not agree, because I do not include gravitation. This is a rather basic point, where I have a different view as you in your papers on unitary field theory. Hopefully, I soon will be able to finalize my idea on gravitation” ([168], p. 167).^{72}
Around the same time as Pauli, B. Hoffmann who had left Princeton for the University of Rochester, had had the same idea. It was couched in the language of projective theory on which he had worked with O. Veblen (cf. Section 6.3.2 of Part I) and on his own [275].^{73} He suggested the Lagrangian \({\mathcal L} = (\sqrt {1 + B}  1)\sqrt { \gamma}\) where γ_{ ij } is the 5dimensional projective metric and B the projective curvature scalar. Due to B = R − g^{ pr } g^{ qs } f_{ rq }f_{ ps }, his Lagrangian corresponds to Kaluza’s. Born & Infeld had remarked that in order to include gravitation in their theory, only Einstein’s Lagrangian must be added to (148). Hoffmann now tried to obtain a static spherically symmetric solution for both theories with a nonvanishing electromagnetic field. In the augmented BornInfeld Lagrangian, the Minkowski metric could be used as a special case. According to Hoffmann this was no longer possible for his Lagrangian because “the electromagnetic field exerts a gravitational influence” ([275], p. 364). As he could not find a solution to his complicated field equations, the “degree of modification of the electrostatic potential by its own gravitational field” could not be determined.
“[…] I always had a lot of understanding for your good Jewish physics, and much amusement with it; however, I myself have produced it only once: the nonlinear electrodynamics, and this is no particular success […].”^{75}
Nevertheless, it had some influence on UFT; cf. Sections 6.1.3, 9.7, and 10.3.4.
Born and Infeld unsuccessfully tried to quantize their nonlinear theory of the electromagnetic field by using the commutation rules of Heisenberg and Pauli for the field strenghts [43, 44]. They noticed that the theory could be presented differently according to whether the pairs \(\vec E,\;\vec B\), or \(\vec D, \vec H; \;\vec D, \vec B; \;\vec E, \vec H\) were chosen as independent variables. The authors took \(\vec D,\;\vec B\) in order to avoid “formal difficulties”. However, a perturbative approach by canonical quantization of either the field or the vector potential could not succeed because the interaction term in the Hamiltonian included higher powers of derivative terms.^{76}
S. Kichenassamy^{77} studied the subcase of an electromagnetic null field with matter tensor: T_{ ij } = A^{2}k_{ i }k_{ j }, k_{ i }k^{ i } = 0 and showed that in this case the BornInfeld theory leads to the same results as Maxwell’s electrodynamics [328, 340].
6 Affine Geometry: Schrödinger as an Ardent Player
6.1 A unitary theory of physical fields
“Schrödinger was as kind as to write to me himself about his work. At the time I was quite enthusiastic about this way of thinking. Its weakness lies in the fact that its construction from the point of view of affine space is rather artificial and forced. Also, the link between skew symmetric curvature and the electromagnetic states of space leads to a linear relation between electrical fields and charge densities. […]” ([168], p. 196.).^{79}
As I suppose, the “At the time” refers to 1923. With “skew symmetric curvature”, the antisymmetric part of the Riccitensor is meant. Schrödinger believed that Einstein had left affine theory because of “aesthetic displeasure” resulting from a mistake in his interpretation of the theory.
6.1.1 Symmetric affine connection
In his first papers on affine geometry, Schrödinger kept to a symmetric connection.^{80} There is thus no need to distinguish between \(\overset + \nabla\) and \(\overset  \nabla\) in this context. Within purely affine theory there are fewer ways to form tensor densities than in metricaffine or mixed geometry. By contraction of the curvature tensor, secondrank tensors K_{ ij } and V_{ ij } are available (cf. Section 2.3.1) from which tensor densities of weight −1 (scalar densities) (cf. Section 2.1.5 of Part I) like \(\sqrt {\det ({K_{ij}})}\) or \(\sqrt {\det ({V_{ij}})}\) can be built. Such scalar densities are needed in order to set up a variational principle.

ϕ_{ ik } corresponds to the electromagnetic field tensor (\({\bf{\vec E}},\;{\bf{\vec B}}\)),

\({\hat f^{ik}}\) corresponds to its conjugate field quantity (\( {\bf{\vec D}},\;{\bf{\vec H}}\)),

j^{ k } corresponds to the electric 4current density,

T_{ ik } corresponds to the “fieldenergytensor of the electromagnetic field”.
We note from (163) that the electric current density is the negative of the electromagnetic 4potential. The meson field is not yet included in the theory.
“\({\mathcal L}\) is essentially Born’s Lagrangian, with ϕ_{ kl } in place of his (\({\bf{\vec B}},\;{\bf{\vec E}}\)) \([ \ldots ]\;{\hat f^{ik}}\) agrees in form with Born’s contravariant tensordensity (\({\bf{\vec H}},\;  {\bf{\vec D}}\)) […].” ([545], p. 52.)
6.1.2 Application: Geomagnetic field
Blackett conjectured “that a satisfactory explanation of (172) will not be found except within the structure of a unified field theory” [28]. M. J. Nye is vague on this point: “What he had in mind was something like Einstein asymmetry or inequality in positive and negative charges.” ([460], p. 105.) Schrödinger seconded Blackett; however, he pointed out that it was “not a very simple thing” to explain the magnetic field generated by a rotating body by his affine theory. “At least a general comprehension of the structure of matter” was a necessary prerequisite ([554], p. 216). The theoretical physicist A. Papapetrou who had worked with Schrödinger joined Blackett in Manchester between 1948 and 1952. We may assume that the experimental physicist Blackett knew of Schrödinger’s papers on the earth’s magnetism within the framework of UFT and wished to use Papapetrou’s expertise in the field. The conceptional link between Blackett’s idea and UFT is that in this theory the gravitational field is expected to generate an electromagnetic field whereas, in general relativity, the electromagnetic field had been a source of the gravitational field.
“As concerns Blackett’s new material on the magnetism of the earth and stars, I have the following difficulty: In case it is an effect of acceleration the dependency of the angular velocity must be different; in the case of an effect resulting from velocity, a translatory movement ought to also generate a magnetic field. Special relativity then requires that the matter at rest possesses an electric field as well. […] I do not know how to escape from this dilemma.”^{86} ([489], p. 543)
Three weeks earlier, in a letter to Leon Rosenfeld, he had added that he “found it very strange that Blackett wrote articles on this problem without even mentioning this simple and important old conclusion.” ([489], p. 539) This time, Pauli was not as convincing as usual: Blackett had been aware of the conclusions and discussed them amply in his early paper ([28], p. 664).
6.1.3 Application: Point charge
Two weeks later, Schrödinger put out another paper in which he wrote down 16 “conservation identities” following from the fact that his Lagrangian is a scalar density and depends only on the 16 components of the Ricci tensor. He also compared his generalization of general relativity with Weyl’s theory gauging the metric (cf. [689]), and also with Eddington’s purely affine theory ([140], chapter 7, part 2). From (158) it is clear that Schrödinger’s theory is not gaugeinvariant.^{88} He ascribed this weakness to the missing of a third fundamental field in the theory, the meson field. According to Schrödinger the absence of the meson field was due to his restraint to a symmetric connection. Eddington’s theory with his general affine connection would house all the structures necessary to include the third field. It should take fifteen months until Schrödinger decided that he had achieved the union of all three fields.
6.1.4 Semisymmetric connection
In this paper, Schrödinger changed the relation between mathematical objects and physical observables:
The variables \((j, \phi, \hat f)\) related to the Ricci tensor correspond to the meson field; whereas \((Y,M,\hat m)\) related to torsion describe the electromagnetic field.
“Now the gravitational field and the mesonic field are actually, to all appearance, universally and jointly produced in the same places, viz. in the heavy nuclear particles. They have at any rate their principal seat in common, while there is absolutely no parallelism between electric charge and mass” ([549], p. 282).
In addition, Schrödinger referred to Einstein’s remark concerning the possibility of exchanging the roles of the electromagnetic fields \({\bf{\vec E}}\) by \({\bf{\vec H}}\) and \( {\bf{\vec D}}\) by \( {\bf{\vec B}}\) ([142], p. 418). “Now a preliminary examination of the wholly nonsymmetrical case gives me the impression that the exchange of roles will very probably be imperative, […]” ([549], p. 282).
“a third field […], of equally fundamental standing with gravitation and electromagnetism: the mesonic field responsible for nuclear binding. Today no fieldtheory which does not embrace at least this triad can be deemed satisfactory at all.” ([549], p. 574.)^{91}
He believed to have reached “a fully satisfactory unified description of gravitation, electromagnetism and a 6vectorial meson.”([547], p. 575.) Schrödinger claimed a further advantage of his approach from the fact that he needed no “special choice of the Lagrangian” in order to make the connection between geometry and physics, and for deriving the field equations.
As to quantum theory, Schrödinger included a disclaimer (in a footnote): “The present article does not touch on it and has therefore to ignore such features in the conventional description of physical fields as are concerned with their quantum character […].” ([549], p. 574.)
“My opinion is that you have the right to speculate, other people including myself have not. […] Honestly, when average people want to procure laws of nature by pure thinking, only rubbish can result. Perhaps Schrödinger can do it. I would love to know what you think about his affine field theories. I find all of it beautiful and full of wit; but whether it is true? […]” ([168], p. 212–213)^{92}
7 Mixed Geometry: Einstein’s New Attempt
After his move to Princeton, Einstein followed quite a few interests different from his later work on UFT. In the second half of the 1930s he investigated equations of motion of point particles in the gravitational field in the framework of his general relativity theory (with N. Rosen L. Infeld, and B. Hoffmann), and the conceptual intricacies of quantum mechanics (with B. Podolsky and N. Rosen). As we have seen in Section 3.2, in 1938–1943 he had turned back to Kaluza’s 5dimensional UFT (with P. G. Bergmann, V. Bargmann, and in one paper with W. Pauli) to which theory he had given his attention previously, in 1926–1928; cf. Section 6.3 of Part I. After joint work with V. Bargmann on bivector (bilocal) fields in 1944 (cf. Section 3.3.1), he took up afresh his ideas on mixed geometry of 1925. He then stayed within this geometrical approach to UFT until the end of his life.
In the period 1923–1933 Einstein had tried one geometry after the other for the construction of UFT, i.e., Eddington’s affine, Cartan’s teleparallel, Kaluza’s 5 dimensional Riemannian geometry, and finally mixed geometry, a blend of affine geometry and Foerster’s (alias Bach’s) idea of using a metric with a skewsymmetric part. Unlike this, after the second world war he stuck to one and the same geometry with asymmetric fundamental tensor and asymmetric affine connection. The problems dealt with by him then were technical at first: what fundamental variables to chose, what “natural” field equations to take, and how to derive these in a satisfactory manner. Next, would the equations chosen be able to provide a set of solutions large enough for physics? Would they admit exact solutions without singularities? In physics, his central interest was directed towards the possibilities for the interpretation of geometrical objects as physical observables. During his life, he believed that the corpus of UFT had not yet become mature enough to allow for a comparison with experiment/observation. His epistemological credo lead him to distrust the probability interpretation of quantum theory as a secure foundation of fundamental physical theory; for him quantum mechanics amounted to no more than a useful “model”. His philosophical position may also have demotivated him to the extent that, already in the late 1930s, he had given up on learning the formalism of quantum field theory in order to be able to follow its further development.^{93} To see him acquire a working knowledge of quantum field theory as a beginner, after World War II, would have been asking too much in view of his age and state of health. That he did not take into account progress in nuclear and elementary particle physics reached in the two decades since he first had looked at mixed geometry, was a further factor isolating him from many of his well known colleagues in theoretical physics.
“But in my work I am more fanatical than ever, and really hope to have solved my old problem of the unity of the physical field. Somehow, however, it is like with an airship with which we can sail through the clouds but not clearly see how to land in reality, i.e on the earth.” ^{94}
Einstein’s first three papers on UFT via mixed geometry ([142, 147, 179]) all employ the metric g and the connection L as independent variables — with altogether 80 available components in local coordinates while just 6 + 10 of them are needed for a description of the gravitational and electromagnetic fields. (The number of the inherent “degrees of freedom” is a more complicated affair.) A strategy followed by Einstein and others seems to have been to remove the superfluous 64 variables in the affine connection by expressing them by the components of the metric, its first derivatives, and the torsion tensor. Since the matter variables were to be included in the geometry, at least in the approach to UFT by Einstein, enough geometrical objects would have to be found in order to represent matter, e.g., 4 components of the electric 4current, 4 components of the magnetic 4current, 5 components for an ideal fluid, more for the unspecified energymomentumtensor in total. In Einstein’s approach, the symmetric part of the metric, h_{ ij }, is assumed to correspond to inertial and gravitational fields while the antisymmetric part k_{ ij } houses the electromagnetic field. The matter variables then are related to derivatives of the metric and connection (cf. Section 10.3.1). The field equations would have to be derived from such a Lagrangian in such a way that general relativity and Maxwell’s equations be contained in UFT as limiting cases. Unlike the situation in general relativity, in metricaffine geometry a twoparameter set of possible Lagrangians linear in the curvature tensor (with the cosmological constant still to be added) does exist if homothetic curvature in (65), (66) of Section 2.3.1 is included. Nevertheless, Einstein used a Lagrangian corresponding (more or less) to the curvature scalar in Riemannian geometry \(\sqrt { {\rm{det(}}{{\rm{g}}_{ik}}{\rm{)}}} \,{g^{lm}}{K_{lm}}(L)\) without further justification.^{95} Such field equations, the main alternatives of which came to be named strong and weak, were used to express the connection as a complicated functional of the metric and its derivatives and to determine the two parts, symmetric and skewsymmetric, of the metric. This was fully achieved not before the 1950s; cf. Section 10.1.
Interestingly, in his second paper of 1945 using mixed geometry, Einstein did not mention his first one of 1925. It seems unrealistic to assume that he had forgotten what he had done twenty years earlier. His papers had been published in the proceedings of the Prussian Academy of Science in Berlin. Possibly, he just did not want to refer to the Prussian Academy from which he had resigned in 1933, and then been thrown out. This is more convincing than anything else; he never ever mentioned his paper of 1925 in a publication after 1933 [312]. There is a small difference between Einstein’s first paper using mixed geometry [142] and his second [147]: He now introduced complexvalued fields on real spacetime in order to apply what he termed “Hermitian symmetry”; cf. (46). After Pauli had observed that the theory could be developed without the independent variables being complex, in his next (third) paper Einstein used “Hermitian symmetry” in a generalized meaning, i.e., as transposition invariance [179]; cf. Section 2.2.2.
7.1 Formal and physical motivation
Once he had chosen geometrical structures, as in mixed geometry, Einstein needed principles for constraining his field equations. What he had called “the principle of general relativity”, i.e., the demand that all physical equations be covariant under arbitrary coordinate transformations (“general covariance”), became also one of the fundamentals of Einstein’s further generalization of general relativity. There, the principle of covariance and the demand for differential equations of 2nd order (in the derivatives) for the field variables had led to a unique Lagrangian \(({1 \over {2k}}R + \Lambda)\sqrt { g}\), with the cosmological constant Λ being the only free parameter. In UFT, with mixed geometry describing spacetime, the situation was less fortunate: From the curvature tensor, two independent scalar invariants could be formed. Moreover, if torsion was used as a separate constructive element offered by this particular geometry, the arbitrariness in the choice of a Lagrangian increased considerably. In principle, in place of the term with a single cosmological constant g_{ ij } Λ, a further term with two constants could be added: h_{ ij } Λ′ and k_{ ij } Λ″.
“Substance, in your sense means the existence of timelike lines of a special kind. This is the translation of the concept of particle to the case of a continuum. […] the necessity of such translatibility, seems totally unreasonable as a theoretical demand. To realize the essential point of atomic thought on the level of a continuum theory, it is sufficient to have a field of high intensity in a spatially small region which, with respect to its “timelike” evolution, satisfies certain conservation laws […].” ([116], p. 95.)^{96}
“[…] writers have occasionally noted the possibility that material particles might be considered as singularities of the field. This point of view, however, we cannot accept at all. For a singularity brings so much arbitrariness into the theory that it actually nullifies its laws.”
As we will see below, an idea tried by Einstein for the reduction of constructive possibilities, amounted to the introduction of additional symmetries like invariance with respect to Hermitian (transposition) substitution, and later, λtransformations; cf. Sections 2.2.2 and 2.2.3. Further comments on these transformations are given in Section 9.8.
Interestingly, the limiting subcase in which the symmetric part of the (asymmetric) metric is assumed to be Minkowskian and which would have lead to a generalization of Maxwell’s theory apparently has been studied rarely as an exact, if only heavily overdetermined theory; cf. however [450, 600, 502].
7.2 Einstein 1945
“What I now do will seem a bit crazy to you, and perhaps it is crazy. […] I consider a space the 4 coordinates x^{1}, … x^{4} which are complex such that in fact it is an 8dimensional space. To each coordinate x^{ i } belongs its complex conjugate x^{ ị }. […] In place of the Riemannian metric another one of the form g_{ ik } obtains. We ask it to be real, i.e., g_{ ik } = ḡ_{ ki } must hold (Hermitian metric). The g_{ ik } are analytical functions of the x^{ i }, and x^{ ị }. […]”^{97}
He then asked for field equations and for complex coordinate transformations. “The problem is that there exist several equations fulfilling these conditions. However, I found out that that this difficulty goes away if attacked correctly, and that one can proceed almost as with Riemann” ([163], p. 367–368).
During the 3 years until he published his next paper in the framework of mixed geometry, Einstein had changed his mind: he stuck to real spacetime and only took the field variables to be complex [147]. He was not the first to dabble in such a mathematical structure. More than a decade before, advised by A. Eddington, Hsin P. Soh^{98}, during his stay at the Massachusetts Institute of Technology, had published a paper on a theory of gravitation and electromagnetism within complex fourdimensional Riemannian geometry with real coordinates. The real part of the metric “[…] is associated with mass (gravitation) and the imaginary part with charge (electromagnetism)” ([581], p. 299).
7.3 EinsteinStraus 1946 and the weak field equations
Einstein and Straus then discussed whether (207) and (208) could be replaced by P_{ ik } = 0. By again looking at the linear approximation, they “get a dependence of the electric from the gravitational field which cannot be brought in accord with our physical knowledge […]” ([179], p. 737).
“really does not yet know, whether this new system of equations has anything to do with physics. What justly can be claimed only is that it represents a consequent generalization of the gravitational equations for empty space.”^{104}
“As long as the Γ cannot be expressed by the g_{ ik,l } in the simplest way, one cannot hope to solve exact problems. Due to the diligence and inventiveness of my assistant Straus, we will have reached this goal, soon.”^{105}
“Schrödinger told me something about you. But I do not know whether you still keep to the field equations which you investigated with Straus at the time of my departure from Princeton (end of February). My personal conviction remains — not the least because of the negative results of your own numerous tries — that classical field theory in whatever form is a completely sucked out lemon from which in no way can spring something new. But I myself do not yet see a path, which leads us further in the principal questions.” ([489], p. 384)
“I labour very hard with my Herr Straus at the verification (or falsification) of my equations. However, we are far from overcoming the mathematical difficulties. It is hard work for which a true mathematician would not at all muster the courage.” ([160], p. 84)^{106}
And, as may be added, for which a genuine true mathematician possibly would not muster enough interest. After all, the task is the resolution of a system of linear equations, wellknown in principle, but hard to control for 64 equations. Nevertheless, Einstein’s assistant in Princeton, E. Straus, in dealing with the weak field equations, continued to work at the problem of solving (206) for the connection. He worked with tensor algebra and presented a formal solution (cf. Eq. (1.9), p. 416 of [592]). However, it was not only unwieldy but useless in practice. Yet, the mathematical difficulty Einstein blamed for the slow progress made, was “the integration of malicious nonlinear equations” (letter to H. Zangger of 28 July 1947 in [560], p. 579).
Another link from Einstein’s Hermitian theory to modern research leads to “massive gravity” theories, i.e., speculative theories describing an empirically unknown spin2 particle (graviton) with mass [76, 255]. However, it is not clear whether these theories are free of ghosts.
8 Schrödinger II: Arbitrary Affine Connection
“I am sending them to nobody else, because you are the only person known to me who is not wearing blinkers in regard to the fundamental questions in our science. The attempt depends on […] the introduction of a nonsymmetric tensor as the only relevant field quantity […]. Pauli stuck out his tongue at me when I told him about it.” (quoted from [446], p. 424.)
In his subsequent letter of 3 March 1946, Einstein pointed to a technical weakness of his theory: “the nonsymmetric tensor is not the most simple structure that is covariant with respect to the group, but decomposes into the independently transforming parts g_{(ik)} and g_{[ik ]}; the consequence of this is that one can obtain a nondescript number of systems of secondorder equations.” ([446], p. 424.)^{108} In both of the preceding papers ([147, 179]), Einstein had not given a single reference to any other publication. Due to the the difficulties concerning transatlantic communication during the war years 1943 and 1944, Einstein possibly might not have seen Schrödinger’s six papers from 1943 and 1944.
“One thing I do know is that my first work [P.R.I.A. 1943] was so imbecilic that it now is repellent to everyone, including you […]. This first work was no advance over ‘Einstein 1923’, but pretended to be.” ([446], p. 426.)
“Because the Lagrangian is left undetermined for the time being, each of the three fields will be represented by two “conjugate” tensorial entities in the field equations, gravitation by ĝ and γ; the skew fields by \(\hat f\) and ϕ and by ŝ and V respectively.”([551], 44.)
In the same approximation, Schrödinger also wrote down gravitational field equations looking like Einstein’s except for the fact that on the side of the matter tensor a number of geometrical objects do appear. They are said to describe the interaction of gravitational and electromagnetic fields as well as of gravitational and mesonic fields. A cosmological term could also be present.
“This encourages one to regard an affine connection of spacetime as the competent geometrical interpretation (from the classical point of view) of the three physical tensor fields we know.” ([551], p. 50)
He questioned, however, that the classical field laws would “be of much help in guessing the true quantum laws of the meson” if they were violently nonlinear.
“We have squandered a lot of time on this thing, and the result looks like a gift from the devil’s grandmother.” ([446], p. 426.)
“I personally am completely convinced — contrary to you as it seems — that for physics nothing reasonable follows from the affine connection without metric. Palatini’s theorem again slams the door. I also believe that each tensor, e.g., the contracted curvature tensor, immediately must be split into a symmetric and a skew part* (* In general: tensors into their irreducible symmetry classes), and to avoid every adding sign between them. What God did separate, humans must not join.) ([489], p. 401)”^{111}
8.1 Schrödinger’s debacle
“Today I can report on a real advance. […] In brief, the situation is this. If in the affine theory, which I have developed in general form in recent years, one takes the special, the only reasonable Lagrange function, namely the square root of the determinant of the Einstein tensor, then one obtains something fabulously good.” ([446], p. 430.)
“The nearer one approaches truth, the simpler things become. I have the honour of laying before you today the keystone of the Affine Field Theory and thereby the solution of a 30 year old problem: the competent generalization of Einstein’s great theory of 1915. The solution was$$\begin{array}{*{20}c} {\quad \quad \quad \quad \quad \delta \int {\mathcal L} = 0} \\ {{\rm{with}}\;{\mathcal L} = \sqrt { \det ({R_{rs}})} \,,} \\ \end{array}$$(229)where Γ is the general affinity of 64 components. That is all. From these three lines my friends would reconstruct the theory, supposing the paper I am handing in got hopelessly lost, and I died on my way home.” ([446], p. 430–432.)$${R_{ik}} \equiv  \underset  K {\,_{ik}} = {{\partial \Gamma _{il}^{l}} \over {\partial {x_k}}}  {{\partial \Gamma _{ik}^{l}} \over {\partial {x_l}}} + \Gamma _{mk}^{l}\Gamma _{il}^{m}  \Gamma _{ml}^{l}\Gamma _{ik}^{m}\,.$$(230)
In the paper submitted together with his presentation, the Lagrangian (229) was given a factor \({2 \over \lambda}\) with a real constant λ ≠ 0 playing an important role ([552], p. 164). Schrödinger first played its occurrence down, unconvincingly though, by saying that it could be transformed to ±1, but in his final field equations, the constant stood for an additional “cosmological” term. In his own words (in a note “added in proof”), his field equations “[…] include ‘the cosmological term’ without containing a cosmological constant.” ([552], p. 171.)
“We now have to endorse the remarkable fact, that the actual content of equations [(236)–(237) …] differs from the theory presented in Einstein’s two papers, quoted above, (i.e., [147, 179]) only by formal λterms.^{113} His theory amounts to putting λ = 0 in (236)–(237). There is a formal difference in that he, from the outset, regards all skew tensors as purely imaginary. […]” ([552], p. 167.)
Schrödinger also confessed that “it may turn out that I have overrated the practical advantage of (233) over (235)–(237).”
“Schrödinger’s latest effort […] can be judged only on the basis of mathematicalformal qualities, but not from the point of view of ‘truth’ (i.e., agreement with the facts of experience). Even from this point of view I can see no special advantages over the theoretical possibilities known before, rather the opposite. […] It seems undesirable to me to present such preliminary attempts to the public in any form.” ([446], p. 432–433.)
and stopped writing to Schrödinger for the next three years. Pauli seemingly had followed the events from Switzerland and wrote to Schrödinger calmly on 9 February 1947:“I was not correct in my objection to your Hamiltonfunction. But your theory does not really differ from mine, only in the presentation and in the ‘cosmological term’ which mine lacks. […] Not your startingpoint but your equations permit a transition to vanishing cosmological constant, then the content of your theory becomes identical with mine” ([446], p. 434),
“Many thanks for your interesting letter of 26. Jan. I would have liked to only respond to it after your first enthusiasm about the new field equations will have given place to a more sober judgment (perhaps the letter is written still too early). Of course, progress is made by your decision to take a specific Lagrangian; also, the mathematical side of your thoughts to me seems extraordinarily clear. Nevertheless, my reservations with regard to a nonirreducible object as a basis continue unabatedly. […]” ([489], p. 415).^{114}
He then expressed in more detail, why for him, only irreducible tensors are the variables to be used. He emphasized that he was not against the “logical possibility” of Schrödinger’s field equations, but could not accept their “necessity and naturalness”. According to Pauli, already before “the next few years”, it would become clear whether these field equations “have something to do with physics, or not.” In a letter to Sommerfeld of 31 October 1947, Pauli agreed with Sommerfeld’s “negative opinion concerning Einstein’s present physics” and supported them with much the same arguments as those given to Schrödinger ([489], p. 475).
“We must not forget, that we are here faced with a truly unitary theory, in which we have to expect all fields to coalesce into an inseparable union, almost as close as that of the electric and magnetic field entailed by Restricted Relativity.” ([552], p. 169.)
The episode differs from Einstein’s repeated claims to have found the final unified field theory, in the 1920s and 30s, in that Einstein did not have to call in the press, and in fact was clever enough not do so. However, his friends in the press were covering his work to the extent, that for each new publication he received the same public attention as Schrödinger in this single case — staged by himself. By his public reaction to Schrödinger, Einstein solidified his position as the opinion leader in research concerning UFT.
8.2 Recovery
“I have found the unitary field equations. They are based on primitive affine geometry, […] Albert did the main job in 1923, but missed the goal by a hair’s breadth. The result is fascinatingly beautiful. I could not sleep a fortnight without dreaming of it.” ([256], p. 168)^{115}
The report of L. Bass that: “After a farcical debacle […], Schrödinger put away the material in a file labeled despondently ‘Die Einstein Schweinerei’ (the Einstein mess)” ([13], p. 120) describes only a momentary halt. After a pause of almost one year, Schrödinger continued his publishing in this topic, i.e., UFT. As a beginning, he surveyed the possibilities for the construction of UFT in a whole class of geometries descending “from the theory of gravitation in empty space by very natural and straightforward generalization without any further artifice.” ([555], p. 205.) He distinguished between the three cases: metrical, affine and mixed geometry depending on whether only g_{ ik }, \({\Gamma _{ik}}^l\), or both can be regarded as independent variables. As a Lagrangian he took ĝ^{ rs }R_{ rs }, with as before \({R_{ik}} =  {\underset  K _{ik}}\). The further classification depended on additional symmetry conditions on the basic variable(s). He dismissed the case of a nonsymmetric metric and symmetric connection “since there is no simple and natural clue” by which the LeviCivita connection should be replaced. It was perhaps this remark which induced J. I. Horváth to suggest “a selection principle for the final theory in the case of the affine theories”, to wit: the field equations must be invariant against changes of the affinities which preserve the parallelism [284]. He derived such transformations (changes) from the weakened condition for autoparallels (cf. Section 2.1.1, after (22)) and arrived, without noting it, at Einstein’s λtransformations (52) introduced before.
“Even if we try to specialize these functions by simplifying arguments, a lot of arbitrariness remains. The impression prevails that the basic geometrical concepts have nothing to do with physics. Einstein did express it like this: ‘the action function then is obtained by leering at another sheet of paper lying next to it, and on which the formulae for another theory can be seen.’” ([489], p. 541–542).^{117}
“[…] an assiduous application of such methods to weak fields is bound to tell us something on the interlacing of three things, gravitational field, electromagnetic field, and electric charges, all three of which spring from one basic conception. […] One may hope that this will provide a better foundation to the quantum mechanical treatment of fields, which at present is based on a number of classical or pseudoclassical field theories of independent origin, cemented together by interaction terms’.” ([558], p. 555.)
As a result he claimed that “a pure chargefree Maxwellian field of radiation is capable of producing a gravitational field which according to the old theory could only be produced by matter other than an electromagnetic field.” By this, the nonvanishing of the trace of a correspondingly defined energymomentum tensor is meant. He also offered three alternatives for an energy (pseudo) tensor which all vanish for a single plane wave. As to physical interpretations, k_{ ij } = g_{[ij ]} is identified with the electromagnetic field with the spacespace components standing for the electrical field. The magnetic 4current vanishes in consequence of the field equations while the electrical 4current is added by hand and given by the expression k_{ ij,l } + k_{ jl,i } + k_{ li,j }. Despite Schrödinger’s going beyond the linear approximation up to quadratic terms, the “[…] influence of both fields [i.e., gravitational and electromagnetic] on the motion of the charges and that of the gravitational field on the electromagnetic” was missing. Thus, the paper contained no new fundamental insights.
8.3 First exact solutions
We shall come back to the demand that exact solutions ought to be free of singularities in Sections 9.6.2 and 10.3.2.“As I’ve said, one does’t get away without singularities in the case of Maxwell’s equations. But no reasonable person believes that Maxwell’s equations can hold rigorously. They are, in suitable cases, first approximations for weak fields. It is now my belief that, for a serious and rigorous field theory, one must insist that the field be free of singularities everywhere.” ([116], p. 93)^{122}
For static metrics, Papapetrou was able to extend a result of Einstein [145], and Einstein & W. Pauli [177] to the strong field equations of Einstein’s UFT: nonsingular static metrics of the strong field equations of UFT which would represent the (gravitational) field of a nonvanishing mass do not exist [474]. Note that this result depends on the identification of the symmetric part of the metric with the gravitational field (potential). A year later, a different proof was given by E. Straus for the weak field equations. In the same paper, Straus concluded: “There exists no static centrally symmetric solution of the field equations which is asymptotically flat and regular throughout” ([592], p. 420). For A. Lichnerowicz’ contribution to the problem of Einstein & Pauli cf. Section 10.5.1.
In view of the research done since, e.g., on black holes or cosmology, the last sentence possibly would not be upheld by him, today.“One may hope that exact solutions, involving strong fields, will reveal the nature of the ultimate particles. I do not believe this, mainly because I do not believe the ultimate particles to be identifiable individuals that could be described in this fashion. Moreover, in the symmetric theory (i.e., in Einstein’s theory of 1916) the exact solutions, involving strong fields, have disclosed the ingenuity of the mathematicians who discovered them, but nothing more. […]” ([558], p. 3)
9 Einstein II: From 1948 on
“Scientifically, I am still lagging because of the same mathematical difficulties which make it impossible for me to affirm or contradict my more general relativistic field theory […]. I will not be able to finish it [the work]; it will be forgotten and at a later time arguably must be rediscovered. It happened this way with so many problems.”^{123}
“Indeed, I am not strongly convinced that this can be achieved with the theory of my continuous field although I have found for it an — until now — apparently reasonable possibility. Yet the calculatory difficulties are so great that I shall bite the dust until I myself have found an assured opinion of it. […]” ^{124}
In spite of such reservations, Einstein carried on unflagging with his research. In his next publication on UFT [148], he again took a complex (asymmetric) metric field. In order to justify this choice in comparison to Schrödinger’s who “has based his affine theory […] on real fields […]”,^{125} he presented the following argument: Just by multiplication and the use of a single complex vector A_{ i } a Hermitian tensor A_{ i }Ā_{ k }. can be constructed. By adding four such terms, the Hermitian metric tensor \({g_{ik}} = \underset \kappa \Sigma \underset \kappa c {\underset \kappa A _i}{\overset {} {\underset \kappa A} _k}\) can be obtained. “A nonsymmetric real tensor cannot be constructed from vectors in such close analogy” ([148], p. 39). Nevertheless, in Einstein’s future papers, the complex metric was dropped.
As in [179], Einstein did not include homothetic curvature into the building of his Lagrangian with the same unconvincing argument: from his (special) field equations and (252) the vanishing of the homothetic curvature would follow.
In his paper, Einstein related mathematical objects to physical observables such that “the antisymmetric density ĝ^{ ikl } plays the role of an electromagnetic vector potential, the tensor ĝ_{[jk ],l} + ĝ_{[kl ],i} + ĝ_{[li ], k} the role of current density.” More precisely, the dual object j^{ s } ∼ ∊^{ sikl } (ĝ_{[ik ],l} + ĝ_{[kl ], i} + ĝ_{[li ],k}) with vanishing divergence \(j_{\,\,\,,s}^s = 0\) is the (electric) current density ([148], p. 39).
“[…] that I have another opinion than you on the question, mentioned in your letter, of the physical usability of singularityfree solutions of classical field equations. To me it deems that, even if such solutions do exist in a suitably chosen field theory, it would not be possible to relate them with the atomic facts in physics in the way you wish, namely in a way that avoids the statistical interpretation, in principle.” ([489], p. 621.)^{128}
9.1 A period of undecidedness (1949/50)
“Now you will ask me: Did God tell this into your ear? Unfortunately, not. But the way of proceeding is: identities between the equations must exist such that they are compatible. […] For their compatibility, i.e., that continuation from a [time] slice is possible, there must be 6 identities. These identities are the means to find the equations. […]” ([163], p. 410).^{130}
“I recently found a very forceful derivation for this system; it shows that the equations follow from the generalized field as naturally as the gravitational equations from the postulate of the symmetric field g_{(ik)}. The examination of the theory still meets with almost unsurmountable mathematical difficulties […]” ([163], p. 423).^{131}
“On the other hand, one is pushed to the stronger system by formal considerations […]. But the compatibility for this stronger system is problematic; i.e., at first one does not know whether the manifold of its solutions is sufficiently large. After many errors and efforts I have succeeded in proving this compatibility” ([163], p. 439).^{132}
“Apart from [coordinate] transformation invariance, invariance also is assumed for the transformations of the nonsymmetric ‘displacement field’ \(\Gamma _{ik}^{\,\,\,\,\,l}:\,{(\Gamma _{ik}^{\,\,\,\,\,\,l})^*} = \Gamma _{ik}^{\,\,\,\,\,l} + \delta _i^l{\lambda _k}\), where λ_{ k } is an arbitrary vector. In this extended group, the old gravitational equations are no longer covariant […].”([163], p. 465)^{133}
We will come back to his final decision in Section 9.2.3.
9.1.1 Birthday celebrations
“The temptation is great to mention here some of the endeavors that have been made to utilize these more general geometries for setting up unified field theories encompassing the electromagnetic field beside the gravitational one or even including not only the photons but also the electrons, nucleons, mesons, and whatnot. I shall not succumb to that temptation.” ([693], p. 539.)
“[…] by abstract reasoning from a few general assumptions. In this he is following the heroic method that proved so successful […] in the theory of relativity. Unfortunately there are many possible approaches, and since each requires a year or more of intensive computation, progress has been heartbreakingly slow.”
That Hoffmann himself was a little outside of mainstream physics can be seen from his remark that quantum theory, now dominating physics, “has developed a stature comparable to that of the theory of relativity.” ([281], p. 54/55.) Hoffmann was also one of the contributors to the special number of Reviews in Modern Physics “in commemoration of the seventieth birthday of Albert Einstein” issued in September 1949. Possibly, the best remembered paper among the 38 articles is Gödel’s “new type of cosmological solutions”, with local rotation and closed timelike world lines, now just named “Gödel’s solution” [227]. Only E. Straus wrote an article about UFT: “Some results in Einstein’s unified field theory” [592]. The others, big names and lesser known contributors except for the mathematician J. A. Schouten, shunned this topic. Schouten’s contribution surveyed classical meson theories in view of their making contact with the conformal group [539]. In connection with Yukawa’s prediction of a meson and with Hoffmann’s similarity geometry (cf. Section 3.1), he boldly stated: “[…] the conformal field theory failed to ask for a meson field, but the meson field came and asked for a conformal theory!” (ibid., p. 423.) Einstein’s oldest son Hans Albert reported on “Hydrodynamic Forces on a Rough Wall” [180].
“that these equations constitute the most natural generalization of the equations of gravitation. The proof of their physical usefulness is a tremendously difficult task, inasmuch as mere approximations will not suffice. The question is: ‘What are the everywhere regular solutions of these equations?’” ([153], p. 93–94.)^{134}
9.2 Einstein 1950
9.2.1 Alternative derivation of the field equations
9.2.2 A summary for a wider circle
“[…] Einstein will set forth what some of his friends say is the longsought unified field theory. The scientist himself has given no public hint of any such extraordinary development, but he is said to have told close associates at the Institute for Advanced Studies that he regards the new theory as his greatest achievement” ([564], p. 26).
“Soon I will also send you the new edition of my little book with the appendix. A few weeks ago, it has caused a loud rustling noise in the newspaper sheets although nobody except the translator had really seen the thing. It’s really drole: laurels in advance” ([160], p. 96).^{138}
 (1)
all equations remain unchanged with respect to simultaneous substitution of the g_{ ik } and \(\Gamma _{ik}^{\,\,\,\,\,l}\) by \({\tilde g_{ik}}\) and \(\tilde \Gamma _{ik}^{\,\,\,\,\,\,l}\) (transposition invariance),
 (2)
all contractions of the curvature tensor (54) vanish,
 (3)
that (30) hold,
 (1)
(269) shows that there is no magnetic current density present (no magnetic monopoles),
 (2)
the electric current density (or its dual vector density) is represented by the tensor g_{[ik ],l} + g_{[kl ],i} + g_{[li ], k}).
The “eminent author” himself confessed in a letter to Max Born of 12 December 1951:“Nevertheless, what has been written here shows how much of the subsequent formulation appears to be entirely arbitrary and how little of it has received physical interpretation. It is clear that a tremendous amount of investigation is required before others than the eminent author himself are enabled to form an opinion of the significance of this work” ([420], p. 129).
“Unfortunaletly, the examination of the theory is much too difficult for me. After all, a human being is only a poor wretch!” ([168], p. 258).^{143}
Einstein’s former assistant and coauthor Leopold Infeld sounded quite skeptical as well when he put the focus on equations of motion of charges to follow from “the new Einstein theory”. By referring to the 3rd Princeton edition of The Meaning of Relativity he claimed that, in 1st approximation, “the equations of motion remain Newtonian and are uninfluenced by the electromagnetic field.” But he offered immediate comfort by the possibility “that this negative result is no fault of Einstein’s theory, but of the conventional interpretation by which it was derived” [303].“Regarding Einstein’s ‘unified’ field theory I am extremely skeptical. It seems not only arbitrary to add a symmetrical and an antisymmetrical tensor together but there is also no reason why Einstein’s system of equations should be compatible (the counting of identities between these equations given in the appendix of the new edition turned out to be incorrect). Certainly no work on similar lines will be done in Zürich.” ([490], p. 137–138)
9.2.3 Compatibility defined more precisely
In a long paragraph (§7) of Appendix II, of this 3rd Princeton edition, Einstein then asked about the definition of what he had termed “compatibility”. This meant that “the manifold of solutions” of the different systems of field equations “is extensive enough to satisfy the requirements of a physical theory” ([150], p. 150), or put differently, the field equations should not be overdetermined. In view of the “System I”’s containing four more equations, i.e., 84, than the 80 unknowns, this might become a difficulty. Starting from the Cauchy problem, i.e., the time development of a solution off an initial hypersurface, he counted differential equations and the variables to be determined from them.^{144} To give an example for his method, he first dealt with general relativity and obtained the result that the general solution contains four free functions of three (spacelike) coordinates — “apart from the functions necessary for the determination of the coordinate system” ([150], p. 155). The corresponding results for “Systems Ia, (I)” according to him turned out to be: 16, (6) arbitrary functions of three variables, respectively. In case “System I” should turn out to be too restrictive to be acceptable as a physical theory, Einstein then would opt for the “weak field equations” (“System Ib”). “However, it must be admitted that in this case the theory would be much less convincing than if system (I) can be preserved” ([150], p. 160).
This discussion calls back into memory the intensive correspondence Einstein had carried on between 1929 and 1932 with the French mathematician E. Cartan on an equivalent problem within the theory of teleparallelism, cf. Section 6.4.3 of Part I. At the time, he had asked whether his partial differential equations (PDEs) had a large enough set of solutions. Cartan had suggested an “index of generality” s_{0} for firstorder systems in involution which, essentially, gave the number of arbitrarily describable free data (functions of 3 spacelike variables) on an initial hypersurface (t = t_{0}). He calculated such indices, for Maxwell’s equations with currents to be s_{0} = 8, and without s_{0} = 4, for Einstein’s vacuum field equations s_{0} = 4, (in this case 4 free functions of 4 variables exist^{145}), and of course, for Einstein’s field equations in teleparallelism theory. Note that Maxwell’s and Einstein’s vacuum field equations according to Cartan exhibit the same degree of generality. It had taken Cartan a considerable effort of convincing Einstein of the meaningfulness of his calculations also for physics ([116], pp. 114, 147, 174). Already in the 3rd Princeton edition of The Meaning of Relativity, in Appendix II [150], Einstein tried to get to a conclusion concerning the compatibility of his equations by counting the independent degrees of freedom but made a mistake. As mentioned above, W. Pauli had noticed this and combined it with another statement of his rejection of the theory. Compatibility was shown later by A. Lichnerowicz [369] (cf. Section 10.5).
The fact that Einstein had to correct himself in his calculations of the “coefficient of freedom” already may raise a feeling that there exists a considerable leeway in redefining field variables, number and order of equations etc. Moreover, he did not prove the independence of the relative order of “strength” for two PDE’s from mathematical manipulations affecting the form of the equations but not their physical content: remember (279), (282), (283). Regrettably, it is to be noted that Einstein’s last attempt to gain a reliable mathematical criterion for singling out one among the many possible choices for the field equations in UFT remained unconvincing.^{149}
9.2.4 An account for a general public
“[…] a detailed account of it before a group of readers interested in science. That should be done only with theories which have been adequately confirmed by experiment.” ([152], p. 14.) ^{150}
“The skeptic will say: ‘It may well be true that this system of equations is reasonable from a logical standpoint. But this does not prove that it corresponds to nature.’ You are right, dear skeptic. Experience alone can decide the truth. Yet we have achieved something if we have succeeded in formulating a meaningful and precise question.” ([152], p. 17.)
Painstaking efforts and probably new mathematics would be required before the theory could be confronted with experiment. The article is illustrated by a drawing of Einstein’s head by the American artist Ben Shahn ([152], p. 17).
There were not only skeptics but people like Dr. C. P. Johnson in the Chemistry Department of Harvard University who outrightly criticized “Dr. Albert Einstein’s recent unified field theory” [311]. He pointed out that the theory permits a class of similarity solutions, i.e., with g_{ ik } (x^{ l }) also ′g_{ ik } (kx^{ l }) solves the field equations. For a system of two charged and one uncharged massive bodies he qualitatively constructed a contradiction with Coulomb’s law. Einstein replied with a letter printed right after Johnson’s by stating that if solutions depending upon a continuous parameter exist, “then the field equations must prevent the coexistence within one system of such elementary solutions pertaining to arbitrary values of their parameters.” The underlying reason was that “for a system of field equations to be acceptable from a physical point of view, it has to account for the atomic structure of reality.” This would entail that regions of space corresponding to a ‘particle’ have discrete masses and charges. The coexistence of similar solutions “in one and the same world” would make the theory unacceptable [161]. As we shall see, the situation of Einstein’s UFT was worse: it did not lead to Coulomb’s law — at least not in the lowest approximations. See Sections 9.3.3, 9.6, and Section 10.3.2.
His Italian colleague Bruno Finzi was convinced that the final aim had been reached:“At long last, the generalization of gravitation from a formal point of view now is fully convincing and unambiguous — unless the Lord has chosen a totally different way which no one can imagine.”^{151}
“[…] all physical laws laws of the macrocosm reduce to two geometrical identities […]. Therefore, the game is over, and the geometric model of the macrocosm has been constructed.”([200], p. 83)^{152}
However, at the end of his article, Finzi pointed out that it might be difficult to experimentally verify the theory, and thought it necessary to warn that even if such an empirical base had been established, this theory would have to be abandoned after new effects not covered by it were observed.
9.3 Einstein 1953
In a discussion covering twelve pages, Einstein again took up the question of “compatibility” from the previous edition and introduced the concept of the “strength” of a system of differential equations in order to bolster up his choice of field equation. A new principle applying to physical theories in general is put forward: “The system of equations is to be chosen so that the field quantities are determined as strongly as possible ” ([156], p. 149). In Section 9.2.3, a detailed discussion of this new principle has been given such that we need not dwell on it. The paucity of physical input into Einstein’s approach to UFT becomes obvious here. May it suffice to say that according to the new principle the weak field equations (277), (278) are called “stronger” than the strong field equations (268). However, this has lead to the misleading labeling of the system II as the “strong system” [704]. The relation of geometrical objects to physical observables remained unchanged when compared to the 3rd edition ([150]). Einstein saw a close relationship to Maxwell’s theory only in the linear approximation where “the system decomposes into two sets of equations, one for the symmetric components of the field, and the other for the antisymmetric components.[…] In the rigorous theory this independence no longer holds.” ([156], p. 147.)
“[…] I see in the present situation no possible way other than a pure field theory, which then however has before it the gigantic task of deriving the atomic character of energy. […] We are […] separated by an as yet insurmountable barrier from the possibility of confronting the theory with experiment. Nevertheless, I consider it unjustified to assert, a priori, that such a theory is unable to cope with the atomic character of energy.” ([156], p. 165.)
“The minority who reject the theory, although led by the great names of Albert Einstein and P. A. M. Dirac, do not yet have any workable alternative to put in its place.” ([137], p. 57.)
Such kind of sober judgment did not bother The New York Times which carried an almost predictable headline: “Einstein Offers New Theory to Unify Law of the Cosmos.” ([469], p. 350.)
Privately, in a letter to M. Solovine of 28 May 1953, Einstein seemed less assured. Referring to the appendix of this 4th edition of “The Meaning of Relativity ”, he said: “[…] Of course, it is the attempt at a theory of the total field; but I did not wish to give the thing such a demanding name. Because I do not know, whether there is physical truth in it. From the viewpoint of a deductive theory, it may be perfect (economy of independent concepts and hypotheses).” ([160], p. 96).^{153}
9.3.1 Joint publications with B. Kaufman
“My endeavours to complete general relativity by a generalization of the gravitational equations owe their origin partially to the following conjecture: A reasonable general relativistic field theory could perhaps provide the key to a more perfect quantum theory. This is a modest hope, but in no way a creed.” ([155], p. 14.)^{156}
Here, the covariant derivative refers to the connection *Γ and \({\underset * K _{(ik)}} \equiv {\underset  K _{ik}}(*\Gamma)\).
In an appendix to the paper with title “Extension of the Relativistic Group” [172], Einstein combined the “group” of coordinate transformations with the λtransformations to form a larger transformation group U. (cf. the letter to Besso mentioned in Section 9.1.) He then discussed the occurring geometric objects as representations of this larger group and concluded: “The importance of the extension of the transformation group to U consists in a practically unique determination of the field equation.”([172], p. 341.)^{158}
The next paper with Bruria Kaufman may be described as applied mathematics [173]. Einstein returned to the problem, already attacked in the paper with E. Straus, of solving (30) for the connection in terms of the metric and its derivatives. The authors first addressed the question: “What are necessary and sufficient conditions for constant signature of the asymmetric metricfield to hold everywhere in spacetime?” At first, it was to be shown “that the symmetric part g_{(ik)} of the tensor g_{ ik } is a Riemannian metric with constant signature”. For a proof, the conditions det(g_{(ik)}) ≠ 0 and a further algebraic inequality were needed. In addition, the connection Γ, calculated from \({\hat g_{\mathop + \limits^i \mathop  \limits^k \left\ l \right.}} = 0\), had to be finite at any point and “algebraically determined”. This is meant in the sense of interpreting \({\hat g_{\mathop + \limits^i \mathop  \limits^k \left\ l \right.}} = 0\) as an inhomogeneous linear equation for the components of Γ. The situation was complicated by the existence of the algebraic invariants of the nonsymmetric g_{ ik } as well as by the difficulty to solve for the connection as a functional of the metric tensor. Although not necessary for a solution of the field equations, according to the authors it is “of interest to give a closed expression for the Γ as a function of the g_{ ik } and its first derivatives.” This problem had been addressed before and partial results achieved by V. Hlavatý [258, 260], and S. N. Bose [52].^{159} The papers by M.A. Tonnelat published earlier and presenting a solution were not referred to at all [622, 623, 630, 629].
9.3.2 Einstein’s 74th birthday (1953)
“A new theory often only gradually assumes a stable, definite form when later findings allow the making of a specific choice among the possibilities given a priori. This development is closed now in the sense that the form of the field laws is completely fixed. — The theory’s mathematical consistence cannot be denied. Yet, the question about its physical foundation still is completely unsettled. This follows from the fact that comparison with experience is bound to the discovery of exact solutions of the field equations which seems impossible at the time being.”^{161} ([570], p. 401–402)
9.3.3 Critical views: variant field equation
Already in 1950, Infeld had pointed to the fact that the equations of motion for particles following from Einstein’s UFT (weak field equations), calculated in the same way as in general relativity, did not lead to the Lorentz equations of motion [304]. This result was confirmed by Callaway in 1953. Callaway identified the skew part of the fundamental tensor with the electromagnetic field and applied a quasistatic approximation built after the methods of Einstein and Infeld for deriving equations of motion for point singularities. He started from Einstein’s weak field equations and showed that (208) could not influence the equations of motion. His conclusion was that he could reduce “Einstein’s new unified field theory to something like Maxwell’s equations in a sufficiently low approximation”, but could not obtain the Lorentz equation for charged particles treated as singularities in an electromagnetic field [69].
In fact, as Bonnor then showed in the lowest approximation (linear in the gravitational, quadratic in the electromagnetic field), the static spherically symmetric solution contains only two arbitrary constants e, m besides p^{2} which can be identified with elementary charge and mass; they are separately selectable [33]. However, in place of the charge appearing in the solutions of the EinsteinMaxwell theory, now for e^{2} the expression e^{2}p^{2}, and for \(e,\,{{me} \over {{p^2}}}\) occurred in the same solution. The definition of mass seemed to be open, now. For vanishing electromagnetic field, the solution reduced to the solution for the gravitational field of general relativity.
In a discussion concerning the relation of matter and geometry, viz. matter as a “source” of geometry or as an intrinsic part of it, exemplified by the question of the validity of Mach’s principle, J. Callaway tried to mediate between the point of view of A. Einstein with his unified field theory already incorporating matter, geometrically, and the standpoint of J. A. Wheeler who hoped for additional relations between matter and spacetime fixing the matter tensor as in the case of the EinsteinMaxwell theory ([70], p. 779). Callaway concluded that “if the approach of field theory is accepted, it is necessary to construct a theory in which spacetime and matter enter as equals.” But he would not accept UFT as an alternative to quantum theory.
9.4 Einstein 1954/55
“For I have succeeded — in part in collaboration with my assistant B. Kaufman — in simplifying the derivations as well as the form of the field equations. The whole theory becomes thereby more transparent, without changing its content” ([158], page before p. 1).^{165}
From a letter to his friend Solovine in Paris of 27 February 1955, we note that Einstein was glad: “At least, yet another significant improvement of the general theory of the gravitational field (nonsymmetric field theory) has been found. However, the thus simplified equations also cannot be examined by the facts because of mathematical difficulties”. ([160], p. 138)^{166} In this edition of The Meaning of Relativity’, he made a “remark on the physical interpretation”. It amounted to assign \(\hat g_{\,\,\,\,\,,s}^{[is]}\) to the (vanishing) magnetic current density and \({1 \over 2}{\eta ^{iklm}}{g_{[ik],l}}\) to the electric current density.
However, it is to be noted that in the 5th Princeton edition the λtransformation is reduced to λ_{ k } = ∂_{ k }λ (Eq. (5) on p. 148). In a footnote, Appendix II of the 4th Princeton edition of The Meaning of Relativity is given as a reference for the concept of “strength” of a system of differential equations (cf. Section 9.2.3). W. Pauli must have raised some critical questions with regard to the construction of the paper’s Lagrangian from irreducible quantities. In her answer of 28 February 1956, B. Kaufman defended the joint work with Einstein by discussing an expression (α_{1}g^{ ik } + α_{2}g^{ ki })R_{ ik } = γ^{ ik }R_{ ik }: “Now the point is here that g^{ ik } was introduced in our paper merely as a multiplying function such as to make, together with R_{ ik }, a scalar. Hence g^{ ik } can just as well be this multiplier. The field equations we would get from this Lagrangian would be identical with the equations in our paper, except that they would be expressed in terms of g^{ ik }.” As to scalars quadratic in curvature she wrote: “ […] our paper does not claim that the system we give is 100% unique. In order to do that one would have to survey all possible additional tensors which could be used in the Lagrangian. We only considered the most ‘reasonable’ ones.” ([492], pp. 526–527.)“In a system with no λinvariance, there are 64 Γ and no counterbalancing identity. This is the deeper reason for the relative weakness of systems which lack λinvariance. We hold to the principle that the stronger system has to be preferred to any weaker system, as long as there are no special reasons to the contrary.” ([174], p. 137.)
Until 1955, more than a dozen people had joined the research on UFT and had published papers. Nevertheless, apart from a mentioning of H. Weyl’s name (in connection with the derivation of the “Bianchi”identities) no other author is referred to in the paper. B. Kaufman was well aware of this and would try to mend this lacuna in the same year, after Einstein had passed away.
At the “Jubilee Conference” in Bern in July 1955,^{167} based on her recent work with A. Einstein [174], B. Kaufman gave an account “[…] of the logical steps through which one goes when trying to set up this generalization”, i.e., of general relativity to the “theory of the nonsymmetric field” ([322], p. 227). After she presented essential parts of the joint paper with Einstein, Kaufman discussed its physical interpretation and some of the consequences of the theory. As in [148, 150], and [156], the electric current density is taken to be proportional to g_{[ik ],l} + g_{[kl ],i} + g_{[li ],k}. From this identification, transposition invariance receives its physical meaning as showing that “all equations of the theory shall be invariant under a change of the sign of electric charge” ([322], p. 229). With (252), i.e., \({\hat g^{[ik]}}_{\,\,\,\,\,\,\,,k} = 0\), holding again in the theory, ĝ^{ i } ^{4} (with i = 1, 2, 3) is identified with the components of the magnetic field. In the linear approximation, the field equations decompose into the linear approximation of the gravitational field equations of general relativity and into the weaker form of Maxwell’s equations already shown in (210), (211) of Section 7.3.
“We have seen how Einstein and Mrs. Kaufman struggled heroically […], and how this fight has been led with the particular weapon of the λtransformation. Certainly, all this is formally very correct; however, I was unable to make sense of the λtransformations, either physically or geometrically.”[486]^{168}
The search for solutions of the weak field equations had begun already with exact spherically symmetric, static solution derived by a number of authors (cf. [475, 31, 32]; see Sections 8.3, and 9.6).
“[…] as anything like a wellestablished theory. It must be confessed that we have as yet no glimpse of how to represent electrodynamic interaction, say Coulomb’s law. This is a serious desideratum. On the other hand we ought not to be disheartened by proofs, offered recently by L. Infeld, M. Ikeda and others, to the effect, that this theory cannot possibly account for the known facts about electrodynamic interaction. Some of these attempts are ingenious, but none of them is really conclusive.” ([557], reprint 1954, p. 119.)
9.5 Reactions to EinsteinKaufman
In 2nd order, the chargecurrent tensor was defined by \({\underset 2 s}\;_{ijk} = {\underset 2 g}\;_{[ij],k} + {\underset 2 g}\;_{[jk],i} + {\underset 2 g}\;_{[ki],j}\), and the wave equation then \({\eta ^{rs}}{\partial _r}{\partial _s}{\underset 2 g}\;_{[ik]} = {\underset 2 s}\;_{ijk} + {\eta ^{rm}}{\eta ^{sn}}{\underset 1 g}\; _{[rs]}{\underset 1 B}\;_{nmik}\). \({\underset 2 B}\;_{nmik}\) is a linear combination of the 2nd derivatives of \({\underset 1 g _{ik}}\). However, Schrödinger rejected this equation: “it is not invariant” ([559], p. 19). Since 1952, Cornelius Lanczos had come to Dublin, first as a visiting, then as a senior professor, and, ultimately, as director at the Dublin Institute for Advanced Studies. In his paper, Schrödinger acknowledged “discussions with my friend professor Cornel Lanczos” ([559], p. 20).
“[…] in the future, no reason exists for connecting the metric more closely to the electromagnetic field, and perhaps also to the meson field.” ([680], p. 141.)^{170}
One year later, consistent with this, and with Einstein’s death “in April 1955, Schrödinger became quite depressed, for he was now convinced that his unified field theory was no longer tenable” ([446], p. 326). In any case, there is no further published research on UFT by him.
In a joint paper with M. L. Abrol, also directed to the EinsteinKaufman version of Einstein’s theory, Mishra claimed: “It is shown […] that Infeld’s method [cf. [304]] of approximation, to find the equations of motion of charged particles from the system of field equations, fails in this particular theory” [437]. This was due to some unknown terms in the 2nd and 3rd order of the approximation. After a modification of the field equations according to the method of Bonnor [cf. [34]], the Coulomb force appeared in 4th order.
9.6 More exact solutions
9.6.1 Spherically symmetric solutions
Wyman also questioned the boundary condition used at spacelike infinity: limes_{r→ ∞} g_{ ab } = η_{ ab }, where η_{ ab } is the Minkowski metric. By looking at his (or Papapetrou’s) sss solutions, he showed that different boundary condition could be set up leading to different solutions: vr^{2} → 0 for r → ∞ or v → 0 for r → ∞.
In her book, M.A. Tonnelat discussed these solutions; her new contribution consisted in the calculation of the components of her connection Δ — Schrödinger’s star connection (27) — and the Ricci W (Δ)tensor formed from it for the more general case of timedependent spherically symmetric fields [629], ([632], p. 71, 73). By help of this calculation, her collaborator Stamatia Mavridès could present a general result: for g_{[23]} = 0, g_{[10]} ≠ 0 (as the nonvanishing components of k_{ ij }) only static exact spherically symmetric solutions do exist [402]. Later in Italy, F. De Simoni published another generalization of Wyman’s and Bonnor’s solutions for the weak field equations; he used the Ricci tensor of Einstein and Straus (73) made Hermitian, i.e., \({P_{ij}} + {\tilde P_{ij}}\). His paper is not referred to in Tonnelat’s book [114]. J. R. Vanstone mistakenly believed he had found timedependent spherically symmetric solutions, but the time dependency can be easily removed by a coordinate transformation [668]. Also B. R. Rao had calculated some, but not all components of the connection for the case of a timedependent spherically symmetric field but had failed to find a timedependent solution [502].
Unfortunately, all this work did not bring further insight into the physical nature of the sss solutions. The only physically “usable” solution remained Papapetrou’s. He also proved the following theorem: “Sphericallysymmetric solutions periodical in time of the “weak” field equations satisfying the boundary conditions ĝ^{ ik } → η^{ ik } for r → ∞ are, in 1st approximation, identical to solutions of the “strong”field equations” [478].
9.6.2 Other solutions
“[…] for a theory based on a nonsymmetric tensor an infinity of tensors of all orders exist. The only hope to extract from this maze the proper mathematical expressions to use for physical quantities would thus have to be physical in nature. So far no such physical assumptions have been put forward” ([710], p. 229).
This result casts into doubt much of the work on exact solutions independently of any specific assignment of mathematical objects to physical variables. It vindicated Schrödinger’s opinion that exact solutions were of useless for a better understanding of the particleaspect of the theory; cf. the quotation at the end of Section 8.2. Nevertheless, the work of assembling a treasure of exact solutions continued. In 1954, it had still been supported by Kilmister & Stephenson in this way: “The true test of this theory [i.e., Einstein’s weak field equations] as an adequate description of the physical world must await exact solutions of the field equations” [331].
“The big difficulty [of UFT] lies in the lack of a method for deriving singularityfree exact solutions which alone are physically interesting. Yet the bit we have been able to calculate has strengthened my trust in this theory.” (Einstein to Pauli, April 1, 1948 quoted from [489])^{176}
How would he have dealt with the fact, unearthed in 1958, that such nonsingular solutions not always offered a convincing physical interpretation, or even were unphysical?
The symmetry of socalled “1dimensional” gravitational fields of general relativity, i.e., those for which the metric components depend on only a single coordinate, is high enough to try and solve for them field equations of UFT. In fact, already in 1951, Bandyopadhyay had found such a solution of the weak equations with g_{[10]} ≠ 0, g_{[23]} = 0, g_{22} = g_{33} and had taken it as describing an infinite charged plate [7]. In 1953, E. Clauser presented another such 1dimensional field as a solution of the weak equations with g_{22} ≠ g_{33} and saw it as representing a magnetostatic field [79]. B. R. Rao in 1959 generalized Bandyopadhyay’s solution to the case g_{[10]} × g_{[23]} ≠ 0 without attempting to provide a physical interpretation [503].
9.7 Interpretative problems
Already up to here, diverse assignments of geometrical objects to physical quantities (observables) were encountered. We now assemble the most common selections.
9.7.1 a) Gravitational Field
9.7.2 b) Electromagnetic Field and Charge Currents
An ambiguity always present is the assignment of the electric and magnetic fields to the components k_{i0} or k_{ ab }, a,b =1, 2, 3, or vice versa in order to arrive at the correct Maxwell’s equations.
Another object lending itself to identification with the electromagnetic field would be homothetic curvature encountered in Section 2.3.1, i.e., \({\underset + V _{kl}} = {\underset + K ^j}_{jkl} = {\partial _k}L_{li}^{\,\,\,\,\,j}  {\partial _l}L_{kj}^{\,\,\,\,\,i}:L_{lj}^{\,\,\,\,i}\) could then play the role of the vector potential. This choice has been made by Sciama, but with a complex curvature tensor \({\overset s K ^j}_{jkl}\) [565]. In this case \({\overset s K ^j}_{jkl} = {\partial _k}L_{[lj]}^{\,\,\,\,\,\,\,i}  {\partial _l}L_{[kj]}^{\,\,\,\,\,\,\,\,j} = 2{S_{[j,k]}}\) The vector potential thus is identified with the torsion vector.
It is obvious that the assumed mapping of geometrical objects to physical variables had to remain highly ambiguous because the only arguments available were the consistency of the interpretation within unified field theory and the limit to the previous theories (EinsteinMaxwell theory, general relativity), thought to be necessarily encased in UFT. As we have seen, the hope of an eventual help from exact solutions had to be abandoned.
9.7.3 c) Matter tensor
“Thus it is as yet undecided what interpretation of the various tensors and densities is most likely to let the theory meet observed facts” ([557], reprinted 1963, p. 115).
9.8 The role of additional symmetries
P. G. Bergmann also discussed Einstein’s λtransformations, but just in the special form used in the 5th edition of The Meaning of Relativity, i.e., with λ_{ k } = ∂_{ k }λ. No wonder that he then concluded: “[…] the λ transformation appears to be closely related in its conception to Weyl’s original gauge transformation” ([23], p. 780).
10 EinsteinSchrödinger Theory in Paris
Research on unified field theory in Paris centered around the mathematician A. Lichnerowicz, a student of Georges Darmois, and the theoretical physicist MarieAntoinette Tonnelat. It followed two main lines: the affine or metricaffine approaches of Einstein and Schrödinger, and the 5dimensional unification originating with G. Nordström and Th. Kaluza. The latter theme was first studied in Paris by Y. Thiry, a former student of A. Lichnerowicz (“JordanThirytheory”), and by students of M.A. Tonnelat; the first topic, “EinsteinSchrödinger theory”, mainly by Tonnelat and her coworkers but no strict divide did exist. Between 1950 and the mid 1960s at least two dozen doctoral theses on topics in unitary field theory were advised by Lichnerowicz and/or by Tonnelat. Whereas the work of Tonnelat’s students could be classified as applied mathematics, Lichnerowicz’s interest, outside of pure mathematics, was directed to mathematical physics with its rigid proofs. This joined attack on unsolved questions and problematic features of classical unified field theory has made clear that (1) the theories under scrutiny were mathematically consistent, but (2) they could not be transformed into an acceptable part of physics.
10.1 MarieAntoinette Tonnelat and Einstein’s Unified Field Theory
“[…] the theory of maximal spin 2 allows to show how a unitary theory presents itself, approximately, but in the framework of wave mechanics.”^{182}
In this paper and in others in the early 1940s she also wrote down the standard commutation relations for the quantized spin2 field [611, 616].
“Despite the ordeal which oppresses the country, the Comptes Rendus attest that scientific research has not bent, and that the Academy of Sciences remains a focus of ardent and fruitful work. […] Let us work.”^{184}
“Einstein’s efforts in this direction, ever characterized by the salient originality of his thought, will not be examined here. Despite their indisputable interest, they have not, to the best of our knowledge, attained any decisive success […]. Moreover, the nature of the electromagnetic field is so intimately bound to the existence of quantum phenomena that any nonquantum unified theory is necessarily incomplete. These are problems of redoubtable complexity whose solution is still ‘in the lap of the gods’ ” ([113], p. 121).
At the 8th Solvay Congress in 1948 in Brussels, Mme. Tonnelat presented a paper by L. de Broglie on the photon as composed of two neutrinos. Schröodinger asked a question afterwards about a tiny mass of the photon ([446], p. 444). During her work on unified field theory Tonnelat continued to study spinparticles, e.g., to regard a spin1particle as composed of two spin1/2particles [625].
10.2 Tonnelat’s research on UFT in 1946–1952
By her, the torsion tensor \(\Lambda _{ij}^{\,\,\,\,\,k}\) is defined through“The author states without proof some formal consequences of a variational principle in which the action function is an unspecified function of a symmetric second order tensor and three (of which two are independent) antisymmetric second order tensors. These four tensors are defined in terms of a general affine connection in a four dimensional space, and its derivatives. The connection is not assumed to be symmetric. The paper does not explain how the invariant element of volume entering into the action principle is defined.”
“[…] the possibility remains of finding, thanks to the exact solution to the equation \(0{= _{g\,\underset + i \,\underset  k \Vert l}}\), a solution valid even in the case of strong fields, an explanation of the nature of the elementary particles. However, as Schrödinger very strongly emphasized, the realization of this hope remains quite problematical despite all efforts.” ([626], p. 832) ^{191}
A comparison of M.A. Tonnelat’s research with respect to Einstein’s and Schroödinger’s shows that, though first generalizing the class of possible Lagrangians enormously by including four tensor fields, in the end she went back to only one: she used Schrödinger’s Lagrangian corresponding to Einstein’s Lagrangian for general relativity. She avoided the additional equation within the field equations which demands that the torsion vector vanish by directly starting with a connection with zero torsion vector. Although this approach was new,^{195} most characteristic and important for her research seems to be that she directed her attention to “metric compatibility” in the sense of (200) of Section 7.2 and succeeded to “solve” it for the connection; cf. Section 10.2.3. She also showed that out of a purely affine theory, by proper definitions and interpretations, a theory within mixed geometry could be made. It was such a theory that she finally adopted.
10.2.1 Summaries by Tonnelat of her work
“In this book the author summarizes and discusses a great body of material on the Einstein and Schrödinger unified field theories. […] The previous work of the author is collected and presented in a logical coherent fashion. The results obtained by other workers are also presented and compared. Thus, in this single volume containing an introduction and seven chapters one can obtain a well written complete and succinct account of the recent work in the field.”
Tonnelat’s associate J. Winogradzki, in her report on the book, gave a condensed list of the contents and found “that the major part of the work is devoted to the mathematical study of the field equations. The two last chapters deal with some physical content of the theory” [705]. Tonnelat clearly drew the line with regard to work by Lichnerowicz, e.g., the initial value problem. For her, the balance between the remaining problems of UFT and the results obtained was positive: “[…] Einstein’s theory binds together the realization of a satisfying synthesis, originating from a very general principle, and the possibility of new provisions.” ([632], p. 11) ^{197} For her, UFT was a fruitful and important theory.
In the following, some of the main aspects of her approach to UFT will be described.“Whatever the future of the unitary theories might be, this book will have reached its objective, if it has somehow shown that the ties between electromagnetism and gravitation form a history of rebouncings the outcome of which is far from being written.” ([641], p. IX)^{198}
10.2.2 Field equations
10.2.3 Removal of affine connection
A first objective was to use the equation \({\hat g^{\overset i + \overset k }}_{\,\,\,\,\,\,\,\,\Vert l} = 0\) or, equivalently, (30) to express the affine connection \(L_{ij}^{\,\,\,\,\,k} = L_{ij}^{\,\,\,\,\,k}({g_{rs}};{\partial _k}{g_{rs}})\) as a functional of the asymmetric metric g_{ ij } and its first derivatives in the same way as the Christoffel symbol had been expressed by the metric and its first derivatives. Now, the system comprises 64 linear equations for 64 variables \(L_{ij}^{\,\,\,\,\,k}\). As an already solved algebraic problem this might not create much interest for “pure” mathematicians: an inverse matrix must be found, if only a large one with functions as its elements. V. Hlavatý called for an “elementary algebraic device” to be invented. As a problem in applied mathematics, even in the computer age, it takes quite an effort to do this by computer algebra. The wish to obtain the solution in tensorial form aggravates matters. According to Hlavatý: “Finding such a device is by no means an easy task” ([269], p. 50).
A reproduction of Tonnelat’s calculations would not bring further insight, the more so as lots of auxiliary symbols were introduced by her including indices with one and two strokes. M.A. Tonnelat has presented the method in detail not only in her books but also in an article [633], and in a talk given at the outstanding Relativity Jubilee Conference in 1955 in Bern ([631], p. 192–197). She was keen on securing priority, i.e., for having found the solution already in 1949–1950. This seemed imperative to her because in the meantime V. Hlavatý [257, 259], and N. S. Bose [52, 50] had also published solutions of \({g_{\underset  i \underset + k \Vert l}} = 0\) by other methods (for Hlavatý cf. Section 12.2). In fact, Hlavatý had reviewed Tonnelat’s paper in Mathematical Reviews [MR0066128], in which she had shown that det(k_{ ij }) = 0 did not affect her solution [630], and he added that “for the solution in the exceptional cases \({g \over h} = 0,2\) ” one should consult a forthcoming paper of his [265]. While the limit k_{ ij } → 0 leads back to the well known results in general relativity, the other limit h_{ ij } → η_{ ij } seemingly has not been discussed intensively by Tonnelat.
Indeed, the whole procedure is drastically shortened and becomes very transparent if h_{ ij } = η_{ ij } is assumed. In this context, apparently, no one did look at this particular case. N. N. Ghosh began with another simplified metric built like the general spherically symmetric metric, i.e., with only h_{00}, h_{11}, h_{22}, h_{33} ≠ 0 and k_{10},k_{23} ≠ 0, but with all components being functions of the four coordinates x^{0}, …, x^{3} However, he managed to solve (30) for the connection only by adding 4 conditions for the first derivatives of h_{ ij } and k_{ ij } in an ad hoc manner [222]. S. N. Bose^{203} rewrote (30) into an inhomogeneous linear equation for tensorial objects Ti_{[jk ]}, U_{i[jk]}, i.e., lh (T_{i[jk]}) = U_{i[jk]} where lh (T) is homogeneous and linear in T [50]. Considered as matrix equation, its solution is T = BU. The matrix \(\check C_i^{\,j}: = {h^{js}}{k_{si}}\), its eigenvalues and eigenvectors play an important role. Although the method is more transparent than Tonnelat’s, the solution is just as implicitly given as hers.
“I am glad to learn that one of my former pupils, Mme. Tonnelat, who really is a remarkable person, has had contact with you with regard to her papers on the unitary theories, and that you have shown an interest in her results”.^{205}
P. G. Bergmann’s report on the Jubilee Conference was noncommittal: “A. Tonnelat of the Sorbonne reported on some mathematical results she had obtained on this theory independently of Einstein and Kaufman.” And a little later: “The papers by Kaufman and Tonnelat are too technical to be reported here.”([22], p. 493.)
In a later approach by M.A. Tonnelat [636], the affine connection is expressed by the metric as above but without a decomposition of g_{ μν } — in a similar but very much more complicated way as in the case of the LeviCivita connection (the Christoffel symbol). This second method does not work if 4h + 12k = 3g. An improvement of it was given by Dautcourt [110] who also showed that 4h + 12k; ≠ 3g does not guarantee a solution. V. Hlavatý used still another method to express the affine connection as a functional of the metric; cf. Section 12.2.
St. Mavridès applied Tonnelat’s method in the case of l_{ ij } and m_{ ij }, i.e., the inverses of l^{ ij },m^{ ij } in g^{ ij } being used as metric and electromagnetic field.^{206} As an existencecondition (364) appeared as well [406]. A plenitude of further work concerning this problem of how to express the affine connection by the asymmetric metric, its derivatives and torsion was done, with the uniqueness proof by Hlavatý & Saenz among them [270]. It amounted to a mathematical discussion of all logically possible cases and subcases without furthering UFT as a physical theory; cf. Sections 12.2 and 13.3.
As a functional of the metric, its first and second derivatives, the Ricci tensor becomes a rather complicated expression. To then find exact solutions of the remaining field equation in (340) is a difficult task. In a paper dealing with approximations of the field equations, M.A. Tonnelat tried to show the superiority of her method by applying a scheme of approximations to her (weak) field equations [634]. However, the resulting equations of 4th order for weak electromagnetic fields k_{ ij } and of 1st order for weak gravitational fields h_{ ij } are still as complicated as to not allow a physical interpretation. In fact, the solution of the problem to remove the connection from the field equations neither helped the search for exact solutions nor contributed to a convincing physical interpretation of the theory. Nonetheless, it was of crucial importance for the proofs given by A. Lichnerowicz that the initial value problem could be well posed in UFT.
Remark:
10.3 Some further developments
From the mathematical point of view, the results of Tonnelat and Hlavatý may be interpreted as having simplified the study of the weak field equations to some degree. For physics, no new insights were gained. In order to make progress, topics like exact solutions, equations of motion of test particles, or the problem how to express continuously distributed “matter” had to be investigated. It is here that the conflict between the “dualistic” approach to UFT separating the fields and their sources, and the “purely geometric” one showed up clearly. In the latter, the (total) field itself defines its own sources.
10.3.1 Identities, or matter and geometry
An example for an electromagnetic energymomentum tensor built from geometric quantities is given in (422) of Section 10.5.4.“The immediate advantage of a unitary theory is that from the theory itself the form of the electromagnetic energymomentum tensor and, perhaps, of the matter tensor can be extracted. The expression of this tensor then would be imposed by the very geometric principles, and not by conclusions from an alien theory as interesting as it might be.” ([635], p. 6)^{209}
Another approach for the introduction of the energymomentum tensor of matter T_{ ik } is the following. First a symmetric metric must be be chosen, e.g., h_{ mn }. Then in the symmetrical part of the Ricci tensor K_{(ij)}(L) a term of the form of the Einstein tensor G_{ ik } (h_{ mn }) is separated out. The field equations of UFT are then rewritten as formal field equations of the type of Einstein’s equations in general relativity plus terms left over. This remainder is identified as T_{ ik } ∼ G_{ ik }. The method is applicable because mixed geometry can always be reinterpreted as Riemannian geometry with many extra fields (geometric objects). Its ambiguity lies in the choice of the Riemannian metric. Taking h_{ ij } as the metric, or the reciprocal of \(\sqrt {{l \over g}} {l^{ij}}\) or \(\sqrt {{g \over l}} {l_{ij}}\) like in [273, 390], or another of the many possible choices, makes a difference. By the formulation within a Riemannian geometry, the unifying strength of a more general geometry is given up, however. Also, according to a remark by M.A. Tonnelat, the resulting equations ∇ _{ s }T^{ is } = 0 are satisfied identically if K_{[ij ]} (L) = 0 holds. Thus, the information about the gravitational field contained in the symmetric part of the field equation K_{(ij)}(L) = 0 does not influence the equations of motions of matter following from ∇ _{ S }T^{ is } = 0 [637]. Related with this is the fact that the matter tensor “seems to vanish together with the electromagnetic field g_{[ij ]}, or at least with a field the properties of which remind of the electromagnetic field” ([635], p. 7).^{210}
In H.J. Treder’s access to a “matter” tensor in “the asymmetric field theory of Einstein”, the subtraction was done not on the level of the Einstein tensor, but for the Lagrangian: from the Lagrangian density of UFT the Einstein Lagrangian was subtracted. An advantage is that the variational principle ensures the existence of an identity [649]. A disadvantage is that the Lagrangian density for the matter part depends not only on the metric but also on its derivatives of 1st and 2nd order. For the metric Treder took the symmetric part h_{ ij } of the asymmetric fundamental tensor g_{ ij }. His references went to Infeld and to Schrödinger’s work, none to Tonnelat’s. We conclude that Tonnelat’s hope presented in the the first quotation above remained unfulfilled.
10.3.2 Equations of motion
Secondly, in correspondence to the vacuum field equations of general relativity, the method of treating the motion of matter as motion of singular point particles as Einstein, Infeld & Hoffmann had done with their approximation scheme in general relativity (EIHmethod) [171, 170], would be in conceptual conflict with the spirit of UFT. Was it possible, here, to only consider the region outside of a worldtube around the moving body where the matter tensor T_{ ij } = 0? The alternative method of Fock with T_{ ij } ≠ 0 included the interior of the moving bodies as well. In order to avoid infinitely many degrees of freedom for extended bodies, some limit procedure had to be introduced. In the EIHmethod, δfunctions, i.e., distributions are used as matter sources, although Einstein’s equations do not admit distributions as exact solutions. Nonetheless, many authors applied the EIHmethod also in UFT. E. Clauser showed in great detail that the method is applicable there for charged particles [83]; cf. also Section 15. Pham Tan Hoang wrote his doctoral thesis by applying this “singularitymethod” to unified field theory [273] (cf. Section 10.4.1). However, we have mentioned already in Section 9.3.3 that both, L. Infeld and J. Callaway were not even able to derive the results of EinsteinMaxwell theory.
With the singularitymethod employed, it turned out to be nontrivial to reach the Lorentzforce, or even the Coulomb force in a “slowmotion” and “weakfield”approximation: h_{ ij } = ^{(0)}h_{ ij } + ϵ^{(1)}h_{ ij } + ϵ^{2} ^{(2)}h_{ ij } + …; k_{ ij } = ϵ^{(1)}k_{ ij } + ϵ^{2} ^{(2)}k_{ ij } + …, with only terms \(\sim {v \over c}\) being retained. In first approximation, only the motion of uncharged particles was described properly by the weak field equations if h_{ ij } is taken as the metric and k_{ ij } as the electromagnetic field [69]. This negative result remained valid up to 4th order in ϵ for l^{ ij } (l_{ ij } ≠ h_{ ij }) chosen as the metric and m^{ ij } (m_{ ij } ≠ k_{ ij }) as the electromagnetic field [271, 272]. Better results in which the Coulomb force could be made to appear were achieved by Treder and Clauser [650, 81, 82], and later by N. P. Chau, in a slightly changed theory, [77]; cf. also Section 15.1.
“It appears impossible to come to a direct phenomenological use of this theory which would allow a satisfactory treatment of macroscopic problems. But this does not prove anything with regard to the applicability of the theory in the microphysical domain.” ([477], p. 203.)^{212}
A review by M. Lenoir [357] of progress made in the papers by Clauser [81], Treder [649] and Papapetrou [477] with regard to the Coulomb force, was reviewed itself by W. B. Bonnor in Mathematical Reviews [MR0119977].
For alternative field equations following from M.A. Tonnelat’s model cf. (383)–(385) in the next Section 10.3.3.
10.3.3 Tonnelat’s extension of unified field theory
“Nevertheless, we are convinced that a modification of the generalization of the theory suggested by Einstein can lead, at least partially, to the goal Einstein himself had set. […] if one wants to cling to the original form of this theory which has caused many hopes and initiated a flood of papers, he would not know how to achieve the objectives which had been proposed at first, within the strict scope of the theories’ principles.” ([302], p. 117–118)^{213}
The choice α = β = σ= p = 0;q = − 1 (q ≠ 0) leads back to Einstein’s strong field equations for a connection without torsion vector. Thus, M.A. Tonnelat always assumed \({{4{\beta ^2}} \over 3}  p  \sigma \ne 0\). The subcase α = β = 0 first has been studied by doctoral students of Tonnelat in [361], and [53, 56]. For this case, (383) leaves Γ_{ i } undetermined.
If the Lagrangian density \(\hat {\mathcal L}\) is augmented by a phenomenological matterLagrangian density \({\hat {\mathcal L}_{{\rm{mat}}}}\), then through \({\Theta _{ij}}: =  {1 \over {\sqrt { g}}}{{\delta {{\mathcal L}_{{\rm{mat}}}}} \over {\delta {g^{ij}}}} = {\theta _{ij}}  {1 \over 2}{g_{ij}}{g^{rs}}{\theta _{rs}}\) a phenomenological mattertensor can be described.

as a metric, the quantity \(\sqrt {{g \over h}} {l_{ij}}\) was chosen; ^{215}

geometrically, \({\overset l \Gamma ^i}: = {l^{is}}{\Gamma _s}\) corresponds to vector torsion S^{ i }; physically, according to (384), it is linked to the current ĝ^{(ir)}Γ_{ r } and proportional to the 4velocity u^{ k } of a particle;

from the equations of motion, in linear approximation, an acceleration term ≃ Γ^{ s }∂_{ s } Γ^{ i } and the Lorentz force showed up as well as further terms characterizing other forces of unknown significance.
 1)As admitted by Tonnelat, it is obtained only after partial neglect of the variational principle: instead of ω_{[ij ]} = 0 which follows from (386) ω_{[ij ]} ≠ 0 must be required in order to give a meaning to an equation like ([382], (108), p. 217):(387) follows from (386) after a longer calculation omitting terms ∼ ∂_{ k } (σK^{2}). The covariant derivative is formed with regard to the LeviCivita connection for the metric \(\sqrt {{g \over g}} {l_{ij}}\) “Hence, in the extended version of the asymmetric theory like in the initial version, the equations of motion can make sense only if at least one of the expressions ω_{[ij ]} or Θ_{[ij ]} does not vanish.”^{216}$${1 \over 2}\sqrt {{g \over h}} \;{g^{[rs]}}{\omega _{\{[ir],s\}}} + {\omega _{[ir]}}{\overset {\rm{Rie}} {\nabla}_s}{g^{[rs]}} = 0\,.$$(387)
 2)As usual, the equations of motion of charged particles were derived in linear approximation in which the electromagnetic field k_{ ij } is taken to be small of first order. It was expressed by a vector potential ϕ_{ k } and an axial potential vector through \({{\mathcal X}_{ij}} = \sqrt { l} \,{\epsilon _{ijlm}}\,{\partial ^{\left[ l \right.}}{{\mathcal X}^{\left. m \right]}}\) with \(l = {\rm{det(}}{l_{ij}}{\rm{)}}\) such that:$${k_{ij}} = {g_{[ij]}} = 2{\partial _{\left[i\right.}}{\phi _{\left.j\right]}} + \sqrt { l} \,{\epsilon_{ijlm}}\;{\partial ^{\left[l\right.}}{\chi ^{\left.m\right]}}.$$
10.3.4 Conclusions drawn by M.A. Tonnelat
 (1)
the dynamics of both the electromagnetic and the gravitational fields are modified such that there appears to be also an influence of the gravitational field on the electromagnetic one;
 (2)
As a nonlinear electrodynamics follows, new effects will appear — as, e.g., “a diffusion of light by light”.
 (3)
The relation between field strengths and inductions is similar as in nonlinear BornInfeld theory ([632], p. 10.).
The choice of words impregnated by ideology like “inveterate” and “totalitarian” speaks for itself.“Nevertheless, the discouraging results obtained from different directions never have definitely compromised the theory; it is the ambiguity of the possible interpretations (choice of metric, interpretation of the skewsymmetric fields, etc) which have set straight issues for the inveterate and totalitarian unifiers.” [382], p. 200).^{219}
As has been remarked before, W. Pauli had criticized unified field theory approached through metricaffine geometry: he demanded that the fundamental objects must be irreducible with regard to the permutation group and also referred to Weyl ([487], Anm. 23, p. 273). In this view, an admissible Lagrangian might be \({\mathcal L} = a{\hat g^{(ik)}}{K_{(ik)}} + b{\hat g^{[ik]}}{K_{[ik]}}\) rather than \({\mathcal L} = {\hat g^{ik}}{K_{ik}}\) By an even stricter application, Pauli’s principle would also rule out this Lagrangian, cf. Section 19.1.1.
“It would be childish to think that, for Einstein, the existence of the unified fields would resolve into an ontological crisscross of torsions and curvatures. It would be likewise improper to reduce such schemes to a pure formalism without any relation to a universe the objectivity of which they propose to present. […] The objective pursued by unitary physics presents itself not as a realization with well defined contours but as a possible direction. […]” ([645], p. 396)^{220}
10.4 Further work on unified field theory around M.A. Tonnelat
Much of the work initiated by M.A. Tonnelat has been realized in doctoral theses, predominantly in the framework of EinsteinSchrödinger field theory. About a dozen will be discussed here. They are concerned with alternative formulations of the field equations, with the identities connected with them, with exact solutions, and with equations of motion for (charged) particles in different approximation schemes.
10.4.1 Research by associates and doctoral students of M.A. Tonnelat
As to the associates of M.A. Tonnelat, it is unknown to me how Stamatia Mavridès got into theoretical physics and the group around M.A. Tonnelat. She had written her doctoral dissertation in 1953 outside of physics [405]. For 5 years, since 1954, she contributed, alone and with Mme. Tonnelat, to EinsteinSchrödinger unitary field theory in many different aspects. Some of her publications have already been encountered. In Section 9.7, her assignment of physical variables to geometrical objects was noted, in Section 10.2.3 her contribution to the removal of the connection. Moreover, her contribution to spherically symmetric exact solutions mentioned in Section 9.6.1 must be kept in mind. Mme. Mavridès also took part in the research on linear field theories; cf. Section 16.1. Since the 1970s, her research interests have turned to astrophysics and cosmology [417].
Then, Pham Tan Hoang applied the EIHmethod to the weak field equations. When \({a_{ij}} = \sqrt {g/l} {l_{ij}}\) with l = det(l_{ rs }) is taken as the metric and \({q^{ij}} = \sqrt {l/g} \,{m^{ij}}\) as electromagnetic field, he obtained the same negative result as J. Callaway [69]: in linear approximation only uncharged particles can be described properly ([274], p. 89). No cure for this failure was found. In the end, the author could only bemoan the ambiguity inherent in the basis of the theory — implying structural richness, on the other side. We shall see that higher approximations had to be calculated in order to get the Coulomb field and the Lorentz force; cf. Section 15.3. Although dependent on the identifications made, another difficulty pointed out by Pham Tan Hoang is the vanishing of the charge current with the vanishing of g_{[ij ]}. Moreover, the identification of a geometric object corresponding to the energymomentum tensor of matter could not be made unambiguously.
The aim of the thesis by Marcel Bray was to study exact spherically and axially symmetric solutions of the weak field equations (417)–(419) and to compare them with exact solutions in general relativity.^{225} For a possible physical interpretation he had to make a choice among differing identifications between mathematical objects and physical observables. For the metric, interpreted as describing the inertialgravitational potentials, he investigated two choices: the metric suggested by MaurerTison, cf. (412) of Section 9.7, and \({g_{ij}} = \sqrt {h/g} \,{l_{ij}}\). Unfortunately, his hope that his research “perhaps could also provide some helpful guiding principles for the choice of the metric” (p. 1) did not materialize. No solutions of physical interest beyond those already known were displayed by him [59].
It thus seems possible that further linear combinations of the 3 quantities used for the definition of the “lightcone”, i.e., h_{ ij },l_{ ij },n_{ ij } may occur.
In the framework of nonlinear electromagnetism which should follow from UFT, he suggested an interpretation different from Maxwell’s theory: the electromagnetic potential ought to be described by a tensor potential identified with the antisymmetric part of the metric k_{ ij }, not just a 4vector. In first approximation □ k_{ ij } = 0. A consequence would be that elementary particles must be described differently; beyond mass and charge, an electron would obtain further characteristics incompatible with spherical symmetry ([456], p. 354).
Further dissertations dealing with the generalization of KaluzaKlein theory and with linear theories of gravitation in Minkowski space are discussed in Sections 11.1.1 and 16.1, 16.2, respectively.
10.5 Research by and around André Lichnerowicz
Within the Institut Henri Poincaryé, a lively interaction between theoretical physicists, mathematicians and natural philosophers took place which tried to grab some of the mysteries from “the lap of the gods”. One of the Paris mathematicians sharing Mme. Tonnelat’s interest in metric affine geometry was André Lichnerowicz^{227}. He looked mainly at problems of interest for a mathematician. In gravitation — both in general relativity and the “nonsymmetric theory” — questions concerning the integration of the systems of partial differential equations representing the field equations were investigated, be it identities for curvature, the Cauchy problem arising from field equations in affine spaces [368, 370], existence and uniqueness of solutions and their global properties, or the compatibility of the field equations of both general relativity and UFT. In his own words: “[…] I could attack what interested me — the global problems of relativity, the keys to a real understanding of the theory.” ([379], p. 104.) This has also been subsumed in Lichnerowicz’s contribution to the Chapel Hill Conference of 1957 on the role of gravitation republished in 2011 ([120], 65–75). For scalartensor theory with its 15th scalar variable ϕ, it is to be noted that Lichnerowicz not only discussed ϕ ∼ κ^{−1}(κ the gravitational “constant”) as a possibility like Ludwig and Just [385] but accepted this relation right away ([371], p. 202).
10.5.1 Existence of regular solutions?
In Section 7.1, it was pointed out that Einstein thought it imperative to banish singular solutions from his theory of the total field. Therefore it was important to get some feeling for whether general relativity theory would allow nonsingular solutions or not. Einstein and Pauli set out to prove theorems in this regard [177]. Their result was that the vacuum field equation R_{ ik }. = 0 did not admit any nonsingular static solution describing a field with nonvanishing mass. For distances tending to infinity, the asymptotic values of the Schwarzschild solution were assumed. The proof held for any dimension of space and thus included the theory by Kaluza and Klein. However, prior to Einstein and Pauli, Lichnerowicz had proven a theorem almost identical to theirs; he had shown the nonexistence of nontrivial regular stationary, asymptotically Euclidean vacuum solutions with^{228} g_{00} = 1 − ϵ_{00}, ϵ_{00} > 0 [363, 362, 364]; cf. also Section 8.3. In a letter of 4 September 1945 ([489], p. 309), a double of which he had sent to Einstein, Lichnerowicz pointed this out to Pauli.^{229} In his response of 21 September 1945, Pauli found the condition on g_{00} unphysical: why should g_{00} > 1 be impossible near infinity? After he had studied the paper of Einstein and Pauli in more detail, Lichnerowicz commented on it in a further letter to Pauli of 11 November 1945. There, he also confessed to be “a bit shocked” about the fact that Einstein and Pauli had only proven “nonexistence” of regular solutions while he had shown that Euclidean space is the only regular solution ([489], p. 325–326). Pauli, in his answer of 15 November 1945, apparently had suggested a related problem. On 15 December 1945, Lichnerowicz wrote back that he had solved this problem, outlined the structure of the proof, suggested a joint publication in Comptes Rendus, and congratulated Pauli for receiving the Nobel prize ([489], p. 333–335). A coauthored paper did not appear but Lichnerowicz published a short note: “W. Pauli signaled me his interest in the possibility to avoid any auxiliary hypothesis: he thought that this could be reached by a synthesis of our respective methods. In fact, this has happened: an important problem in relativity theory has been solved” [367]. A further proof of the occurrence of singularities for static gravitational fields in general relativity was given by A. Lichnerowicz and Y. FourèsBruhat [380]. That Pauli was impressed by Lichnerowicz’s theorem is shown by his detailed discussion of it in his special lectures on relativity in 1953 as reported in ([194], p. 389–390).
As already mentioned in Section 8.3, A. Papapetrou, working at the time in Dublin with Schrödinger, extended the theorem of Einstein and Pauli to a nonsymmetric metric, i.e., to UFT with the strong field equations \({R_{ik}} = 0,\,{g_{\underset + i \underset  k \Vert l}} = 0,\,\,{{\hat g}^{[is]}}{\,_{,s}} = 0\) [474].
10.5.2 Initial value problem and discontinuities
“I believe, the most important else we have heard, was the report by Lichnerowicz on the Cauchy initial value problem in the nonlinear field equations of general relativity. I attach great importance to the study of such problems, because I suppose that it also will play an essential role with field quantization.”^{233} [486].
10.5.3 Characteristic surfaces
When two further approaches to the discontinuities of curvature tensors within the framework of UFT were published in 1961 in Comptes Rendus, the respective authors did not take notice of each other. In the first half of the year, L. Mas and A. Montserrat presented their three papers on “wave fronts” in unified field theory, while in the second half J. Vaillant published on “discontinuities” of the curvature tensor in EinsteinSchr ödinger theory. Both continued the work of Lichnerowicz and MaurerTison.
10.5.4 Some further work in UFT advised by A. Lichnerowicz
The doctoral theses inspired by A. Lichnerowicz are about equally directed to EinsteinSchrödinger and JordanThiry (Kaluza) theory. As interesting as the study of the Cauchy problem initiated by Lichnerowicz was, it also could not remove the ambiguities in the choice for the metric.
In her thesis “Aspects mathématiques de la théorie du champ unifié d’EinsteinSchrödinger”, Françoise MaurerTison first wrote an introductory part on the geometrical background of unified field theory; she developed the concept of “coaffine connection”, i.e., an infinitesimal connection on the fiber bundle of affine reference frames. In Part 2, MaurerTison investigated in detail the Cauchy initial value problem. The last Part 3 of her thesis is devoted to the “physical interpretation”.
“The unified field theory of EinsteinSchrödinger is attractive by its apparent simplicity and repellent by the finicky calculations it requires: it is a young theory with moderate baggage as long as it is investigated with rigour, but an immense load when the efforts are taken into account which have been tried to explore its possibilities” ([398], p. 187).^{241}
The following doctoral thesis by Marcel Lenoir constitutes a link with the next Section 11.1. In it, he gave as his aim the introduction of a geometrical structure which permits the incorporation of Bonnor’s supplementary term into the Lagrangian of UFT (cf. Section 13.1) resulting from the contraction of a suitable Ricci tensor ([359], p. 7). This is achieved by the introduction of spacetime as a hypersurface of a 5dimensional space V_{5} with metric tensor and asymmetric linear connection. The wanted supplementary terms follow from the curvature of V 5. Lenoir’s approach to Bonnor’s field equations is an alternative to (and perhaps more convincing) than the earlier derivation, in spacetime, by F. de Simoni (cf. Section 15.1).
In the last chapter, Lenoir investigated whether Lichnerowicz’s theorem on the nonexistence of regular solutions could also be proven for his extended unitary theory but did not arrive at a conclusive result.
The doctoral thesis of another student of Lichnerowicz, Albert Crumeyrolle, contained two different topics [93]. In the larger part, research on the equations of motion of charged particles and on the energymomentum tensor (corresponding to the “matter” tensor in UFT) was resumed in the framework of EinsteinSchrödinger theory. As to the equations of motion for charged particles, Pham Tan Hoang had simplified calculations by a more complete use of the isothermal condition (392) and by further improvements as mentioned in Section 10.4. Yet the negative result remained the same as the one already obtained by E. Clauser and H.J. Treder, cf. Section 10.3.2. The same applies to Crumeyrolle’s approximative calculation of the equations of motions in ([94], p. 390).
Because the energymomentum tensor he constructed had to contain a metric field, Crumeyrolle investigated which of the three possibilities for the metric, i.e., h_{ ij },l_{ ij }, and γ_{ ij } emerging from the Cauchyproblem (cf. (421) of Section 10.5.4) would be best for reaching the special relativistic energymomentum tensor of the electromagnetic field. In fact, none was good enough. In first approximation, h_{ ij } fared best [94].
With these tensors, modified field equations which contained the “weak” Einstein equations including additional terms in \({{\mathcal P}_{jk}}\) then could be introduced ([96], p. 103–128). But a number of extra field equations had to be joined such as, still among others, \(\Lambda _{\left[ {s\left. j \right]} \right.}^s = 0,\,d{\overline {\mathcal P} _{[jk]}} = 0,\,{\overline {\mathcal P} _{(jk)}} = 0\) cf. ([96], p. 126). Another approach by Crumeyrolle using a field of numbers different from the real numbers will be discussed in Section 11.2.2. In its Section XV ([97], pp. 126–130), it contains a new attempt at a unified field theory with a slightly changed formalism. As Crumeyrolle’s aim was to regain the old EinsteinSchrödinger theory from a theory with additional field variables and field equations, his approach could not bring progress for an eventual physical interpretation of UFT.
11 HigherDimensional Theories Generalizing Kaluza’s
11.1 5dimensional theories: JordanThiry theory
“Lichnerowicz is a pure mathematician who is occupied with the integration of Einstein’s field equations. One of his students, Ives Thiry now has looked into the (not mutilated) Kaluzatheory (with g_{55}) and, so I believe, has simplified very much the calculational technique.” ^{245}
“As to unitary field theories, it seems that their mathematical study has been quite neglected […]. We thought it useful to try a systematic mathematical study of a unitary field theory, and to find out whether such a theory is able to present the same coherence like general relativity.” ([606], p. 3)^{249}
In the third chapter, Thiry aimed at showing that his unitary field theory possessed the same mathematical coherence with regard to its global aspects as general relativity. By partially using methods developed by Lichnerowicz, he proved theorems on the global regularity of solutions such as: “A unitary field with normal asymptotic behavior (i.e., tending uniformly to Minkowski space at spatial infinity) cannot be regular everywhere.” Thiry compared the proofs on the existence and regularity of solutions in O. Klein’s version of Kaluza’s theory and in his generalization and found them much simpler in his theory. These results are of a different nature than what Jordan had achieved; they are new and mathematically exact.“The introduction of a fifth coordinate […] thus shall justify itself by the fact that it imparts the role of geodesics to the trajectories of charged particles which they lost in spacetime […]”^{251}
To Pauli, Thiry’s global theorems might not have been “interesting novelties”, because in his corresponding paper with Einstein on the nonregularity of solutions, the proof had been independent of the dimension of space [177]. Pauli, at first, also did not read Thiry’s thèse, but responded arrogantly:“By the way, in his thèse published in 1951, Thiry has studied systematically and extensively the theory with variable gravitational constant; […] I received it only after my book appeared and, at present, I have not read it very closely. It thus is not really clear to me whether it contains interesting novelties.” ([490], p. 799/800)^{252}
“The Thèses by Thiry are laying on my desk; however they are so appallingly thick (do not contain a reasonable abstract) such that it is so much simpler to not open the book and reflect about what must be inside.” (W. Pauli to P. Jordan 8. 6. 1953, [491], p. 176)^{253}
Somewhat later, Pauli corrected himself and wrote to Jordan that in the preparation for a course “he nevertheless had read around in Thiry’s Thèse” W. Pauli to P. Jordan 3 February 1954 ([491], p. 442). Note that neither of these two eminent theoretical physicists discussed Thiry’s paper as regards its valuable content.
As to the field equations corresponding to (112) to (114), Thiry had calculated them in great detail with Cartan’s repère mobile for both a Euclidean or Lorentz metric of V_{5}, and also with a 5dimensional matter tensor of the form of dust pu^{ α }u^{ β }. From the 15th equation, he even had obtained “a new physical effect”: uncharged dustmatter could generate an electromagnetic field [[606], p. 79, footnote (1)]. He linked this effect to Blackett’s search for the magnetic field of a rotating body, described in Section 6.1.2. As to the interpretation of the fifteenth variable, the scalar field denoted by V: for him _{ χTh }:= V^{2}G_{0} with G_{0} = 8πG_{Newton}/c^{2} was the gravitational coupling factor (“facteur de gravitation”) and put in front of the matter tensor [606], p. 72, 75, 77). In 1951, Y. FourèsBruhat proved existence and uniqueness theorems of “the unitary theory of JordanThiry” [216, 218].
11.1.1 Scientists working at the IHP on the JordanThiry unified field theory
P. Pigeaud used two metrics, the “natural” one γ following from the projection of R_{5} to R_{4}, and another one conformal to it ξγ. The first is employed for the calculation of the potentials (up to 2nd approximation), the 2nd for the study of the equations of motion. The reason is that for the “natural” metric uncharged test particles do not follow geodesics, yet for the conformal metric they do. At some point, Pigeaud had to make an ad hoc change of the energymomentum tensor of matter (perfect fluid) not justified by empirical physics [495]. In a later development, Pigeaud did interpret the scalar fifteenth field variable as the field of a massive meson [387, 496]. The investigations within JordanThiry theory were carried on by Aline SurinParlange to the case of perfect fluid matter with an equation of state ρ = f (p) where ρ,p are mass density and pressure, f is an arbitrary function. The Cauchy problem for this case was solved, the existence of hydrodynamical waves shown, and their propagation velocity determined. A. Surin compared both, the “singularity” method and the method using the vanishing of the divergence of the matter tensor in 1st order approximation: they gave the same results. Unlike F. Hennequin who had assumed for the metric components^{255} that γ_{04} and γ_{0A} are of the same order \({1 \over {{c^2}}}\), A. Surin assumed γ_{04} of order \({1 \over {{c^2}}}\), and γ_{0A} of the order \({1 \over {{c^3}}}\). Unfortunately, as in previous work, the resulting equation of motion for charged particles still were in conflict with the classical electrodynamic equations. Surin thus had to change her choice of metric to a conformal metric in spacetime; she found that the equations of motion in first approximation are independent of the conformal metric. Nevertheless, she had to admit that this did not help: “These last results, particularly those concerning the first approximation to the equations of motion seem to necessitate a modification of the field equations.”^{256} ([597], thèse, p. 126). A. Surin went on to calculate the 2nd approximation, an arduous task indeed, but which did not make the theory physically more acceptable. Moreover, in 2nd approximation, as in general relativity the results from both methods did not agree. Surin noted, that a modification of the field equations already had been suggested by H. Leutwyler [360], but did not comment on it. He had started from a variational principle mixing 5 and 4dimensional quantities:
“It seems preferable to us to interpret ϕ as a scalar field without mass. JordanThiry theory then will be a unitary theory of three massless fields: a scalar field, a vector field, and a 2tensor field of spin 0, 1, 2, respectively.”^{257}
11.1.2 Scalartensor theory in the 1960s and beyond
Around Y. Thiry in Paris, the study of his theory continued with J. Hély producing static spherically symmetric solutions of the field equations [253, 252, 251]. They describe a point particle with charge and mass and include the Schwarzschild metric. Due to the different field equations, a direct comparison with the earlier solution by Heckmann et al. of Jordan’s theory (cf. Section 3.1.2) seems not appropriate. However, H. Dangvu also contributed by looking at static and nonstatic spherically symmetric solutions to the JordanThiry field equations in spacetime.^{258} The solutions carry mass and some of them also electrical charge. Dangvu could compare them with those by Heckmann, Just, and Schücking in the Hamburg group around P. Jordan [108, 108].
From the full field (particle) content of KaluzaKlein theory, mainstream physics became interested most in the scalar field. In spite of the investigations within the framework of the ideas of Kaluza and Klein, and of Jordan’s approach within projective relativity, attention to the scalar field evolved in complete separation from its origin in unitary field theory. Soon, scalartensor theories were understood strictly as alternative theories of gravitation. We comment briefly on the loss of this historical perspective the more so as current publications on scalartensor theory are more interested in the subsequent modern developments than in the historical record [219, 58]. In the AngloSaxon literature, scalartensortheories run under the name of “BransDicke theory”, or, at best, BransDickeJordan theory, i.e., two authors being late with regard to Jordan and Thiry are given most of the credit; cf. standard references like ([242], pp. 59, 64, 71, 77, 362); ([438], pp. 1068, 1070, 1098; ([698], pp. 125, 126, 341).^{259} True, the three groups of successive authors departed from different physical or mathematical ideas; Jordan from a varying gravitational constant as a hypothetical consequence of Dirac’s large number hypothesis, Thiry from a mathematical study of KaluzaKlein theory and its global aspects, and Brans & Dicke from an implementation of their interpretation of Mach’s principle. A fourth author, W. Scherrer, must be included who was the first of all, and who considered the scalar field as a matter field coupled to gravitation; vid. Section 3.1.2 and [230]. Yet, one should keep in mind that, together with YangMills theories and, perhaps nonlocal field theory, scalartensor theories of gravitation may be also considered as one surviving offspring of unitary field theory. Hence a total loss of historical memory with regard to the origins of scalartensor theory appears unjustified.^{260}
11.2 6 and 8dimensional theories
Since Kaluza had proposed a 5dimensional space as a framework for a unitary field theory of the gravitational and electromagnetic fields in 1919, both experimental elementary particle physics and quantum field theory had progressed. Despite the difficulties with divergencies, since the mid 1950s renormalization procedures had been stable enough to make quantum field theory acceptable and needed as a proper formalism for dealing with the known elementary particles. Nevertheless, in some approaches to unified field theory, it still was thought useful to investigate classical theory. Thus, in this context theories with new degrees of freedom for the new fields (π and μmesons, neutrino) had to be constructed. We noticed that both in the EinsteinSchrödinger affine field theory and in the JordanThiry extension of Kaluza’s theory such attempts had been made. The increase in the dimension of space seemed to be a handy method to include additional fields. In Section 7.2.2 of Part I, papers of Rumer, Mandel, and Zaycoff concerning 6dimensional space have been mentioned. None of them is referred to by the research described below. It involved theorists working independently in the USA, Great Britain, and France.
11.2.1 6dimensional theories
The first of these, B. Hoffmann in Princeton, wanted to describe particles with both electric charge and magnetic charge μ. ^{261} Because the paths of electrically charged particles could be described as geodesics of Kaluza’s 5dimensional space, he added another spacedimension. The demand now was that his particles with both kinds of charge follow geodesics in a 6dimensional Riemannian (Lorentzian) space R^{6}. [277]. The coordinates in R^{6} are denoted here by A,B = 0,1, 2, 3, 4, ∞, in R^{5} by α, β, … = 0,1, …, 4, and in spacetime i, j, = 1, 2, 3, 4. The metric k_{ ab } of R^{6} contains, besides the metric g_{ ij } of spacetime, two vector fields k_{k0} = ϕ_{ k }, k ∞j = ψ_{ k } and three scalar fields k_{00},k_{0∞},k_{∞,∞}. The scalar fields are disposed of immediately: k_{00} = 1 (cylinder condition), k_{0∞} = 0, k_{∞,∞} = −1, while ϕ_{ k } is taken to be the electrical 4potential:\({\phi _{ij}} = 1/2({{\partial {\phi _i}} \over {\partial {x^j}}}  {{\partial {\phi _j}} \over {\partial {x^i}}})\) and ψ_{ k } the corresponding quantity following from the dual of the electrical field tensor \({\psi ^{ij}} = {\phi ^{\ast ij}} = 1/2{(\sqrt { g})^{ 1}}{\epsilon ^{ijrs}}{\psi _{rs}}\)
Pauli’s interest really went toward a paper of Pais who had suggested a 6dimensional ωspace consisting of spacetime and an internal spherespace affixed at any point [468]. From here, a direct line leads to Pauli’s (unpublished) derivation of the nonAbelian SU (2) gauge theory presented in letters to Pais. For a detailed history cf. O’Raifeartaigh’s book ([463], Chapter 7). cf. also the section “A vision of gauge field theory” in [194], pp. 476–488.“In it, also Kaluza’s 5th dimension did occur, and per se it is quite satisfying if now, in place of it, two additional dimensions with the 3dimensional rotation group are introduced (this has already been suggested, if only formalistic, by e.g., Podolanski who has a 6dimensional space).” ([491], p. 186–187.) ^{264}
When Josette Renaudie wrote her dissertation with A. Lichnerowicz and M.A. Tonnelat on 6dimensional classical unitary field theory in 1956, she also used elementary particles as her motivation [505, 506], ([507], p. 3; [508]). However, unlike Podolanski [498] and Yano & Ogane [715], she worked in a 6dimensional Lorentz space: 2 spacedimensions are added. While Yano & Ogane employed a general 2parameter isometry group and claimed to have the most general formalism, Renaudie admitted a general Abelian 2dimensional isometry group, thus keeping 3 arbitrary scalar functions. Her two Killing vectors with regard to which the projection from 6dimensional V_{6} along the trajectories of one Killing vector first to to V_{5}, and then with the 2nd Killing vector to spacetime is performed, are spacelike. Again, an Einstein field equation S_{ ab } = P_{ ab }, (A, B = 0,1, …, 5) is written down with the “matter tensor” P_{ ab } getting its interpretation backwards from the corresponding 4 and 5dimensional quantities. The Einstein tensor now has 21 independent components and can be split into a rank 2 symmetric tensor, two 4vectors and 3 scalar functions. Renaudie gave two interpretations: (1) these variables stand for a hyperfield composed of gravitational, electromagnetic and mesonic fields (with the mesonic field a complex vector field), and (2) the field of a particle of maximal spin 1 in interaction with the gravitational field (p. 68/69). Note that in both interpretations the scalar fields remain unrelated to physical quantities. The terms in the field equations describing the interaction of the mesonic and electromagnetic fields are independent of the geometrical objects in spacetime. The Cauchy initial value problem can be set up and solved properly. ^{265}
In 1963, Mariot & Pigeaud again took up the 6dimensional theory in a paper^{266} [388]. After the introduction of a conformal metric in V_{4}, \(g_{ij}^{\ast} = \eta \xi {g_{ij}}\), i,j = 1, 2, 3,4 with ξ^{2}, η^{2} being the norms squared of the 2 Killing vectors from the additional dimensions, they studied the linear approximation of the theory with incoherent matter mu_{ i }u_{ j }. They were able to identify in V_{4} matter tensors belonging to the electromagnetic field, to a neutral vector meson field and to both a massive neutral and a charged scalar meson field. Yet, some remaining terms were still not amenable to a physical interpretation.
11.2.2 Eight dimensions and hypercomplex geometry
As early as 1934, an eightdimensional space with two timedimensions was introduced in order to describe the gravitational field corresponding to an accelerated electromagnetic field. It then turned out that the two time coordinates were related by the eight geodesic equations such that, essentially, a 7dimensional Lorentz space remained. Einstein’s vacuum field equations in 8 dimensional space were assumed to hold [421, 422]. In view of the substantial input, the results, reached by approximate calculations only, were meagre: an approximate solution of the gravitational twobody problem; only static electric and constant magnetic fields could be described.
In the 1950s, the idea of using an 8dimensional space as the stage for UFT seemingly arose by an extension of the mapping of connections in the same 4dimensional space to a second copy of such a space. In Section 2.1.2 we have noted F. MaurerTison’s interpretation of (30): The transport of a covariant vector ν_{ i } with regard to the connection \({{\tilde L}_{ij}}^k\) corresponds to the transport of the contravariant vector \(\underset v {{\rightarrow ^k}} = {g^{ks}}{v_s}\) with regard to \({L_{ij}}^k\). This situation was turned into a geometry named semimetric by PierreV. Grosjean^{267} in which two identical 4dimensional spaces (“distinct universes”) were introduced with the connection acting in one and its Hermitian conjugated connection \({\tilde L}\) in the other [235]. Inner products of vectors were allowed only if one vector is in the one space, the other in the second. His conclusion that the semimetric geometry would be “the key to the unitary theory, in the same way as metric geometry was the key to general relativity” is more than overbearing. All that remains is a (physically empty) mathematical formalism the only consequence of which was to eventually motivate the scheme of applied mathematics to be discussed next.
12 Further Contributions from the United States
12.1 Eisenhart in Princeton
12.2 Hlavatý at Indiana University
Hlavatý^{272} is the fourth of the main figures in UFT besides Einstein, Schrödinger, and Tonnelat. His research was published first in a sizeable number of articles in the Journal of Rational Mechanics and Analysis of Indiana University^{273} and in other journals; they were then transformed into a book [269]. According to its preface, his main intent was “to provide a detailed geometrical background for physical application of the theory”. As he was very optimistic with regard to its relation to physics, he went on: “It so happens that the detailed investigation of Einstein’s geometrical postulates opens an easy way to a physical interpretation”([269], p. X). We have noticed in Section 9.7 that this possibly could not have been the case. In the preface of his book, Hlavatý became more explicit; his program was to encompass: (1) an investigation of the structure of the curvature and torsion imposed on spacetime by the field equations, equations which he claimed to be “of a purely geometrical nature” without physical interpretation being “involved in them a priori”. The two further points of his program, i.e., (2) an attempt to identify the gravitational field and the electromagnetic field by means of the field equations, and (3) an investigation of the physical consequences of his theory, were treated only in “an outline of the basic ideas” ([269], p. XVIII). In comparison with Einstein, Schrödinger and Tonnelat who followed their physical and mathematical intuition, Hlavatý’s investigations were much more systematical and directed first to what could be proven by mathematics; whether a relation to physics could be established became secondary to him. Although mostly working and publishing alone, he corresponded with about 40 scientists working on UFT. He also was a frequent reviewer for Mathematical Reviews (cf. Section 18.1).
According to Hlavatý, the first two classes cannot be handled simultaneously with the third class ([266], p. 421). This makes it more involved to read his papers, because the results proven by him must now be distinguished according to the special class of k_{ ij }.
12.2.1 i) Fields of third class.
“In the unified theory the electromagnetic field is always present; hence we might look upon it as a primary field which […] creates the gravitational field. However, there is at least one known electromagnetic field which does not create a gravitational field (i.e., the field of the plane wave in the electromagnetic theory of light).” ([266], p. 420.)
12.2.2 ii) Fields of class 1 and 2.
For paths of photons Eq. (453) still holds. If gravitation is neglected, i.e., h_{ ij } = η_{ ij }, Hlavatý found a discrepancy with the special relativistic explanation of the MichelsonMorley experiment. Although he referred to the judgment of Shankland et al. that Miller’s result is erroneous [575], he concluded: “From the point of view of the unified field theory Miller’s result, properly interpreted, is not necessarily at variance with the assumption of the constant velocity of light.” ([266], p. 471).
Hlavatý’s research will be appealing to some by its logical guideline concerning mathematical structures. His many special cases and set up “agreements” in proving results are somewhat bemusing for a physicist. An example is given by his publications dealing with the special case h = 0, g ≠ 0 when the symmetric part h_{ ij } of the metric g_{ ij } is degenerated [267, 268]. It is a purely mathematical exercise meant to fill a gap, but is without physical meaning. For the cases in which the theory could be applied to physical systems, in principle, Hlavatý was also forced to alter the original field equations in order to avoid objections against the unphysical results following from them. It is not unfair to conclude that he did not succeed in making a breakthrough in the sense of his physical interpretations being more convincing than those suggested by others.
The investigations of his doctoral student R. Wrede^{275} were directed to the mathematical structure of the theory: He partially extended Hlavatý’s theory to an ndimensional space by adhering to the two principles: A.) The algebraic structure of the theory is imposed on the space by a general real tensor g_{ ab }; B.) The differential geometrical structure is imposed on the space by the tensor g_{ ab } by means of a connection defined by (30). Hlavatý’s third principle, i.e., the existence of the constraints R_{ ab } = ∂_{[a}X_{b]}, \({R_{ab}} = {\partial _{\left[ a \right.}}{X_{\left. b \right]}},\,{S_{ar}}^r = L_{\left[ {a\left. r \right]} \right.}^r = 0\) with an arbitrary vector field X_{ a } is left out [708]. The paper solves (30) in n dimensions for the various possible cases.
12.3 Other contributions
 (1)
Any unified field theory should reduce to EinsteinMaxwell theory in a first approximation for weak electromagnetic fields.
 (2)
Firstorder corrections to the Coulomb field of the electron should not become appreciable for r ≥ 10^{−13} cm.
 (3)
The affine connection has the form \({\Gamma _{ij}}^k = {C_{ij}}^k + {\Gamma _j}\delta _i^k\), where Γ_{ j } is related to the vector potential of the Maxwell field. Also \({C_{ij}}^k = {C_{ji}}^k\) is assumed.
 (4)
The Lagrangian must be invariant under the combined gauge transformation \({\Gamma _{ij}}^k \rightarrow {\Gamma _{ij}}^k + {\Lambda _{,j}}\delta _i^k\) and (for the metric) g_{ ij } → exp[2Λ(x)] g_{ ij }. The metric tensor is also symmetric.
The auxiliary field F_{ ij } satisfies the vacuum Maxwell equations.
13 Research in other English Speaking Countries
13.1 England and elsewhere
The assignment of g_{(ij)} to the gravitational potentials and of g_{[ij ]} to the electromagnetic field was upheld while the electric current became defined as J_{ ijk } = g_{{[ij],k}}.
“[…] that a correct and unified quantum theory of fields, without the need of the socalled renormalization of some physical constants, can be reached only through a complete classical field theory that does not exclude gravitational phenomena.” ([343], p. 1396.)
As to the equations of motion, Kursunŏglu assumed that (30) is satisfied and, after some approximations, claimed to have obtained the Lorentz equation, in lowest order with an inertial mass \({m_0} = {1 \over {2{c^2}}}\kappa \,{e^2}\) (cf. his equations (7.8)–(7.10)); thus according to him, inertial mass is of purely electromagnetic nature: no charge — no mass! Whether this result amounted to an advance, or to a regress toward the beginning of the 20th century is left open.
G. Stephenson from the University College in London altered Einstein’s field equation as given in Appendix II of the 4th London edition of The Meaning of Relativity^{286} by replacing the constraint on vector torsion S_{ j } = 0 by S_{ r }S^{ r } = a with constant a, and by introducing a vectorpotential A_{ j } for the electromagnetic field tensor k_{ ij } [588]. His split of (30) for the symmetric part coincided with Tonnelat’s (363), but differed for the skewsymmetric part from her (362) of Section 10.2.3. The missing term is involved in Stephenson’s derivation of his result: Dirac’s electrodynamics \({\overset {\{_{ij}^k\}} \nabla _s}{{\check k}^{is}} = {A^i}\). Hence the validity of this result is unclear.
“The Einstein paper contains three separate sections. In the first section the author expresses the symmetric part \(\Lambda _{\lambda \mu}^v = \Gamma _{(\lambda \mu)}^v\) of the unified field connection \(\Gamma _{\lambda \mu}^v\) by means of its skew symmetric part \(S_{\lambda \mu}^v = \Gamma _{\left[ {\lambda \left. \mu \right]} \right.}^v\), and vice versa. Then he identifies the electromagnetic field with k_{ λμ } = g_{[λμ ]} and imposes on it the first set of Maxwell conditionsThe second set of Maxwell conditions is equivalent to the Einstein condition \(S_{\lambda \alpha}^\alpha = 0\) However, according to the author, there appears to be no definite reason for imposing the additional condition (1). In the second part the author considers Einstein’s condition$${{{d^2}{x^\nu}} \over {d{s^2}}} + \left\{{\begin{array}{*{20}c} {\nu \;} \\ {\lambda \,\mu} \\ \end{array}} \right\}{{d{x^\lambda}} \over {ds}}{{d{x^\mu}} \over {ds}} + {F_\mu}^\nu {{d{x^\mu}} \over {ds}} = 0$$(1)coupled with$${R_{[\mu \lambda ]}} = 2{\partial _{\left[\mu\right.}}{X_{\left.\lambda \right]}}$$(2)(where D_{ α } denotes the covariant derivative with respect to \(\Lambda _{\lambda \mu}^v\)). Hence$${R_{[\mu \lambda ]}} =  {D_\alpha}S_{\mu \lambda}^\alpha$$where \(T_{\mu \lambda}^v = T_{\left[ {\mu \left. \lambda \right]} \right.}^v\), is a solution of$$S_{\mu \lambda}^\nu = 2{X_{\left[\mu \right.}}\delta _{\left.\lambda \right]}^\nu + T_{\mu \lambda}^\nu ,$$(3)Therefore \({S_{\lambda \alpha}}^\alpha = 0\) is equivalent to$${D_\alpha}T_{\mu \lambda}^\alpha = 0.$$(4)and the field equations reduce to 16 equations (4) and R_{(μλ)}^{=} 0. In the third part the author considers all possibilities of defining \(\Gamma _{\lambda \mu}^v\) by means of the derivatives of g_{ λμ } with all possible combinations of Einstein’s signs (+ +), (+ −), (−−). He concludes that in both cases (i.e. for real or complex g_{ λμ }) only the (+ −) derivation leads to a connection \(\Gamma _{\lambda \mu}^v\) without imposing severe restrictions on g_{ λμ }.”$${X_\mu} =  {1 \over 3}{T_{\mu \alpha}}^\alpha$$(5)
13.1.1 Unified field theory and classical spin
Each of the three scientists described above introduced a new twist into UFT within the framework of — real or complex — mixed geometry in order to cure deficiencies of Einstein’s theory (weak field equations). Astonishingly, D. Sciama^{287} at first applied the full machinery of metric affine geometry in order to merely describe the gravitational field. His main motive was “the possibility that our material system has intrinsic angular momentum or spin”, and that to take this into account “can be done without using quantum theory” ([565], p. 74). The latter remark referred to the concept of a classical spin (point) particle characterized by mass and an antisymmetric “spin”tensor s_{ αβ }, α, β = 1,2,3. Much earlier, Mathisson (1897–1940) [392, 391, 393], Weyssenhoff (1889–1972) [695, 694] and Costa de Beauregard (1911–2007) [87], had investigated this concept. For a historical note cf. [584]. Sciama did not give a reference to C. de Beauregard, who fifteen years earlier had pointed out that both sides of Einstein’s field equations must become asymmetric if matter with spin degrees of freedom is generating the gravitational field. Thus an asymmetric Ricci tensor was needed. It also had been established that the deviation from geodesic motion of particles with charge or spin is determined by a direct coupling to curvature and the electromagnetic field \(R_{jkl}^i{{d{x^j}} \over {ds}}{F^{kl}}\) or, analogously, curvature and the classical spin tensor \(R_{jkl}^i{{d{x^j}} \over {ds}}{s^{kl}}\). The energymomentum tensor of a spinfluid (Cosserat continuum), discussed in materials science, is skewsymmetric. What Sciama attempted was to geometrize the spintensor considered before just as another field in spacetime.
Perhaps, Sciama had convinced himself that mixed geometry was too rich in geometrical objects for the description of just one, the gravitational interaction. In any case, in his next five papers in which he pursued the relation between (classical) spin and geometry, he went into UFT proper [565]. He first dealt with the electromagnetic field which he identified with an expression looking like homothetic curvature: \({V_{kl}} = \overset s K {\,^j}_{jkl}\, = \,{\partial _k}{\Gamma _l}  {\partial _l}{\Gamma _k}\). However, here \({\Gamma _k}: = {L_{\left[ {k\left. r \right]} \right.}}^r = {S_k}\) In order to reach this result he had introduced a complex tetrad field^{288} and defined a complex curvature tensor \(\overset s K {\,^i}_{jkl}\), skewsymmetric in one pair of its indices and “skewHermitian” in the other. In analogy to Weyl’s second attempt at gauge theory [692], he arrived at the trace of the tetradconnection as his “gaugepotential” without naming it such. He also introduced a “principle of minimal coupling” as an equivalent to the “equivalence principle” of general relativity: matter must not directly couple to curvature in the Lagrangian of a theory. M.A. Tonnelat and L. Bouche [646] then showed that Sciama’s nonsymmetric theory of the pure gravitational field [565] “implies that the streamlines of a perfect fluid (T^{ μν } = ρv^{ μ }v^{ ν }) are geodesics of the Riemannian space with metric g^{(ij)} These streamlines are not geodesics of the metric g_{(ij)}, but deviate from them by an amount which, in first approximation, agrees with a heuristic formula occurring in Costa de Beauregard’s theory of the gravitational effects of spin [89]”.^{289}
“The majority of physicists considers with some reserve unified field theory. In this article, my intention is to suggest that such a reserve is not justified. I will not explain or defend a particular theory but rather discuss the physical importance of nonRiemannian theories in general.” ([566], p. 1.)^{290}
Sciama’s main new idea was that the holonomy group plays an important rôle with its subgroup, Weyl’s U (1), leading to electrodynamics, and another subgroup, the Lorentz group, leading to the spin connection. Although he gave the paper of Yang & Mills [712] as a reference, he obviously did not know Utiyama’s use of the Lorentz group as a “gauge group” for the gravitational field [661]. C. de Beauregard‘s reaction to Sciama’s paper was immediate: he agreed with him as to the importance of embedding spin into geometry but did not like the two geometries introduced in [565]. He also suggested an experiment for measuring effects of (classical) spin in spacetime [88].
“We may note in passing that the result (7) [here Eq. (488)] suggests that unified field theories based on a nonsymmetric connection have nothing to do with electromagnetism.” ([569], p. 467)
C. de Beauregard had expressed this opinion three years earlier; moreover his doubts had been directed against the “unified theory of EinsteinSchrödingertype” in total [90]. In the 1960s, the subject of classical spin and gravitation was taken up by F. W. Hehl [245] and developed into “Poincaré gauge theory” with his collaborators [246].
13.2 Australia
13.3 India
14 Additional Contributions from Japan
We already met Japanese theoreticians with their contributions to nonlocal field theory in Section 3.3.2, to wave geometry as presented in Section 4.3, and to many exact solutions in Section 9.6.1. The unfortunate T. Hosokawa showed, in the paper mentioned in Section 4.3, that “a group of motions of a Finsler space has at most 10 parameters” [287]. Related to the discussion about exact solutions is a paper by M. Ikeda on boundary conditions [299]. He took up Wyman’s discussion of boundary conditions at spacelike infinity and tried to formulate such conditions covariantly. Thus, both spacelike infinity and the approach to it were to be defined properly. He expressed the (asymmetric) metric by referring it to an orthormal tetrad tetrad \({g_{ij}} \rightarrow {a_{AB}} = {g_{ij}}\xi _A^i\xi _B^j\) where \(\xi _A^i\) are the tetrad vectors, orthonormalized with regard to \({e_A}{\delta _{AB}},{e_A} = \pm 1\). The boundary condition then was \({a_{AB}} \rightarrow {e_A}{\delta _{AB}}\) for ρ → ∞, where \(\rho (PQ): = \int\nolimits_P^Q {\sqrt {{\gamma _{ij}}d{u^i}d{u^j}}}\) and the integral is taken over a path from P to Q on a spacelike hypersurface x^{ k } = x^{ k } (u^{1}, u^{2}, u^{3}, σ) with parameters u^{ i } and metric γ.
Possibly, in order to prepare a shorter way for solving (30), S. Abe and M. Ikeda engaged in a systematic study of the concomitants of a nonsymmetric tensor g_{ ab }, i.e., tensors which are functionals of g_{ ab } [301, 300]. A not unexpected result are theorems 7 and 10 in ([301], p. 66) showing that any concomitant which is a tensor of valence 2 can be expressed by h_{ ab }, k_{ab}, \({k_{ar}}{k^r}_b,{k_{ar}}{k^r}_s{k^s}_b\) and scalar functions of \({g \over h},{g \over k}\) as factors. In the second paper, pseudotensors (e.g., tensor densities) are considered.
It can be viewed as “the common tangential segment of the oriented sphere with center ξ^{ k } and radius \(r = \int [{\phi _s}{x^k}){{d{\xi ^s}} \over {dt}} + {\phi _4}({x^k})]dt \ldots\)” ([599].
15 Research in Italy
15.1 Introduction
“The charm of this theory lies in its generality, its simplicity, and, let’s say it clearly, in its beauty, attributes which the utmost Einsteinian synthesis possesses, more than any other noted today.” ([473], p. 306)^{301}
In this spirit, contributions of mathematicians like I. Gasparini Cattaneo (1920–2011) [75], or A. Cossu (1922–2005) [85, 86], and more or less formal mathematical manipulations by Italian researchers played an important part in the work on UFT. The other leading elder figure Maria Pastori^{302} and some of her students present a main example. Already in the 1930s, she had published on anisotropic and “conjugated” skewsymmetric tensors. Hattori’s paper sparked her interest: she intended to reduce his two assumptions concerning skewsymmetric tensors to one [480, 482, 481]. Now in the 1950s, she studied the properties of the new tensorial objects appearing within UFT [483, 484, 485]. Elisa Brinis considered parallel transports conserving the scalar product of two vectors with a nonsymmetric fundamental tensor [61].
B. Todeschini arrived at an inhomogeneous d’Alembert’s equation in which the electromagnetic tensor coupled to torsion [609]. F. Graiff derived expressions for the commutation of the ±− derivatives; she also studied alternative forms for the electromagnetic tensor (k_{ ij }, R [ij ], or its duals) in first and second approximation according to the scheme (499) given below [232, 233]. By adding a term to the EinsteinStraus Lagrangian depending only on the metric tensor, F. de Simoni showed how to trivially derive from a variational principle all the different systems of field equations of UFT including those suggested by Bonnor, or Kursunŏglu (cf. Section 13.1) [115]. Laura Gotusso generalized a theory suggested by Horváth with a Riemannian metric and the connection \({L_{ik}}^l = {L_{(i}}_{k)}^l + {F_{\left[ i \right.}}^l{j_{\left. k \right]}}\) where j_{ k } is the electrical current vector \({j^i} = \sigma {{d{x^i}} \over {ds}}\). The torsion vector thus is proportional to the Lorenz force \({S_i} = {1 \over 2}{F_i}^s{j_s}\) and the autoparallels describe the motion of a charged point particle^{303} [285]. Gotusso generalized Horváth’s theory by adding another tensor to the connection: \(\overset {{\rm{Got}}} L {\,_{ik}}^l = {L_{(i}}_{k)}^l + {F_{\left[ i \right.}}^l{j_{\left. k \right]}} + {U_{ik}}^l\) satisfying \({U_{ik}}^l{F^{ik}} = 0\). With regard to this connection \({g_{\underset + i \underset + {k\Vert l}}} = 0\) [231]. The Ricci tensor belonging to the connection introduced was not calculated.
15.2 Approximative study of field equations
15.3 Equations of motion for point particles
“In the equations for the unified field, no energytensor has been introduced: only the external problem outside the sources of the unified field (masses and charges), assumed to be singularities, exists” ([659], p. 74).^{306}
As mentioned in Section 10.3.2, Emilio Clauser (1917–1986) used the method of Einstein & Infeld in order to derive equations of motion for point particles. In [81], he had obtained an integral formula for a 2dimensional surface integral surrounding the singularities. With its help, Clauser was able to show that from Einstein’s weak field equations for two or more “particles” all classical forces in gravitation and electromagnetism (Newton, Coulomb, and Lorentz) could be obtained [82]. In his interpretation, g_{(ik)} stood for the gravitational, g_{[ik ]} for the electromagnetic field.
In a subsequent paper by Clauser, Einstein’s weak system for the field equations of UFT was developed in every order into a recursive Maxwelltype system for six 3vectors corresponding to electric and magnetic fields and intensities, and to electric and magnetic charge currents [83]. Quasistationarity for the fields was assumed.
C. Venini expanded the weak field equations by help of the formalism generated by Clauser and calculated the components of the fundamental tensor g_{ ik }, or rather of \({\gamma _{ik}}: = ({g_{(ik)}}  {\eta _{ik}}  {1 \over 2}{\eta _{ik}}{\eta ^{rs}}({g_{(rs)}}  {\eta _{rs}})\), directly up to 2nd approximation: \({\overset 4 \gamma _{ik}},{\overset 4 \gamma _{00}}\), and \({\overset 5 \gamma _{0m}}\) [672]. He applied it to calculate the inertial mass in 2nd approximation and obtained the corrections of special and general relativity; unfortunately the contribution of the electrostatic field energy came with a wrong numerical factor [673]. He also calculated the field of an electrical dipole in 2nd approximation [674]. Moreover, again by use of Clauser’s equation of motion, Venini derived the perihelion precession for a charged point particle in the field of a second one. It depends on both the charges and masses of the particles. However, his formula is not developed as far as that it could have been used for an observational test [675]. In hindsight, it is astonishing how many exhausting calculations Clauser and Venini dedicated to determining the motion of point particles in UFT in view of the ambiguity in the interpretation and formulation of the theory.
16 The Move Away from EinsteinSchrödinger Theory and UFT
Toward the end of the 1950s, we note tendencies to simplify the EinsteinSchrödinger theory with its asymmetric metric. Moreover, publications appear which keep mixed geometry but change the interpretation in the sense of a deunification: now the geometry is to house solely alternative theories of the gravitational field.
Examples for the first class are Israel’s and Trollope’s paper ([308] and some of Moffat’s papers [440, 441]. In a way, their approach to UFT was a backward move with its use of a geometry Einstein and Schrödinger had abandoned.
“If, then, grouptheoretical considerations are accepted as a basic guiding principle in the construction of a unified field theory, it will be logically most economical and satisfactory to retain the symmetry of the fundamental tensor g_{ ik }, while admitting nonsymmetrical Γ_{ ij }k.” ([308], p. 778)
In the lowest order of an expansion g_{ ij } = η_{ ij } + ϵγ_{ ij }, it turned out that the 3rd equation of (512) becomes one of Maxwell’s equations, i.e., γ^{ is },_{ s } = 0, and the first equation of (513) reduces to \(\underset  K {\,_{(ij)}} = 0\). In an approximation up to the 4th order, the Coulomb force and the equations of motion of charged particles in a combined gravitational and electromagnetic field were obtained.
16.1 Theories of gravitation and electricity in Minkowski space
“Madame Tonnelat, whose papers on the unitary theories you know well, is interested with Mr. Vigier^{308} and myself in these aspects of the quantum problem, which of course are very difficult.”^{309}
“[…] a theory of this type is much less natural and, in particular, much less convincing than general relativity. It can only arrive at a more or less efficient formalism with regard to the quantification of the gravitational field.”^{311}
Because (518) is used to derive the equations of motion for particles or continua, this answer is important. In the papers referred to and in further ones, equations of motion of (test) point particle without or within (perfectfluid) matter were studied. Thus, a link of the theory to observations in the planetary system was established [411, 413, 414]. In a paper summing up part of her research on Minkowskian gravity, S. Lederer also presented a section on perihelion advance, but which did not go beyond the results of Mme. Mavridès ([354], pp. 279–280). M.A. Tonnelat also pointed to a way of making the electromagnetic field influence the propagation of gravitational waves by introducing an induction field \({{\hat F}^{pq,r}}\) for gravitation [639]. In the presence of matter, she defined the Lagrangian“In an Euclidean theory of the gravitational field, the motion of a test particle can be associated to conservation of mass and energymomentum only if the latter is defined through the metrical tensor, not the canonical one” [647], p. 373).^{317}
“These, obviously formal, conclusions allow in principle to envisage the influence of an electromagnetic field on the propagation of the ‘gravitational rays’, i.e., a phenomenon inverse to the 2nd effect anticipated by general relativity” ([639], p. 227).^{318}
16.2 Linear theory and quantization
This would have to be compared to the GuptaBleuler formalism in quantum electrodynamics.
These papers differ in their assumptions; e.g., DrozVincent worked with the traceless quantity \({k_{AB}}  {1 \over 2}{\eta _{AB}}{\eta ^{MN}}{k_{MN}}\), for a = 0,\(\tilde b = 0\), and thus for d = −1 his results agree with those of S. Lederer. In his earlier paper, A. Capella had taken η^{ MN }k_{ MN } = 0, and μ = 0. Claude Roche applied the methods of Ph. DrozVincent to the case of mass zero fields and quantized the gravitational and the electromagnetic fields simultaneously.
16.3 Linear theory and spin1/2particles
O. Costa de Beauregard applied the linear approximation of Souriau’s theory for a field variable H_{ αβ } to describe the equations for a spin1/2 particle coupled to the photongraviton system. He obtained the equation \({\partial _\alpha}{\partial ^\alpha}{H_{5k}} =  2i{k_5}{{\hbar \chi} \over c}\bar \phi \gamma k\phi (\alpha, \beta = 1,2, \ldots 5;k = 1,2, \ldots 4)\), where the wave function ϕ again depends on the coordinate x^{5} via exp(k_{5}x^{5}); as before, χ is the coupling constant in Einstein’s field equations. Comparison with electrodynamics led to the identification \({k_5} = {e \over {\hbar \sqrt {2\chi}}}\) with e the electric charge. Costa de Beauregard also suggested an experimental test of the theory with macroscopic bodies [91].
16.4 Quantization of EinsteinSchrödinger theory?
“The endeavour to establish such a program is, to be sure, a bit premature in view of the missing secure physical interpretation of the objects to be quantized.”^{325}
Ph. DrozVincent sketched how to write down Poisson brackets and commutation relations for the EinsteinSchrödinger theory also in the framework of the “theory of the varied field” ([130]. In general, the main obstacle for quantization is formed by the constraint equations, once the field equations are split into timeevolution equations and constraint equations. DrozVincent distinguished between proper and improper dynamical variables. The system \({R_{ij}} = 0,{D_k}{{\hat g}^{\underset + i \underset  j}} = 0\), where D signifies covariant derivation with respect to the star connection (27), led to 5 constraint equations containing only proper variables arising from general covariance and λinvariance. By destroying λinvariance via a term \({{\hat g}^{rs}}{\Gamma _r}{\Gamma _s}\), one of the constraints can be eliminated. The Poisson brackets formed from these constraints were well defined but did not vanish. This was incompatible with the field equations. By introducing a nondynamical timelike vector field and its first derivatives into the Lagrangian, Ph. DrozVincent could circumvent this problem. The physical interpretation was left open [133, 135]. In a further paper, he succeeded in finding linear combinations of the constraints whose Poisson brackets are zero modulo the constraints themselves and thus acceptable for quantization [136].
17 Alternative Geometries
“[…] this theory does not pursue the hidden aim of substituting general relativity but of exploring in a rather heuristic way some specifically tough and complex domains resulting from the adoption of the principles of a nonEuclidean theory […]”. ([382], p. 327)^{326}
We will now describe theories with a different geometrical background than affine or mixed geometry and its linearized versions.
17.1 Lyra geometry
In relying on a weakened criterion for a theory to qualify as UFT suggested by Horváth (cf. Section 19.1.1), after he had added the Lagrangian for the electromagnetic field, Sen could interpret his theory as unitary. In later developments of the theory by him and his coworkers in the 1970s, it was interpreted just as an alternative theory of gravitation (scalartensor theory) [573, 574, 310]. In both editions of his book on scalartensor theory, Jordan mentioned Lyra’s “modification of Riemannian geometry which is close to Weyl’s geometry but different from it” ([319], p. 133; ([320], p. 154).
17.2 Finsler geometry and unified field theory
“The general feeling today is that in fact the nonsymmetric theory is not the correct means for unifying the two fields.”^{327}
Stephenson dropped his plan to calculate the curvature scalar R from (557) and (551), and to use it then as a Lagrangian for the field equations, because he saw no possibility to arrive at a term R (g) + F_{ st }F^{ st } functioning as a Finslerian Lagrangian. His negative conclusion was: “It so seems that this particular generalization of Riemannian geometry is not able to lead to a correct implementation of the electromagnetic field.”^{331}
18 Mutual Influence and Interaction of Research Groups
18.1 Sociology of science
18.1.1 Princeton and UFT
Einstein’s unified field theory makes a good example for showing that the influence of a model scientist may be as important in driving research as ideas coming from physics or mathematics themselves. A realistic impression seems to be that most in the group of young workers busy with Einstein’s UFT after the second world war were enticed by Einstein’s fame and authority — perhaps mediated through the prestige of their professorial advisers. They went into the field despite its being disdained by mainstreamphysicists. In view of the success of quantum field theory, it could hardly have been the methods and conceptions used in UFT that attracted them. In fact, many of those who wrote a doctoral thesis in the field dropped the subject quickly afterwards in favor of general relativity proper, or of some other field. It would be unfair to give too much importance to J. R. Oppenheimer’s reckless rating of 1935 (unpublished at the time), i.e., “Princeton is a madhouse: its solipsistic luminaries shining in separate & helpless desolation. Einstein is completely cuckoo” (Letter of J. R. Oppenheimer to his brother Frank from 11 January 1935 in [462], p. 190.) After all, much later during his time as director of the Institute for Advanced Study there, he supported people working on versions of UFT like R. L. Arnowitt who stayed at the Princeton Institute for Advanced Study from 1954 to 1956 and expressly thanked him in a paper ([5], p. 742). Nevertheless, while highly respected, Einstein and his theories lived there in splendid scientific isolation.
18.1.2 Mathematics and physics
“[…] in the end there is nothing but an X_{4} [a 4dimensional manifold] with only two fields g_{(ij)} and g_{[ij ]} and that […] the differential concomitants of these fields are ordinary concomitants of g_{(ij)},g_{[ij ]}, the curvature tensor of the symmetric connection belonging to g_{(ij)} and the covariant derivatives of these quantities with respect to this connection. […] It may be useful to introduce this new connection in order to get some heuristic principles […]. But nothing really new can ever arise from following this course.” ([540], p. 184.)
Unified field theory has stimulated the fantasy of mathematicians such that they investigated conformal geometry [670] and projective geometry [538], or even ventured to apply odd geometries to geometrize physics. We have met some of them (Finsler, Sphere, Lyra geometries). Certainly, mathematicians also both helped theoretical physicists to solve their equations and invented new equations for UFT. Unfortunately, to the exact solutions of such equations found, in most cases no physical meaning could be given. In physics, the next step in unification would be taken only in the 1960s through the joining of the weak and electromagnetic interactions in electroweak theory with gravitation being left aside, however. In mathematics, important developments leading to differential topology as well as to the theory of fiber bundles originated with E. Cartan among others — not with any of the theoretical physicist connected to the unified field theories of Einstein and Schrödinger. Gauge theory, seen as a development starting from KaluzaKlein theory, and, more recently, stringtheory (Mtheory) then presented further examples for a fruitful interaction between physics and mathematics.
18.1.3 Organization and funding
Apart from the creative abilities of the individual scientist, knowledge production depends on the institutional organization of research and the ways of communication among researchers. Thus, within a review of the history of physics, beside the conceptional developments, questions pertaining to the sociology of science cannot easily be omitted.
During the period looked at, both, changes in the funding of scientific research, and in the institutional organization toward team work and less interaction with teaching, i.e., away from university research along Humboldt’s idea of a close link between teaching and research, still were going on. In Germany, this development had begun with the establishment of the Kaiser Wilhelm Institutes (KWI, now Max Planck Institutes) since 1911, a combination of private and state funding, together with “HelmholtzGesellschaft”, “Stifterverband”, and most influential, the “Notgemeinschaft der deutschen Wissenschaft” predecessor of “Deutsche Forschungsgemeinschaft” since 1920 [676]. In France, where statefunding was the rule, the apparently not very successful “Caisse des recherches scientifiques” (1910–1934) was replaced by CNRS (Centre National de la Recherche Scientifique) only in 1939, right after the beginning of World War II. Of particular interest for the present review is the foundation of the Institut Henri Poincaré (IHP) in 1926 [579] which was built with private money but did not provide salaries. In the United States mostly private sponsors were involved; e.g., the Carnegie Corporation (1911), the Rockefeller Foundation (1913) and Einstein’s “home” institution, the Princeton Institute for Advanced Study (PIAS) (since the 1930s). The Dublin Institute for Advanced Studies in Ireland (DIAS) was founded in 1935, but funded by the government (Department of Education).^{332} In Great Britain, after the Great War a mixture of state and private funding (University grants commission, Science Research Council) persisted, coordinated in part by the Royal Society. After the 2nd world war 35 Research Associations have sprung up.
Looking at four main figures in UFT, Einstein’s and Hlavatý’s research thus depended on private donors while Schrödinger in Dublin, Lichnerowicz, and Mme. Tonnelat in Paris were statefunded. Within this external framework, there existed microstructures pertaining to the inner workings of the particular research groups.
18.2 After 1945: an international research effort
During the second world war, communications among the various scientists investigating UFT came to a halt — with the exception of the correspondence between nobel prize winners Einstein and Schrödinger. That the exchange between libraries then only slowly began to resume is clear from a letter of A. Lichnerowicz to W. Pauli of 6 October 1945: “In France, only one copy of Mathematical Review and one copy of the series Annals [of Mathematics] exist. Concerning the paper of Einstein published in 1941 at Tucuman, I know about it only through the report in Mathematical Review; it never reached France” ([489], p. 317).^{333}
From hindsight, after the second world war, research in classical unified field theory developed into a worldwide research effort. However, at the time it was no concerted action with regard to funding and organization; the only agreement among researchers consisted in the common use of scientific methods and concepts. As described in detail above in Sections 7 to 9, since the 1920s, it had been mainly Albert Einstein (1879–1955) who had pursued research on UFT if we put aside H. Weyl and A. S. Eddington. Since his separation from Berlin, most important actors of the 1930s and 1940s were Einstein himself in Princeton, and somewhat later (from 1943 on) Erwin Schrödinger (1887–1961) in Dublin. After the war, in Paris the mathematician André Lichnerowicz (1915–1998) became interested in mathematical problems related with UFT, and a student of Louis de Broglie, and later professor MarieAntoinette Tonnelat (1912–1980), built a sizable group working in the field. From the 1950s on Vavlav Hlavatý (1894–1969) in Bloomington/Indiana and his collaborators contributed prominently to the field. The Italian groups around Bruno Finzi (1899–1974) and Maria Pastori (1895–1975) were not as influential in the 1960s, perhaps because they wrote exclusively in Italian, published mostly in Italian journals and seemingly deemed networking less important although there were connections to France and the USA (cf. Sections 10 to 15).
18.2.1 The leading groups
The internal structure of the research “groups” differed greatly: in Princeton (as in Berlin), Einstein never had doctoral students but worked with postdocs like Peter Bergmann, Bannesh Hoffmann, Valentine Bargmann, Leopold Infeld, Bruria Kaufman, and Ernst Straus^{334}. At the time, further people interested in unified field theory came to the Princeton Institute for Advanced Study (PIAS), e.g., Tullio LeviCivita (1936), Vaclav Hlavatý (1937), Luis A. Santaló (1948–49) from Argentina, M. S. Vallarta from Mexico (1952), Wolfgang Pauli (1940–1946; 1949–50), Richard Lee Ingraham (1952–53), R. L. Arnowitt (1954–55), both from the USA, and D. W. Sciama (1954–55) from Great Britain. Schrödinger also worked with scientifically advanced people, mostly independent scholars at DIAS (J. McConnell, A. Papapetrou, O. H. Hittmair, L. Bass, F. Mautner, B. Bertotti). Unlike this, A. Lichnerowicz at the Collège de France and Mme. Tonnelat at the Institut Henri Poincaré (IHP) worked with many doctoral students. M.A. Tonnelat had also two experienced collaborators whom she had not advised for her doctoral theses: Stamatia Mavridès [1954–57] and Judith Winogradzki [1954–59]. None of these scientists in Paris have been scholars at the Princeton Institute; possibly, they did not belong to the proper network.
The following students and young scientists wrote their PhD theses in M.A. Tonnelat’s group at the IHP or were interacting intensely with her on the EinsteinSchrödinger type of Unified Field Theories, and on linear theories of gravitation: Jacques Lévy — Thèse 1957; Pham Tan Hoang — Thèse 1957; Dipak K. Sen — Thèse, 1958; Jean Hély — Thèse 1959; Marcel Bray — Thèse 1960; Liane Bouche, née Valere — Thèse 1961; Huyen Dangvu — Thèse 1961; Mme. Aline Surin, née Parlange — Thèse 1963; Nguyên, PhongChau; — Thèse 1963; Philippe DrozVincent — Thèse 1963;^{335} Sylvie Lederer — Thèse 1964; Huyen Dangvu — Thèse 1966, Rudolphe Bkouche — Thèse 196?. S. Kichenassamy wrote his thesis with her on general relativity in 1958. That M.A. Tonnelat advised doctoral students also on subjects outside of unified field theory is shown by the thesis of 1974 on classical renormalization by Th. Damour [99].
Scientists and PhD students closer to A. Lichnerowicz besides Yvonne Bruhat/Fourès(Bruhat)/ChoquetBruhat — Thèse 1951, but unlike her working on Unified Field Theories, were: Yves Thiry — Thèse 1950; Pham Mau Quan — Thèse 1954; Josette Charles, née Renaudie — Thèse 1956; Françoise Maurer, née Tison — Thèse 1957; Françoise Hennequin, née Guyon — Thèse 1958; Pierre V. Grosjean — Thèse 1958; Robert Vallée — Thèse 1961; Albert Crumeyrolle — Thèse 1961; Marcel Lenoir — Thèse 1962; Jean Vaillant — Thèse 1964;^{336} Claude Roche — Thèse 1969; Alphonse Capella — Thèse 1972?. Eliane Blancheton wrote her thesis in general relativity in 1961.
Y. Thiry also supervised doctoral students. Among them is P. Pigeaud with a thesis on the application of approximations in JordanThiry theory [495]. The thesis adviser for Monique SignorePoyet was Stamatia Mavridès (Thesis 1968).^{337}
The relationship between M.A. Tonnelat and A. Lichnerowicz must have been friendly and cooperative; he seems to have been the more influential in the faculty: having become a professor at the prestigious Collège de France while her application had not been successful ([92], p. 330). For the examination of Tonnelat’s PhDstudents, since 1960, Lichnerowicz was presiding the commission; she belonged to the two (or rarely three) examiners. Before Lichnerowicz, G. Darmois had been presiding several times. Tonnelat was backed by L. de Broglie; she became his successor as director of the “Centre de physique théorique” of Paris University (Sorbonne) in 1972.
From the Italian groups around Bruno Finzi and Maria Pastori (Milano) also a sizable number of doctoral degrees resulted. People working in UFT were: Paulo Udeschini (Pavia); Emilio Clauser (Milano); Elisa BrinisUdeschini (Milano); Laura Gotusso (Milano); Bartolomeo Todeschini (Milano); Franco De Simoni (Pisa); Franca Graiff (Milano) [Student of M. Pastori]; Laura Martuscelli [Student of M. Pastori]; Angelo Zanella (Milano); Luigia Mistrangioli.
Of the seven doctoral students of V. Hlavatý,^{338} two were involved in work on UFT: Robert C. Wrede [PhD 1956] and Joseph Francis Schell [PhD 1957].
We notice the considerable number of female collaborators and PhD students both in France and Italy in comparison with all the other countries. In Germany, in particular, no woman scientist has worked in UFT in the period studied.
18.2.2 Geographical distribution of scientists
Research on UFT was done on all continents but essentially centered in Europe and in the United States of America. The contributing scientists came from more than 20 different countries. The largest number of researchers in UFT, between 1955 and 1956, worked in Paris. To a lesser extent work on UFT was done also in Asia, notably in Japan since the 1930s and in India since the 1950s. In the 1970s and 1980s, many papers on exact solutions of the EinsteinSchrödinger theory and alternatives were published by Indian scientists. In the early 1960s, V. Hlavatý had a coworker from India (R. S. Mishra). Of the six papers on UFT published by S. N. Bose in Calcutta from 1953–1955, five appeared in French journals, perhaps due to his contacts established during his previous stay in Paris in 1924/25.
Did those involved in UFT move from one place to another one? During the period considered, the mobility of scientists in Europe was seriously hampered by the Naziregime, the second world war and the ensuing occupation of Eastern Europe by the USSR (“cold war”). While doing their main work, the group leaders Einstein, Schrödinger, Lichnerowicz, and Tonnelat remained at the same place, respectively. After the war had ended, Tonnelat visited Schrödinger in Dublin. Because of the political situation, Hlavatý left his native Czechoslovakia, went to Paris as a guest professor at the Sorbonne (1948). He also spent some time in Princeton following an invitation by Einstein before obtaining a position at the University of Indiana. A. Papapetrou (1907–1997) who had obtained a doctoral degree at the Polytechnical University of Stuttgart, Germany, and had been professor in Athens, Greece, during 1940–1946, was the only major contributor to UFT who changed his positions several times. He first worked in Dublin until 1948, then at the University of Manchester until 1952 when he went to Berlin. There, he headed a group in general relativity at the Academy of Sciences of the GDR until 1962. From then on he stayed in Paris at the Institut Henri Poincaré until his retirement as a “Directeur de recherche” of the CNRS. Of course, there had always been exchanges between Paris and other centers, but they were not concerned with research on UFT. In 1946/47, the French mathematician Cecile Morette[DeWitt] (1922–) originally affiliated with the JoliotCurie group, spent a year at the Institute for Advanced Studies in Dublin. She did not work with Schrödinger on UFT but with P. H. Peng on mesons. Another wellknown French mathematician, Yvonne ChoquetBruhat, who had written her dissertation with A. Lichnerowicz, in 1951/52 was at PIAS.
John Archibald Wheeler must have spent some time in Paris in 1949; he worked on atomic and nuclear physics, though. In 1957 the quantum field theorist Arthur Wightman was a guest scientist at the University of Paris and in 1963/64 and 1968/69 at IHES (Institut des Hautes Etudes Scientifiques) near Paris. Stanley Deser was a Guggenheim fellow and guest professor at the Sorbonne (University of Paris) in 1966/67, and in 1971/72 as a Fulbright Fellow. Tonnelat and Mavridès visited the Pontifical Catholic University of Rio de Janeiro (PUCRio), Brazil, as guest professors in 1971. A. Lichnerowicz visited Princeton University in 1974, i.e., outside the period considered here.
18.2.3 Ways of communications
In 1923 to 1925, when research in classical unified field theory started, the only ways of communication among scientists apart from personal visits or encounters during the rare international conferences, were notes on paper in the form of personal letters by surface mail (Einstein and Pauli are famous for their postcards), and publications in scientific journals. Correspondence sometimes included manuscripts or proof sheets. Both, Einstein and Hlavatý left an enormous correspondence.^{339} The most notable change in available services during the period 1930–1960 was the introduction of air mail and wireless services (world wide telex, since the 1930s) including radio broadcasts.^{340} Radio broadcast as well as gramophone records (Einstein in Berlin!) were used mainly for educational purposes. In principle, telegrams also would have been available but they were unwieldy and too expensive for the communication of scientific content. To a lesser degree, the same may apply to the still costly telephone calls even within the same town.

Séminaire Janet (Séminaire de mécanique analytique et de mécanique céleste);

Séminaire de l’école normale supérieure;

Séminaire Théories physiques Institut Henri Poincaré [with invited talks by Einstein, Bonnor, Stephenson, Sciama, and others];

Séminaire de Physique théorique (Séminaire Louis de Broglie).

Séminaire de Physique mathématique du Collège de France (A. Lichnerowicz).

Séminaire sur la Mécanique quantique et les particules élémentaires (J. Winogradzki).
In London, a seminar on unified field theory existed at the University College (Imperial College) (ca. 1945–ca. 1955) [G. Stephenson, C. Kilmister (1924–2010)], and continued at Kings College as a seminar on general relativity and cosmology initiated by the group around H. Bondi [F. Pirani, W. Bonnor, P. Higgs] (late 1950s–1977).
18.2.3.1 Publications
In total, about 150–170 scientists did take part in research on UFT between 1930 and 1965. If we distinguish three age cohorts according to year of birth, then in the group born until 1900 we find 27 people, from 1901–1920 three more (30), and after 1921 18 persons. This is a biased preliminary survey among less than 50 % of all involved, because the birth dates of the then doctoral students, of not so well know