Advertisement

On the History of Unified Field Theories. Part II. (ca. 1930–ca. 1965)

Abstract

The present review intends to provide an overall picture of the research concerning classical unified field theory, worldwide, in the decades between the mid-1930 and mid-1960. 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 Born-Infeld electromagnetic theory or scalar-tensor theory. Some comments on the structure and organization of research-groups are also made.

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 mid-1930s to the mid-1960s. 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 non-symmetric (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’Einstein-Schrödinger), 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 non-symmetric field”, “einheitliche Feldtheorie” etc. However, we should not forget that other types of unitary field theory were investigated during the period studied, among them Kaluza-Klein theory and its generalizations. In France, one of these ran under the name of “Jordan-Thiry” 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 mathematically-oriented part of the UFT-community 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.

At the time, the mainstream in theoretical physics had shifted to quantum mechanics and its applications in many parts of physics and physical chemistry. Quantum field theory had been invented as a relevant tool for describing the quantum aspects of atoms, molecules and their interactions with P. Jordan, M. Born and W. Heisenberg having made first steps in 1926. Dirac had put forward his “second” quantization in 1927 which was then interpreted and generalized as field quantization by Jordan, Heisenberg, Klein, Pauli, and Wigner in 1927/28. Expert histories of quantum electrodynamics and its beginning have been presentedFootnote 1 by S. Schweber [562], O. Darrigol [109], and A. Pais [470]. Around the time when Einstein left Berlin, Heisenberg and others set up theories of the strong nuclear force. Fermi had introduced a theory of weak interactions in connection with beta-decay. Since 1932/33, besides electron, photon, and proton, three new particles, namely the neutron, positron and neutrino had come into play with the last two already having been found, empirically. Anyone doubting the existence of the neutron, had to give in after nuclear fission had been discovered and nuclear reactors been built. At the 1933 Solvay conference, L. de Broglie had proposed a neutrino theory of light, i.e., with the photon as a composite particle made up by two neutrinos [111, 112], and others like P. Jordan or G. Wentzel had followed suit [314, 315, 687]. For a while, this became a much debated subject in theoretical physics. Another great topic, experimentally, was the complicated physics of “cosmic rays” containing at least another new particle with a mass about 200 times that of the electron. It was called alternatively “heavy electron”, “mesot(r)on”, and “meson” and became mixed up with the particle mediating the nuclear force the name of which was “U-quantum”, or “Yukon” after Yukawa’s suggestion in 1934/35 concerning nuclear interactions. For the history cf. [63]. When the dust had settled around 1947, the “mesotron” became the muon and the pions were considered to be the carriers of the nuclear force (strong interaction). Since 1937, the muon had been identified in cosmic rays [455, 593]. The charged pion which decays into a muon and a (anti-)neutrino via the weak interaction was detected in 1947, the uncharged one in 1950. In the 1940s, quantum electrodynamics was given a new kick by Feynman, Schwinger and Tomonaga. Up to the mid fifties, nuclear theory had evolved, the strong and weak nuclear forces were accepted with the neutrino observed only in 1957, after Einstein’s death. Thus, the situation had greatly changed during the two decades since Einstein had started to get involved in unified field theory: in the 1920s only two fundamental interactions had been known, both long-range: the electromagnetic and the gravitational. Before 1926, neither non-relativistic quantum theory, nor relativistic quantum electrodynamics had been developed. In 1928, with Dirac’s equation, “spin” had appeared as a new property of elementary particles. After a brief theoretical venture into spinors and the Dirac equation (cf. Section 7.3 of Part I and Section 4.1), against all of the evidence concerning new particles with half-integer spin and new fundamental interactions obtained in the meantime, Einstein continued to develop the idea of unifying only the electromagnetic and gravitational fields via pure geometry, cf. Section 7 below. His path was followed in much of the research done in classical UFT. Occasionally, as in Schrödinger’s and Tonnelat’s work, meson fields, treated as classical fields, were also included in the interpretation of geometric objects within the theory. The state of affairs was reflected, in 1950, in a note in the Scientific American describing Einstein’s motivation for UFT as:

“to relate the physical phenomena in the submicroscopic world of the atom to those in the macroscopic world of universal space-time, 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 Einstein-Schrödinger 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 semi-classical 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).

In their demands on UFT, Einstein and Schrödinger differed: while the first one never gave up his hope to find a substitute, or at least a needed foundation for quantum theory in his classical unified field theory, Schrödinger saw his theory as a strictly classical groundwork for an eventual alternative to quantum field theory or, as he expressed it himself, as “‘the classical analogue’ of the true laws of Nature” ([551], p. 50).Footnote 2 Einstein in particular followed his way towards UFT unwaveringly in spite of failing success. Shortly before his death, he even reinterpreted his general relativity, the central concept of which had been the gravitational and inertial potentials encased in the (pseudo-)Riemannian metric tensor, through the lens of unified field theory:

“[…] 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 Γ-field can be deduced from a Riemannian metric […].”Footnote 3 (A. Einstein, 4 April 1955, letter to M. Pantaleo, in ([473], pp. XV–XVI); English translation taken from Hehl and Obuchov 2007 [244].)

To me, this is not a prophetic remark pointing to Abelian and non-Abelian gauge theories which turned out to play such a prominent role in theoretical physics, a little later.Footnote 4 Einstein’s gaze rather seems to have been directed backward to Levi-Civita, Weyl’s paper of 1918 [688], and to Eddington.Footnote 5 The Institute for Advanced Study must have presented a somewhat peculiar scenery at the end of the 1940s and early 50s: among the senior faculty in the physics section as were Oppenheimer, Placzek and Pais, Einstein remained isolated. That a “postdoc” like Freeman Dyson had succeeded in understanding and further developing the different approaches to quantum electrodynamics by Schwinger and Feynman put forward in 1948, seemingly left no mark on Einstein. Instead, he could win the interest and help of another Princeton postdoc at the time, Bruria Kaufman, for his continued work in UFT [587]. We may interpret a remark of Pauli as justifying Einstein’s course:

“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)Footnote 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 (four-fermion 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.)

About a month after Einstein’s death, the mathematician A. Lichnerowicz had the following to say concerning his unified field theory:

“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.)Footnote 7

In Bern, Switzerland, three months after Einstein’s death, a “Jubilee Conference” took place commemorating fifty years of relativity since the publication of his famous 1905 paper on the electrodynamics of moving bodies. Unified field theory formed one of its topics, with 34 contributions by 32 scientists. In 1955, commemorative conferences were also held in other places as well which included brief reviews of UFT (e.g., by B. Finzi in Bari [203] and in Torino [203]). Two years later, among the 21 talks of the Chapel Hill Conference on “the role of gravitation in physics” published [119], only a single one dealt with the “Generalized Theory of Gravitation” [344]. Again five years later, after a conference on “Relativistic Theories of Gravitation”, the astronomer George C. McVittie (1904–1988) could report to the Office of Naval Research which had payed for his attendance: “With the death of Einstein, the search for a unified theory of gravitation and electro-magnetism has apparently faded into the background.” (Quoted in [523], p. 211.) This certainly corresponded to the majority vote. At later conferences, regularly one contribution or two at most were devoted to UFT [302, 382]. From the mid-1960s onward or, more precisely, after the Festschrift for V. Hlavatý of 1966 [282], even this trickle of accepted contributions to UFT for meetings ran dry. “Alternative gravitational theories” became a more respectable, but still a minority theme. Not unexpectedly, some went on with their research on UFT in the spirit of Einstein, and some are carrying on until today. In particular, in the 1970s and 80s, interest in UFT shifted to India, Japan, and Australia; there, in particular, the search for and investigation of exact solutions of the field equations of the Einstein-Schrödinger unified field theory became fashionable. Nevertheless, Hlavatý’s statement of 1958, although quite overdone as far as mathematics is concerned, continues to be acceptable:

“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 Einstein-Schrödinger non-symmetric 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 four-page résumé of both the Einstein-Schrödinger and the Kaluza-Klein approaches to unified field theory [12].Footnote 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 non-technical 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 Einstein-Schrödinger 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.Footnote 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.Footnote 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.

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.

Metrical structure

First, a definition of the distance ds between two infinitesimally close points on a D-dimensional 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 dxi 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.

In the following, ds is linked to a non-degenerate bilinear form g (X, Y), called the first fundamental form; the corresponding quadratic form defines a tensor field, the metrical tensor, with D2 components g ij such that

$$ds = \sqrt {{g_{ij}}\;d{x^i}\;d{x^j}} ,$$
(1)

where the neighboring points are labeled by xi and xi + dxi, respectivelyFootnote 11. Besides the norm of a vector \(\vert X \vert := \sqrt {{g_{ij}}{X^i}{X^j}}\), the “angle” between directions X, Y can be defined by help of the metric:

$$\cos (\angle (X,Y)): = {{{g_{ij}}{X^i}{Y^j}} \over {\vert X\parallel Y\vert}}.$$

From this we note that an antisymmetric part of the metrical’s tensor does not influence distances and norms but angles.

We are used to g being a symmetric tensor field, i.e., with g ik = g(ik) with only D (D + 1)/2 components; in this case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is ±(D − 2)Footnote 12. In this case, the norm is \(|X|: = \sqrt {|{g_{ij\,}}{X^i}{X^j}|}\). In space-time, i.e., for D = 4, the Lorentzian signature is needed for the definition of the light cone: g ij dxidxj = 0. The paths of light signals through the cone’s vertex are assumed to lie in this subspace. In unified field theory, the line element (“metric”) g ik is an asymmetric tensor, in general. When of full rank, its inverse gik is defined throughFootnote 13

$${g_{mi}}{g^{mj}} = \delta _i^j\,,\;\;\;{g_{im}}{g^{jm}} = \delta _i^j\,.$$
(2)

In the following, the decomposition into symmetric and antisymmetric parts is denoted byFootnote 14:

$${g_{ik}} = {h_{ik}} + {k_{ik}},$$
(3)
$${g^{ik}} = {l^{ik}} + {m^{ik}}.$$
(4)

h ik and lik have the same rank; also, h ik and lik have the same signature [27]. Equation (2) looks quite innocuous. When working with the decompositions (3), (4) however, eight tensors are floating around: h ik and its inverse hik (indices not raised!); k ik and its inverse kik; likhik and its inverse l ik h ik , and finally mikkik and its inverse m ik .

With the decomposition of the inverse gjm (4) and the definitions for the respective inverses

$${h_{ij}}{h^{ik}} = \delta _j^k;\;\;{k_{ij}}{k^{ik}} = \delta _j^k;\;\;{l_{ij}}{l^{ik}} = \delta _j^k;\;\;{m_{ij}}{m^{ik}} = \delta _j^k\,,$$
(5)

the following relations can be obtained:Footnote 15

$${l^{ik}} = {l^{(ik)}} = {h \over g}{h^{ik}} + {k \over g}{k^{im}}{k^{kn}}{h_{mn}}$$
(6)

and

$${m^{ik}} = {m^{[ik]}} = {k \over g}{k^{ik}} + {h \over g}{h^{im}}{h^{kn}}{k_{mn}}$$
(7)

where g =: det(g ik ) ≠ 0, k =: det(k ik ) ≠ 0, h =: det(h ik ) ≠ 0. We also note:

$$g = h + k + {h \over 2}{h^{kl}}{h^{mn}}{k_{km}}{k_{ln}}\,,$$
(8)

and

$$g{g^{ij}} = h{h^{ij}} + k{k^{ij}} + h{h^{ir}}{h^{js}}{k_{rs}} + k{k^{ir}}{k^{js}}{h_{rs}}\,.$$
(9)

Another useful relation is

$${g^2} = {h \over l}\,,$$
(10)

with l = det(l ij ). From (9) we see that unlike in general relativity even invariants of order zero (in the derivatives) do exist: \({k \over h}\), and hklhmnk km k ln ; for the 24 invariants of the metric of order 1 in space-time cf. [512, 513, 514].

Another consequence of the asymmetry of g ik is that the raising and lowering of indices with g ik now becomes more complicated. For vector components we must distinguish:

$$\overset{\rightarrow}{y}_{\dot k} : = {g_{kj}}{y^j}\,,\quad \overset{\leftarrow}{y}_{\dot k} : = {y^j}{g_{jk}}\,,$$
(11)

where the dot as an upper index means that an originally upper index has been lowered. Similarly, for components of forms we have

$$\overset{\rightarrow}{w}_{.}^{k} : = {g^{kj}}{w_j}\,,\quad \overset{\leftarrow}{w}_{.}^{k} : = {w_j}{g^{jk}}\,.$$
(12)

The dot as a lower index points to an originally lower index having been raised. In general, \(\overset{\rightarrow}{y}_{\overset{.}{k}}\neq \overset{\leftarrow}{y}_{\overset{.}{k}} \,, \overset{\rightarrow}{w}^{\underset{.}{k}}\neq \overset{\leftarrow}{w}^{\underset{.}{k}}\). Fortunately, the raising of indices with the asymmetric metric does not play a role in the following.

An easier possibility is to raise and lower indices by the symmetric part of g jk , i.e., by h jk and its inverse hij.Footnote 16 In fact, this is often seen in the literature; cf. [269, 297, 298]. Thus, three new tensors (one symmetric, two skew) show up:

$$\begin{array}{*{20}c} {{{\check k}^{ij}}: = {h^{is}}{h^{jt}}{k_{st}} \neq {k^{ij}},\quad {{\check l}_{ij}}: = {h_{is}}{h_{jt}}{l^{st}} \neq {l_{ij}}\,,\quad \quad \quad} \\ {{{\check m}_{ij}}: = {h_{is}}{h_{jt}}{m^{st}} \neq {m_{ij}},\quad {{\check h}^{ij}}: = {h^{ij}}\,.} \\ \end{array}$$

Hence, Ikeda instead of (9) wrote:

$$g{g^{ij}} = h[{h^{ij}}(1 + {1 \over 2}{k_{ij}}{\check k ^{ij}}) + {\check k ^{ij}} - {\check k ^{ir}}\check k _{\dot s}^{j} + {\rho \over {2h}}{\epsilon ^{ijrs}}{k_{rs}}]\,,$$

with \(\rho := {1 \over 8}{\epsilon ^{ijlm}}{\overset \wedge k_{ij}}{\overset \wedge k_{lm}}\). For a physical theory, the “metric” governing distances and angles must be a symmetric tensor. There are two obvious simple choices for such a metric in UFT, i.e., h ik and lik. For them, in order to be Lorentz metrics, h < 0 (l:= det(lij) < 0) must hold. The light cones determined by h ik and by lik are different, in general. For further choices for the metric cf. Section 9.7.

The tensor density formed from the metric is denoted here by

$${\hat g^{ij}} = \sqrt {- g} {g^{ij}},{\hat g_{ij}} = {(\sqrt {- g})^{- 1}}{g_{ij}}\,.$$
(13)

The components of the flat metric (Minkowski-metric) in Cartesian coordinates is denoted by \({\eta _{ik}}\):

$${\eta _{ik}} = \delta _i^0\delta _k^0 - \delta _i^1\delta _k^1 - \delta _i^2\delta _k^2 - \delta _i^3\delta _k^3\,.$$

.

Affine structure

The second structure to be introduced is a linear connection (affine connection, affinity) L with D3 components \({L_{ij}}^k\); it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations.Footnote 17 The connection is a device introduced for establishing a comparison of vectors in different points of the manifold. By its help, a tensorial derivative ∇, called covariant derivative is constructed. For each vector field and each tangent vector it provides another unique vector field. On the components of vector fields X and linear forms ω it is defined by

$$\overset{+}{\nabla}_k{X^i} = X^{\underset{+}{i}}_{\;\;\;\parallel k}: = {{\partial {X^i}} \over {\partial {x^k}}} + L_{kj}^{\;\;i}{X^j},\overset{+}{\nabla}_k{\omega _i}: = \omega_{\underset{+}{i}{\parallel k}}= {{\partial {\omega _i}} \over {\partial {x^k}}} - L_{ki}^{\;\;j}{\omega _j}.$$
(14)

The expressions \({\overset +\nabla _k}{X^i}\) and \({{\partial {X^\iota}} \over {\partial {x^k}}}\) are abbreviated by \(X_{\vert\vert k}^{\overset {i}{+}}\) and \(X_{,k}^i = {\partial _k}{X^i}\). For a scalar f, covariant and partial derivative coincide: \({\nabla _i}f = {{\partial f} \over {\partial {x_i}}} \equiv {\partial _i}f \equiv f{,_i}\). The antisymmetric part of the connection, i.e.,

$$S_{ij}^{\;\;k} = L_{[ij]}^{\;\;k}$$
(15)

is called torsion; it is a tensor field. The trace of the torsion tensor \({S_i} =: \,{S_{il}}^l\) is called torsion vector or vector torsion; it connects to the two traces of the linear connection \({L_i} =: {L_{il}}^l\); \({\tilde L_j} =: {L_{il}}^l\) as \({S_i} = 1/2({L_i} - {\tilde L_i})\). Torsion is not just one of the many tensor fields to be constructed: it has a very clear meaning as a deformation of geometry. Two vectors transported parallelly along each other do not close up to form a parallelogram (cf. Eq. (22) below). The deficit is measured by torsion. The rotation \({\overset {+}{\nabla} _k}{\omega _i} - {\overset {+}{\nabla} _i}{\omega _k}\) of a 1-form now depends on torsion \({S_{ki}}^r\):

$$\overset{+}{\nabla}_{{k}}{\omega _i} -\overset{+}{\nabla}_{{i}}{\omega}_k= {{\partial {\omega _i}} \over {\partial {x^k}}} - {{\partial {\omega _k}} \over {\partial {x^i}}} - 2S_{ki}^{\;\;\;r}{\omega _r}\,.$$

We have adopted the notational convention used by Schouten [537, 540, 683]. Eisenhart and others [182, 438] change the order of indices of the components of the connection:

$$\overset{-}{\nabla}_{{k}} X^i = X^{\underset{-}{i}}_{\;\;\;\parallel k}: = {{\partial {X^i}} \over {\partial {x^k}}} + L_{jk}^{\;\;\;i}{X^j},\quad \overset{-}{\nabla}_{{k}}{\omega}_i:=\omega_{\underset{-}{i}{\parallel k}}= {{\partial {\omega _i}} \over {\partial {x^k}}} - L_{ik}^{\;\;\;j}{\omega _j}\,,$$
(16)

whence follows

$$\overset{-}{\nabla}_{{k}}{\omega _i} -\overset{-}{\nabla}_{{i}}{\omega _k} = {{\partial {\omega _i}} \over {\partial {x^k}}} - {{\partial {\omega _k}} \over {\partial {x^i}}} + 2S_{ki}^{\;\;r}{\omega _r}\,.$$

As long as the connection is symmetric this does not make any difference because of

$$\overset{+}{\nabla}_{{k}}{X^i} - \overset{-}{\nabla}_{{k}}{X^i} = 2S_{[kj]}^{\;\;\;\;\;\;i}{X^j} = 0.$$
(17)

for both kinds of derivatives we have:

$$\overset{+}{\nabla}_{{k}}({v^l}{w_l}) = {{\partial ({v^l}{w_l})} \over {\partial {x^k}}};\;\;\overset{-}{\nabla}_{{k}}({v^l}{w_l}) = {{\partial ({v^l}{w_l})} \over {\partial {x^k}}}.$$
(18)

Both derivatives are used in versions of unified field theory by Einstein and others.Footnote 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.

The covariant derivative with regard to the symmetrical part of the connection \(L_{(kj)}^l = {\Gamma _{jk}}^l\) is denoted by \({\overset {0}{\nabla}_k}\) such thatFootnote 19

$$\overset{0}{\nabla}_k {X^i} = X_{\underset{0}{\parallel}{k}}^i = {{\partial {X^i}} \over {\partial {x^k}}} + \Gamma _{kj}^{\;\;\;\;i}{X^j},\;\;\;\overset{0}{\nabla}_{{k}}{\omega _i} = \omega_{i\underset{0}{\parallel}{k}} = {{\partial {\omega _i}} \over {\partial {x^k}}} - \Gamma _{ki}^{\;\;\;\;j}{\omega _j}\,.$$
(19)

In fact, no other derivative is necessary if torsion is explicitly introduced, because ofFootnote 20

$$\overset{+}{\nabla}_k{X^i} = \overset{0}{\nabla}_k{X^i} + S_{km}^{\;\;\;\;\;i}{X^m}\,,\;\;\overset{+}{\nabla}_{{k}}{\omega _i} = \overset{0}{\nabla}_{{k}}{\omega _i} - S_{ki}^{\;\;\;\;m}{\omega _m}\,.$$
(20)

In the following, \({\Gamma _{ij}}^k\) always will denote a symmetric connection if not explicitly defined otherwise. To be noted is that: \({\lambda _{[i,k]}} = {\overset {+}{\nabla} _{[k\lambda i]}} + {\lambda _s}{S_{ki}}^s = {\overset {0}{\nabla} _{[k\lambda i]}}\).

For a vector density of coordinate weight \(z\,{\hat X^i}\), the covariant derivative contains one more term (cf. Section 2.1.5 of Part I):

$$\overset{+}{\nabla}_{{k}}{\hat X^i} = {{\partial {{\hat X}^i}} \over {\partial {x^k}}} + L_{kj}^{\;\;\;\;\;i}{\hat X^j} - z\;\;\;L_{kr}^{\;\;\;\;\;r}{\hat X^i},\;\;\;\overset{-}{\nabla}_{{k}}{\hat X^i} = {{\partial {X^i}} \over {\partial {x^k}}} + L_{jk}^{\;\;\;\;\;i}{\hat X^j} - z\;\;\;L_{rk}^{\;\;\;\;\;r}{\hat X^i}.$$
(21)

The metric density of Eq. (13) has coordinate weight z = 1.Footnote 21 For the concept of gauge weight cf. (491) of Section 13.2.

A smooth vector field Y is said to be parallelly transported along a parametrized curve λ(u) with tangent vector X iff for its components \(Y_{\Vert k}^i{X^k}(u) = 0\) holds along the curve. A curve is called an autoparallel if its tangent vector is parallelly transported along it in each point:Footnote 22

$$X_{\;\;\parallel k}^i{X^k}(u) = \sigma (u){X^i}.$$
(22)

By a particular choice of the curve’s parameter, σ = 0 may be reached. Some authors use a parameter-invariant condition for auto-parallels: XlXikXk (u) − XiXlkXk (u) = 0; cf. [284].

A transformation mapping auto-parallels to auto-parallels is given by:

$$L_{ik}^{\;\;\;j} \rightarrow L_{ik}^{\;\;\;j} + \delta _{\;(i}^j{\omega _{k)}}.$$
(23)

The equivalence class of auto-parallels defined by (23) defines a projective structure on M D [691], [690]. The particular set of connections

$$_{(p)}L_{ij}^{\;\;\;\;k} = :L_{ij}^{\;\;\;\;k} - {2 \over {D + 1}}\delta _{\;(i}^k{L_{j)}}$$
(24)

with \({L_j} =: {L_{jm}}^m\) is mapped into itself by the transformation (23), cf. [608].

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.

From the connection \({L_{ij}}^k\) further connections may be constructed by adding an arbitrary tensor field T to it or to its symmetrized part:

$$\bar L_{ij}^{\;\;\;\;k} = L_{ij}^{\;\;\;\;\;k} + T_{ij}^{\;\;\;\;\;k},$$
(25)
$$\bar \bar L_{ij}^{\;\;\;\;\;k} = L_{(ij)}^{\;\;\;\;\;\;k} + T_{ij}^{\;\;\;k} = \Gamma _{ij}^{\;\;\;k} + {\tilde T_{ij}}\,.$$
(26)

By special choice of T or \(\tilde T\) we can regain all connections used in work on unified field theories. One case is given by Schrödinger’s “star”-connection:

$$^\ast L_{ij}^{\;\;\;\;\;k} = L_{ij}^{\;\;\;\;\;k} + {2 \over 3}\delta _i^k{S_j}\,,$$
(27)

for which \({}^\ast{L_{ik}}^k = {}^\ast{L_{ki}}^k\) or *S i = 0. The star connection thus shares the vanishing of the torsion vector with a symmetric connection. Further examples will be encountered in later sections; cf. (382) of Section 10.3.3.

Metric compatibility, non-metricity

We now assume that in affine space also a metric tensor exists. In the case of a symmetric connection the condition for metric compatibility reads:

$$^\Gamma {\nabla _k}{g_{ij}} = {g_{ij,k}} - {g_{rj}}\Gamma _{ki}^{\;\;\;r} - {g_{ir}}\Gamma _{kj}^{\;\;\;r} = 0\,.$$
(28)

In Riemannian geometry this condition guaranties that lengths and angles are preserved under parallel transport. The corresponding torsionless connectionFootnote 23 is given by:

$$\Gamma _{ij}^{\;\;\;\;k} = \{_{ij}^k\} = {1 \over 2}{g^{ks}}({\partial _j}{g_{si}} + {\partial _i}{g_{sj}} - {\partial _s}{g_{ij}}){.}$$
(29)

In place of (28), for a non-symmetric connection the following equation was introduced by Einstein (and J. M. Thomas) (note the position of the indices!)Footnote 24

$$0 = g_{\underset{+}{i}\underset{-}{k}{\parallel l}} = :{g_{ik,l}} - {g_{rk}}L_{il}^{\;\;\;r} - {g_{ir}}L_{lk}^{\;\;r}\,.$$
(30)

As we have seen in Section 2.1.1, this amounts to the simultaneous use of two connections: \(L_{ij}^ + {}^k =: {L_{(ij)}}^k + {S_{ij}}^k = {L_{ij}}^k\) and \({{\bar L}_{ij}}^k =: {L_{(ij)}}^k - {S_{ij}}^k = {L_{ij}}^k\).Footnote 25 We will name (30) “compatibility equation” although it has lost its geometrical meaning within Riemannian geometry.Footnote 26 In terms of the covariant derivative with regard to the symmetric part of the connection, (30) reduces to

$$0 = g_{\underset{+}{i}\underset{-}{k}{\parallel l}} = :{g_{ik {\underset{0}{\parallel}l}}} - 2S_{(i\vert l\vert k)}^{\quad \;\;.} + 2{k_{r\left[ i \right.}}S_{\left. k \right]l}^{\;\;\;\;r}\,.$$
(31)

In the 2nd term on the r.h.s., the upper index has been lowered with the symmetric part of the metric, i.e., with h ij . After splitting the metric into its irreducible parts, we obtainFootnote 27

$$0 = g_{\underset{+}{i}\underset{-}{k}{\parallel l}} = :{h_{ik {\underset{0}{\parallel}l}}} +{k_{ik {\underset{0}{\parallel}l}}}- 2S_{(i\vert l\vert k)}^{\quad \;\;.} + 2{k_{r\left[ i \right.}}S_{\left. k \right]l}^{\;\;\;\;r}\,,$$

or (cf. [632], p. 39, Eqs. (S1), (A1)):

$$h_{ik {\underset{0}{\parallel}l}} + 2{h_{r(i}}S_{k)l}^{\;\;\;\;\;r} = 0\,,\;\;k_{ik\underset{0}{\parallel}{l}} + 2{k_{r\left[ i \right.}}S_{\left. k \right]l}^{\;\;\;\;\;r} = 0\,.$$
(32)

Eq. (32) plays an important role for the solution of the task to express the connection L by the metric and its first partial derivatives. (cf. Section 10.2.3.)

In place of (30), equivalently, the ±-derivative of the tensor density ĝik can be made to vanish:

$$\hat{g}^{\underset{+}{i}\underset{-}{k}}_{\;\;\;\;\parallel l}= \hat g_{\;\,,l}^{ik} + {\hat g^{sk}}L_{sl}^{\;\;\;\;i} + {\hat g^{is}}L_{ls}^{\;\;\;\;k} - {\hat g^{ik}}L_{(ls)}^{\;\;\;\;s} = 0\,.$$
(33)

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.Footnote 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].Footnote 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, \({\overset 0 \nabla _k}\delta _i^j = 0\).

A clearer presentation of (30) is given in Koszul-notation:

$$\overset{\pm}{\nabla}_{Z} g(X,Y): = Zg(X,Y) - g(\overset{+}{\nabla}_{Z}X,Y) - g(X,\overset{-}{\nabla}_{Z}Y)\,.$$
(34)

The l.h.s. of (34) is the non-metricity tensor, a straightforward generalization from Riemannian geometry:

$$\overset{\pm}{Q}(Z,X,Y): = {\overset{\pm}{\nabla}_Z}g(X,Y) = {Z^l}{X^i}{Y^k}{g_{\underset{+}{i}}{\underset{-}{k}}{\parallel l}} = - {Z^l}{X^i}{Y^k}\;{\overset{\pm}{Q}}_{lik}\,.$$
(35)

(34) shows explicitly the occurrence of two connections; it also makes clear the multitude of choices for the non-metricity tensor and metric-compatibility. In principle, Einstein could have also used:

$${\overset{++}{\nabla}_Z}g(X,Y): = Zg(X,Y) - g{(\overset{+}{\nabla} _Z}X,Y) - g(X,{\overset{+}{\nabla} _Z}Y)\,,$$
(36)
$${\overset{--}{\nabla}_Z}g(X,Y): = Zg(X,Y) - g{(\overset{-}{\nabla} _Z}X,Y) - g(X,{\overset{-}{\nabla} _Z}Y)\,,$$
(37)
$${\overset{00}{\nabla}_Z}g(X,Y): = Zg(X,Y) - g{(\overset{0}{\nabla} _Z}X,Y) - g(X,{\overset{0}{\nabla} _Z}Y)\,.$$
(38)

and further combinations of the 0- and ±-derivatives. His adoption of (30) follows from a symmetry demanded (Hermitian or transposition symmetry); cf. Section 2.2.2.

An attempt for keeping a property of the covariant derivative in Riemannian geometry, i.e., preservation of the inner product under parallel transport, has been made by J. Hély [249]. He joined the equations 0 = gikl; 0 = gikl to Eq. (30). In the presence of a symmetric metric h ij , in place of Eqs. (25), (26) a decomposition

$$L_{ij}^{\;\;\;\;k} = {\{_{ij}^k)_h} + u_{ij}^{\;\;\;\;k}$$
(39)

with arbitrary \(u_{ij}^{\,\,\,\,k}\) can be made.Footnote 30 Hély’s additional condition leads to a totally antisymmetric \(u_{ij}^{\,\,\,\,k}\).

We will encounter another object and its derivatives, the totally antisymmetric tensor:

$${\epsilon _{ijkl}}: = \sqrt {- g} \;{\eta _{ijkl}}\,,\quad {\epsilon ^{ijkl}}: = (1/\sqrt {- g})\;{\eta ^{ijkl}}\,,$$
(41)

where ηijkl is the totally antisymmetric tensor density containing the entries 0, ±1 according to whether two indices are equal, or all indices forming an even or odd permutation. For certain derivatives and connections, the object can be covariantly constant [473, 484]:

$$\epsilon _{\underset{0}{i} \underset{0}{j} \underset{0}{k} \underset{0}{l}\parallel r} = 0\,,\quad \epsilon^{\underset{+}{i} \underset{+}{j} \underset{+}{k} \underset{+}{l}}_{\;\;\;\;\;\;\;\; \;\;\parallel r} = \epsilon^{\underset{-}{i} \underset{-}{j} \underset{-}{k} \underset{-}{l}}_{\;\;\;\;\;\; \;\;\;\parallel r} = \epsilon^{ijkl} L_{[rs]}^{\;\;\;\;\;s} \,.$$
(42)

Symmetries

Transformation with regard to a Lie group

In Riemannian geometry, a “symmetry” of the metric with regard to a C-generator \(X = {\xi ^a}{\partial \over {\partial {x^a}}}\) of a Lie algebra (corresponding, locally, to a Lie-group)

$$[{X_{(i)}},{X_{(j}})] = c_{ij}^{\;\;\;\;l}{X_{(l)}}\,,$$

is defined by

$${{\mathcal L}_\xi}{g_{ab}} = 0 = {g_{ab,c}}\;{\xi ^c} + {g_{cb}}\;\xi _{\,,a}^c + {g_{ac}}\;\xi _{\,,b}^c\,.$$
(43)

The vector field ξ is named a Killing vector; its components generate the infinitesimal symmetry transformation: \({x^i} \to {x^{i\prime}} = {x^i} + {\xi ^i}\). Equation (43) may be expressed in a different form:

$${{\mathcal L}_\xi}{g_{ab}} = 2{\overset{g}{\nabla} _{(a}}{\xi _{b)}} = 0.$$
(44)

In (44), \(\overset 0 \nabla\) is the covariant derivative with respect to the metric g ab [Levi-Civita connection; cf. (29)]. A conformal Killing vector η satisfies the equation:

$${{\mathcal L}_\eta}{g_{ab}} = f({x^l}){g_{ab}}\,.$$
(45)

Hermitian symmetry

This is a generalization (a weakening) of the symmetrization of a real symmetric metric and connection:Footnote 31 Hermitian “conjugate” metric and connection are introduced for a complex metric and connection by

$${\tilde g_{ik}}: = {g_{ki}};\;\tilde L_{ij}^{\;\;\;\;k}: = L_{ji}^{\;\;\;k}.$$
(46)

In terms of the real tensors h ik , k ik , \(L_{ij}^{\,\,\,\,k}\), \(S_{ij}^{\,\,\,\,k}\), i.e., of g ik = h ik + i kik, \(L_{ij}^{\,\,\,\,k} = \Gamma _{ij}^{\,\,\,k} + i\,S_{ij}^{\,\,\,k}\) obviously \({\tilde g_{ik}} = {\overset - g _{ik}}, \, \tilde L_{ij}^{\,\,\,\,k} = \overset - L _{ij}^{\,\,\,\,k}\) holds, if the symmetry of h ik and the skew-symmetry of k ik are taken into account. For a real linear form \({\omega _i}:\,({\overset + \nabla _k}{\omega _i}){}^ \sim = {\overset - \nabla _i}\omega k\). Hermitian symmetry then means that for both, metric and connection, \({\tilde g_{ik}} = {g_{ik}}, \, \tilde L_{ij}^{\,\,\,\,k}: = L_{ij}^{\,\,\,\,\,k}\) is valid. For the determinant g of a metric with Hermitian symmetry, the relation \(g = \overset - g\) holds.

The property “Hermitian” (or “self-conjugate”) can be generalized for any pair of adjacent indices of any tensor (cf. [149], p. 122):

$${\tilde A_{\ldots ik \ldots}}({g_{rs}}): = {A_{\ldots ki \ldots}}({g_{sr}})\,.$$
(47)

à ij is called the (Hermitian) conjugate tensor. A tensor possesses Hermitian symmetry if Ãik (g rs ) = Aik (g rs ). Einstein calls a tensor anti-Hermitian if

$${\tilde A_{\ldots ik \ldots}}({g_{rs}}): = {A_{\ldots ki \ldots}}({g_{sr}}) = - {A_{\ldots ik \ldots}}({g_{rs}})\,.$$
(48)

As an example for an anti-Hermitian vector we may take vector torsion \({L_i} = L_{[il]}^{\,\,\,\,\,\,\,l}\) with \({\tilde L_i} = - {L_i}\). The compatibility equation (30) is Hermitian symmetric; this is the reason why Einstein chose it.

For real fields, transposition symmetry replaces Hermitian symmetry.

$${\tilde g_{ij}}: = {g_{ji}}\overset{!}{=} {g_{ij}}\,,\;\;\tilde L_{ij}^{\;\;\;\;k}: = L_{ji}^{\;\;\;\;k}\overset{!}{=} L_{ij}^{\;\;\;\;k}\,,$$
(49)

with à ij = A ji .

In place of (47), M.-A. Tonnelat used

$${\tilde A_{\ldots ik \ldots}}(L_{rs}^{\;\;\;\;t}): = {A_{\ldots ki \ldots}}(\tilde L_{rs}^{\;\;\;\;t})$$
(50)

as the definition of a Hermitian quantity [627]. As an application we find \({\tilde g_{\underset + i \,\underset - k \Vert l}} = {g_{\underset + i \,\underset - k \Vert l}}\) and \({\tilde \hat g^{\underset + i \,\underset - k}}_{\,\,\,\,\,\,\,\Vert l} = {\hat g^{\underset - i \,\underset + k}}_{\,\,\,\,\,\,\,\Vert l}\).

λ-transformation

In (23) of Section 2.1.1, we noted that transformations of a symmetric connection \(\Gamma _{ij}^{\,\,\,\,k}\) which preserve auto-parallels are given by:

$$\prime \Gamma _{ij}^{\;\;\;k} = \Gamma _{ij}^{\;\;\;k} + {\lambda _i}\delta _j^{\;\;k} + {\lambda _j}\delta _i^k\,,$$
(51)

where λ i is a real 1-form field. They were named projective by Schouten ([537], p. 287). In later versions of his UFT, Einstein introduced a “symmetry”-transformation called λ-transformation [156]:

$$\prime \Gamma _{ij}^{\;\;\;k} = \Gamma _{ij}^{\;\;\;k} + {\lambda _j}\delta _i^k.$$
(52)

Einstein named the combination of the “group” of general coordinate transformations and λ-transformations the “extended” group U. For an application cf. Section 9.3.1. After gauge-(Yang-Mills-) theory had become fashionable, λ-transformations with λ i = i λ were also interpreted as gauge-transformations [702, 23]. According to him the parts of the connection irreducible with regard to diffeomorphisms are “mixed” by (52), apparently because both will then contain the 1-form λ i . Under (52) the torsion vector transforms like \(\prime{S_k} = {S_k} - {3 \over 2}{\lambda _i}\), i.e., it can be made to vanish by a proper choice of λ.

The compatibility equation (30) is not conserved under λ-transformations 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 λ-transformations is known.

Affine geometry

We will speak of affine geometry in particular if only an affine connection exists on the 4-manifold, not a metric. Thus the concept of curvature is defined.

Curvature

In contrast to Section 2.1.3 of Part I, the two curvature tensors appearing there in Eqs. (I,22) and (I,23) will now be denoted by the ±-sign written beneath a letter:

$$\underset{+}{K}_{\;jkl}^i = {\partial _k}L_{lj}^{\;\;\;\;i} - {\partial _l}L_{kj}^{\;\;\;\;i} + L_{km}^{\;\;\;\;\;i}L_{lj}^{\;\;\;\;m} - L_{lm}^{\;\;\;\;\;i}L_{kj}^{\;\;\;\;m},$$
(53)
$$\underset{-}{K}_{\;jkl}^i = {\partial _k}L_{jl}^{\;\;\;\;i} - {\partial _l}L_{jk}^{\;\;\;\;i} + L_{mk}^{\;\;\;\;\;i}L_{jl}^{\;\;\;\;m} - L_{ml}^{\;\;\;\;\;i}L_{jk}^{\;\;\;\;m}.$$
(54)

Otherwise, this “minus”-sign and the sign for complex conjugation could be mixed up.

Trivially, for the index pair \(j,k,{\underset - K ^i}_{jkl} \ne {\underset + {\tilde K} ^i}_{jkl}\). The curvature tensors (53), (54) are skew-symmetric only in the second pair of indices. A tensor corresponding to the Ricci-tensor in Riemannian geometry is given by

$$\underset{+}{K}_{{jk}}: = \underset{+}{K}_{\;jkl}^{{l}}= {\partial _k}L_{lj}^{\;\;\;l} - {\partial _l}L_{kj}^{\;\;\;l} + L_{km}^{\;\;\;\;l}L_{lj}^{\;\;\;m} - L_{lm}^{\;\;\;\;l}L_{kj}^{\;\;\;m}\,.$$
(55)

On the other hand,

$$\underset{-}{K}_{{jk}}: = \underset{-}{K}_{\;jkl}^{{l}}= {\partial _k}L_{jl}^{\;\;\;l} - {\partial _l}L_{jk}^{\;\;\;l} + L_{mk}^{\;\;\;\;l}L_{jl}^{\;\;\;m} - L_{ml}^{\;\;\;\;l}L_{jk}^{\;\;\;m}\,.$$
(56)

Note that the Ricci tensors as defined by (55) or (56) need not be symmetric even if the connection is symmetric, and also that \({\underset - K_{jk}} \ne {\underset + {\tilde K} _{jk}}\) when \(\tilde K\) denotes the Hermitian (transposition) conjugate. Thus, in general

$${\underset{-}{K} _{[jk]}}: = {\partial _{\left[ k \right.}}{S_{\left. j \right]}} + {\overset{-}{\nabla}_l}S_{kj}^l.$$
(57)

If the curvature tensor for the symmetric part of the connection is introduced by:

$$\underset{0}{K} _{\;jkl}^i = {\partial _k}\Gamma _{lj}^{\;\;\;\;i} - {\partial _l}\Gamma _{kj}^{\;\;\;\;i} + \Gamma _{km}^{\;\;\;\;\;i}\Gamma _{lj}^{\;\;\;\;m} - \Gamma _{lm}^{\;\;\;\;\;i}\Gamma _{kj}^{\;\;\;m}\,,$$
(58)

then

$$\underset{-}{K}_{\;jkl}^i(L) = \underset{0}{K}_{\;jkl}^i(\Gamma) + S_{jl\underset{0}{\parallel}{k}}^{\quad \;\;i} - S_{jk\underset{0}{\parallel}{l}}^{\quad \;\;\;i} + S_{mk}^{\;\;\;\;i}S_{jl}^{\;\;\;m} - S_{ml}^{\;\;\;\;i}S_{jk}^{\;\;\;m}\,.$$
(59)

The corresponding expression for the Ricci-tensor is:

$${\underset{0}{K}_{jk}}: = \underset{0}{K}_{\;jkl}^l = {\partial _k}\Gamma _{lj}^{\;\;\;l} - {\partial _l}\Gamma _{kj}^{\;\;\;l} + \Gamma _{km}^{\;\;\;\;\;l}\Gamma _{lj}^{\;\;\;m} - \Gamma _{lm}^{\;\;\;\;l}\Gamma _{kj}^{\;\;\;m}\,,$$
(60)

whence follows:

$$\underset{0}{K}_{[jk]}: = {\partial _{\left[ k \right.}}{\Gamma _{\left. j \right]}}$$
(61)

with \({\Gamma _k} = \Gamma _{kl}^l\). Also, the relations hold (for (63) cf. [549], Eq. (2,12), p. 278)):

$${{\mathop K\limits_ +} _{jk}} = {{\mathop K\limits_0} _{jk}} + S_{_{jk\mathop {||}\limits_0 l}}^{\;\;l} - {S_{_{jk\mathop {||}\limits_0 k}}} - {S_{jk}}^m{S_m} - {S_{jl}}^mS_{km}^{\;\;l},$$
(62)
$${{\mathop K\limits_ -} _{jk}} = {{\mathop K\limits_0} _{jk}} + S_{_{jk\mathop {||}\limits_0 l}}^{\;\;l} - {S_{_{jk\mathop {||}\limits_0 k}}} - {S_{jk}}^m{S_m} - {S_{jl}}^mS_{km}^{\;\;l}.$$
(63)

A consequence of (62), (63) is:

$${\underset{+}{K}_{jk}} - {\underset{-}{K}_{jk}} = - 2{S_{j {\underset{0}{\parallel}}k}} + 2S_{jk\;{\underset{0}{\parallel}} l}^{\;\;l}\,,\;{\underset{+}{K}_{jk}} + {\underset{-}{K}_{jk}} = 2{\underset{0}{K}_{jk}} + 2S_{km}^{\;\;\;l}S_{lj}^{\;\;m} - 2S_{jk}^{\;\;m}{S_m}\,.$$
(64)

Another trace of the curvature tensor exists, the so-called homothetic curvatureFootnote 32:

$${\underset{+}{V}_{kl}} = \underset{+}{K}_{\;jkl}^j = {\partial _k}L_{lj}^{\;\;j} - {\partial _l}L_{kj}^{\;\;j}\,.$$
(65)

Likewise,

$${\underset{-}{V}_{kl}} = \underset{-}{V}_{\;jkl}^j = {\partial _k}L_{jl}^{\;\;\;\;j} - {\partial _l}L_{jk}^{\;\;\;\;j}\,,$$
(66)

such that \({\underset - V _{kl}} - {\underset + V _{kl}} = 2{\partial _l}{S_k} - 2{\partial _k}{S_l}\). For the curvature tensor, the identities hold:

$$\underset{-}{K}_{\;\{jkl\}}^i - 2{\underset{-}{\nabla}_{\{j}}S_{kl\}}^{\;\;\;\;\;\;i} + 4{S_{\{jk}}^{\;r}S_{l\} r}^{\;\;\;\;\;\;i} = 0\,,$$
(67)
$${\underset{-}{\nabla}_{\{k}}\underset{-}{K} _{\;\vert j\vert lm\}}^i + 2\underset{-}{K}_{jr\{k}^iS_{lm\}}^{\;\;\;\;\;r} = 0\,.$$
(68)

where the bracket {…} denotes cyclic permutation while the index |j | does not take part.

Equation (68) generalizes Bianchi’s identity. Contraction on i, j leads to:

$${V_{jk}} + 2{K_{[jk]}} = 2{\nabla _l}S_{jk}^{\;\;\;l} + 4S_{jk}^{\;\;\;r}{S_r} + 4{\nabla _{\left[ j \right.}}{S_{\left. k \right]}}\,,$$
(69)

or for a symmetric connection (cf. Section 2.1.3.1 of Part I, Eq. (38)):

$${V_{jk}} + 2{K_{[jk]}} = 0.$$

These identities are used either to build field equations without use of a variational principle, or for the identification of physical observables; cf. Section 9.7.

Finally, two curvature scalars can be formed:

$$\underset + K = {g^{ij}}{\underset + K _{ij}}\;\,,\;\underset - K = {g^{ij}}{\underset - K _{ij}}\,.$$
(70)

For a symmetric connection, an additional identity named after O. Veblen holds:

$$\underset 0 K_{\;jkl,m}^i + \underset 0 K _{\;ljm,k}^i + \underset 0 K _{\;mlk,j}^i + \underset 0 K _{\;kmj,l}^i = 0\,.$$
(71)

The integrability condition for (30) is ([399], p. 225), [51]:

$${g^{ri}}R_{\;rlm}^k + {g^{kr}}R_{\;rlm}^i = 0$$
(72)

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].

A list of “Ricci”-tensors

In many approaches to the field equations of UFT, a generalization of the Ricci scalar serves as a Lagrangian. Thus, the choice of the appropriate “Ricci” tensor plays a distinct role. As exemplified by Eq. (64), besides \({\underset + K _{jk}}\) and \({\underset - K _{jk}}\) there exist many possibilities for building 2-rank tensors which could form a substitute for the unique Ricci-tensor of Riemannian geometry. In ([150], p. 142), Einstein gives a list of 4 tensors following from a “single contraction of the curvature tensor”. Santalò derived an 8-parameter set of “Ricci”-type tensors constructed by help of \({\underset - K _{jk}},S_{ik\Vert l}^{\,\,\,\,l},\,{\underset - V_{jk}}(\Gamma),\,{S_{i\Vert k}},\,S_{ik}^{\,\,\,\,m}{S_m},{S_i}{S_k},\,S_{im}^{\,\,\,\,\,l}S_{kl}^{\,\,\,\,m}\) ([524], p. 345). He discusses seven of them used by Einstein, Tonnelat, and Winogradzki.Footnote 33 The following collection contains a few examples of the objects used as a Ricci-tensor in variational principles/field equations of UFT besides \({\underset + K _{ik}}\) and \({\underset - K _{ik}}\) of the previous section.Footnote 34 They all differ in terms built from torsion. Among them are:

$$\begin{array}{*{20}c} {{\underset{-}{\overset{\mathrm{Her}}{K}}\;_{{ik}}} = - {1 \over 2}({{\underset - K}\;_{ik}} + {{\underset - {\tilde K}}\;_{ik}}) = {P_{ik}}} \\ {= L_{ik\;\,,l}^{\;\;\;\;\;l} - L_{im}^{\;\;\;\;l}L_{lk}^{\;\;\;\;\;\;m} - {1 \over 2}(L_{il\;\,,k}^{\;\;\;\;l} + L_{lk\;\,,i}^{\;\;\;\;l}) + {1 \over 2}L_{ik}^{\;\;\;m}(L_{ml}^{\;\;\;\;\;l} + L_{lm}^{\;\;\;\;\;l})([147],{\rm{p}}.\;581)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(73)
$$= {1 \over 2}({\underset 0 K_{ik}} + {\underset 0 {\tilde K}_{ik}}) + S_{ik\underset 0 \parallel l}^{\;\;\;l} + S_{im}^{\;\;\;l}S_{kl}^{\;\;\;m};$$
(74)
$$P_{ik}^\ast = L_{ik\;\,,l}^{\;\;\;\;l} - L_{im}^{\;\;\;\;l}L_{lk}^{\;\;\;\;m} - {1 \over 2}(L_{(il)\;\,,k}^{\;\;\;\;\;l} + L_{(lk)\;\,,i}^{\;\;\;\;\;l}) + {1 \over 2}L_{ik}^{\;\;\;\;m}(L_{ml}^{\;\;\;\;l} + L_{lm}^{\;\;\;\;l})([150],{\rm{p}}.\;142)$$
(75)
$$= {\underset{-}{\overset{\mathrm{Her}}{K}}_{{ik}}} + {S_{[i,k]}}([371],{\rm{p}}.\;247 - 248)$$
(76)
$$^{(1)}{R_{ik}} = - {\underset - K _{ik}} + {2 \over 3}({\partial _i}{S_k} - {\partial _k}{S_i})([632],{\rm{p}}.\;129);$$
(77)
$$\begin{array}{*{20}c} ^{(2)}{R_{ik}} = {\partial _l}L_{ik}^{\;\;\;l} - {\partial _k}L_{(il)}^{\;\;\;\;\;l} + L_{ik}^{\;\;\;\;l}L_{(lm)}^{\;\;\;\;m} - L_{im}^{\;\;\;\;l}L_{lk}^{\;\;\;m} + {1 \over 3}({\partial _i}{S_k} - {\partial _k}{S_i}) - {1 \over 3}{S_i}{S_k}({\rm{[632]}},{\rm{p}}.\;129)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\ {= - {{\underset 0 K}_{ik}} - {2 \over 3}{S_{[i\underset 0 \parallel k]}} + S_{im}^{\;\;\;\;l}S_{kl}^{\;\;\;m} + {1 \over 3}S_{ik}^{\;\;\;m}{S_m} - {1 \over 3}{S_i}{S_k};\;} \\ \end{array}$$
(78)
$$^{(3)}{R_{ik}} = {\;^{(2)}}{R_{ik}} - {1 \over 2}{\underset + V_{ik}}\,,([632],{\rm{p}}.\;129);$$
(79)
$${U_{ik}} = \underset{-}{\overset{\mathrm{Her}}{K}}_{{ik}} - {1 \over 3}[{S_{i,k}} - {S_{k,i}} + {S_i}{S_k}]$$
(80)
$$= {\underset 0 K_{ik}} - S_{ik\underset 0 \parallel l}^{\;\;\;l} + S_{im}^{\;\;\;\;l}S_{kl}^{\;\;m} - {2 \over 3}{S_{[i\underset 0 \parallel k]}} - {2 \over 3}S_{ik}^{\;\;\;m}{S_m} - {1 \over 3}{S_i}{S_k},([151],\;{\rm{p}}.\;137;$$
(81)
$$R_{ik}^\ast = - {\underset - K_{ik}} + {\overset + \nabla _k}{S_i} = L_{ik\;\,,l}^{\;\;\;\;\;l} - L_{im}^{\;\;\;\;\;l}L_{lk}^{\;\;\;\;m} - L_{(il)\;\,,k}^{\;\;\;\;l} + {1 \over 2}L_{ik}^{\;\;\;m}(L_{ml}^{\;\;\;\;l} + L_{lm}^{\;\;\;\;l})([156],\;{\rm{p}}.\;144)$$
(82)
$$R_{ik}^{\ast\ast} = R_{ik}^\ast - {[{(\log (\sqrt {- g}))_{,i}}]_{{{\underset - \parallel}}_{k}}}([156],{\rm{p}}.\;144)$$
(83)

Further examples for Ricci-tensors are given in (475), (476) of Section 13.1.

One of the puzzles remaining in Einstein’s research on UFT is his optimism in the search for a preferred Ricci-tensor 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 6-parameter object could be formed. The additional symmetries without physical support suggested by Einstein did not help. Possibly, he was too much influenced by the quasi-uniqueness of his field equations for the gravitational field.

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.

Curvature and λ-transformation

The effect of a λ-transformation (52) on the curvature tensor \(\underset - K _{jkl}^i\) is:

$$\underset - K_{\;jkl}^i \rightarrow \underset - K_{\;jkl}^i + 2\;{\partial _{\left[ k \right.}}{\lambda _{\left. l \right]}}\delta _j^{\;\;i}\,.$$
(84)

In case the curvature tensor \(\underset + K _{jkl}^i\) is used, instead of (52) we must take the form for the λ-transformation:Footnote 35

$$\prime \Gamma _{ij}^{\;\;\;k} = \Gamma _{ij}^{\;\;\;k} + {\lambda _i}\delta _j^{\;\;k}\,.$$
(85)

Then

$$\underset + K_{\;jkl}^i \rightarrow \underset + K _{\;jkl}^i + 2\;{\partial _{\left[ k \right.}}{\lambda _{\left. l \right]}}\delta _j^i\,.$$
(86)

also holds. Application of (52) to \(\underset + K _{jkl}^i\), or (85) to \(\underset - K _{jkl}^i\) results in many more terms in λ k on the r.h.s. For the contracted curvatures a λ-transformation leads to (cf. also [430]):

$${\underset - K _{jk}} \rightarrow {\underset - K _{jk}} - 2{\partial _{\left[ k \right.}}{\lambda _{\left. j \right]}}\,,\;{\underset + V _{jk}} \rightarrow {\underset + V _{jk}} + 2{\partial _{\left[ j \right.}}{\lambda _{\left. k \right]}}\,,\;\;{\underset - V _{jk}} \rightarrow {\underset - V _{jk}} + 8{\partial _{\left[ j \right.}}{\lambda _{\left. k \right]}}\,.$$
(87)

If λ i = i λ, the curvature tensors and their traces are invariant with regard to the λ-transformations of Eq. (52). Occasionally, \(\prime\Gamma _{ij}^{\,\,\,\,k} = \Gamma _{ij}^{\,\,\,\,k} + ({\partial _i}\lambda)\delta _j^{\,\,k}\) is interpreted as a gravitational gauge transformation.

Differential forms

In this section, we repeat and slightly extend the material of Section 2.1.4, Part I, concerning Cartan’s one-form formalism in order to make understandable part of the literature. Cartan introduced one-forms θâ (â = 1, …, 4) by \({\theta ^{\hat a}}: = h_l^{\hat a}d{x^l}\). The reciprocal basis in tangent space is given by \({e_{\hat j}} = h_{\hat j}^l{\partial \over {\partial {x^l}}}\). Thus, \({\theta ^{\hat a}}({e_{\hat j}}) = \delta _{\hat j}^{\hat a}\). An antisymmetric, distributive and associative product, the external or “wedge” (Λ)-product is defined for differential forms. Likewise, an external derivative d can be introduced.Footnote 36 The metric (e.g., of space-time) is given by \({\eta _{\hat i\hat k}}{\theta ^{\hat i}} \otimes {\theta ^{\hat k}}\), or \({g_{lm}} = {\eta _{\hat i\hat k}}h_l^{\hat i}h_m^{\hat k}\). The covariant derivative of a tangent vector with bein-components \({X^{\hat k}}\) is defined via Cartan’s first structure equations,

$${\Theta ^i}: = D{\theta ^{\hat\iota}} = d{\theta ^{\hat\iota}} + \omega _{\;\hat l}^{\hat\iota} \wedge {\theta ^{\hat l}},$$
(88)

where \(\omega _{\,\,\hat k}^{\hat i}\) is the connection-1-form, and Θî is the torsion-2-form, \({\Theta ^{\hat i}} = - {S_{\hat l\hat m}}^{\hat i}{\theta ^{\hat l}}\wedge{\theta ^{\hat m}}\). We have \({\omega _{\hat i\hat k}} = - {\omega _{\hat k\hat i}}\). The link to the components \(L_{[ij]}^{\,\,\,\,\,\,k}\) of the affine connection is given by \(\omega _{\hat k}^{\hat i} = h_l^{\hat i}h_{\hat k}^m{L_{\hat rm}}^l\theta \hat r\)Footnote 37. The covariant derivative of a tangent vector with bein-components \({X^{\hat k}}\) then is

$$D{X^{\hat k}}: = d{X^{\hat k}} + \omega _{\;\;\hat l}^{\hat k}{X^{\hat l}}.$$
(89)

By further external derivation on Θ we arrive at the second structure relation of Cartan,

$$D{\Theta ^{\hat k}} = \Omega _{\;\;\hat l}^{\hat k} \wedge {\theta ^{\hat l}}.$$
(90)

In Eq. (90) the curvature-2-form \(\Omega _{\,\,\hat l}^{\hat k} = {1 \over 2}R_{\,\,\hat l\hat m\hat n}^{\hat k}{\theta ^{\hat m}}\wedge{\theta ^{\hat n}}\) appears, which is given by

$$\Omega _{\;\;\hat l}^{\hat k} = d\omega _{\;\;\hat l}^{\hat k} + \omega _{\;\;\hat l}^{\hat k} \wedge \omega _{\;\;\hat l}^{\hat k}.$$
(91)

\(\Omega _{\,\,\,\hat k}^{\hat k}\) is the homothetic curvature.

A p-form in n-dimensional space is defined by

$$\omega = {\omega _{{{\hat i}_1}{{\hat i}_2} \ldots {{\hat i}_p}}}d{x^{{{\hat i}_1}}} \wedge d{x^{{{\hat i}_2}}} \wedge \ldots \wedge d{x^{{{\hat i}_p}}}$$

and, by help of the so-called Hodge *-operator, is related to an (n-p)-form)Footnote 38

$$\overset \ast \omega : = {1 \over {(n - p)!}}\epsilon _{{{\hat k}_1}{{\hat k}_2} \ldots {{\hat k}_{n - p}}}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\hat i}_1}{{\hat i}_2} \ldots {{\hat i}_p}}\;{\omega _{{{\hat i}_1}{{\hat i}_2} \ldots {{\hat i}_p}}}\,d{x^{{{\hat k}_1}}} \wedge d{x^{{{\hat k}_2}}} \wedge \ldots \wedge d{x^{{{\hat k}_{n - p}}}}\,.$$

Classification of geometries

A differentiable manifold with an affine structure is called affine geometry. If both, a (possibly non-symmetric) “metric” and an affine structure, are present we name the geometry “mixed”. A subcase, i.e., metric-affine geometry demands for a symmetric metric. When interpreted just as a gravitational theory, it sometimes is called MAG. A further subdivision derives from the non-metricity tensor being zero or ≠ 0. Riemann-Cartan geometry is the special case of metric-affine geometry with vanishing non-metricity tensor and non-vanishing torsion. Weyl’s geometry had non-vanishing non-metricity tensor but vanishing torsion. In Sections 2.1.3 and 4.1.1 of Part I, these geometries were described in greater detail.

Generalized Riemann-Cartan geometry

For the geometrization of the long-range fields, various geometric frameworks have been chosen. Spaces with a connection depending solely on a metric as in Riemannian geometry rarely have been considered in UFT. One example is given by Hattori’s connection, in which both the symmetric and the skew part of the asymmetric metric enter the connectionFootnote 39 [240]:

$$^HL_{ij}^{\;\;\;k} = 1/2\;{h^{\;kl}}({g_{li,j}} + {g_{jl,i}} - {g_{ji,l}})$$
(92)
$$= {\{_{ij}^k\} _h} + 1/2\;{h^{kl}}({k_{li,j}} + {k_{jl,i}} + {k_{ij,l}})\,,$$
(93)

where hkl is the inverse of h kl = g(kl). As described in Section 6.2 of Part I, its physical content is dubious. As the torsion tensor does not vanish, in general, i.e.,

$$^HS_{ij}^{\;\;\;k} = {h^{\;kl}}({k_{l[i,j]}} + 1/2{k_{ij,l}})$$
(94)

this geometry could be classified as generalized Riemann-Cartan geometry.

Mixed geometry

Now, further scalars and scalar densities may be constructed, among them curvature scalars (Ricciscalars):

$$\underset + K : = {g^{jk}}{\underset + K _{jk}} = {l^{jk}}{\underset + K _{(jk)}} + {m^{jk}}{\underset + K _{[jk]}}\,,$$
(95)
$$\underset - K : = {g^{jk}}{\underset - K _{jk}}: = {l^{jk}}{\underset - K _{(jk)}} + {m^{jk}}{\underset - K _{[jk]}}\,.$$
(96)

Here, ljk and mjk come from the decomposition into irreducible parts of the inverse of the non-symmetric metric g jk . Both parts on the r.h.s. could be taken as a Lagrangian, separately. The inverse hjk of the symmetric part, i.e., of g(jk) = h jk could be used as well to build a scalar: \({h^{jk}}{\underset + K_{(jk)}}\). Mixed geometry is the one richest in geometrical objects to be constructed from the asymmetric metric and the asymmetric connection. What at first may have appeared as an advantage, turned out to become an ‘embarras de richesses’: defining relations among geometric objects and physical observables abound; cf. Section 9.7.

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, non-metricity tensor, skew-symmetric 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.

Conformal geometry

This is an “angle preserving” geometry: in place of a metric g ij (xm) a whole equivalence class γ ij (xl) = ρ2(xn)g ij (xm) with a function ρ (xn) obtains. Geometrical objects of interest are those invariant with regard to the transformation: g ij (xm) → ij = ρ2(xn)g ij (xm). One such object is Weyl’s conformal curvature tensor:

$$C_{jkl}^i: = \underset + K _{\;jkl}^i - {1 \over {n - 2}}(\delta _k^i{\underset + K _{jl}} - \delta _l^i{\underset + K _{jk}} + {g_{jl}}\underset + K _{\;k}^i - {g_{jk}}\underset + K _{\;l}^i) + {1 \over {(n - 1)(n - 2)}}\underset + K (\delta _l^i{g_{jk}} - \delta _k^i{g_{jl}})\,,$$
(97)

where n is the dimension of the manifold (n = 4: space-time). \(C_{jkl}^i\) is trace-free. For n > 3, \(C_{jkl}^i = 0\) is a necessary and sufficient condition that the space is conformally flat, i.e., γ ij (xl) = ρ2(xn)η ij (xm) ([191], p. 92).

If ξk is a Killing vector field for g ij , then ξk is a conformal Killing vector field for ij ; cf. Eq. (45) in Section 2.1.2.

A particular sub-case of conformal geometry is “similarity geometry”, for which the restricted group of transformations acts g ij (xm) → γ ij (xl) = k2g ij (xm), with a constant k, cf. Section 3.1.

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).

As hypercomplex numbers are less well known, we briefly introduce them here. Let z = x + ϵy, x, y real and consider the algebra with two elements I, ϵ, where I is the unit element and ϵ2 = I. z is called a hypercomplex number. A function f (z) = P (x, y) + ϵQ (x, y) will be differentiable in z if

$${{\partial P} \over {\partial x}} = {{\partial Q} \over {\partial y}}\,,\qquad {{\partial P} \over {\partial y}} = {{\partial Q} \over {\partial x}}\,.$$
(98)

The product of two identical real manifolds of dimension n can be made into a manifold with hypercomplex structure.

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

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

Projective and conformal relativity theory

Projective relativity theory had been invented expressly in order to avoid the fifth dimension of Kaluza-Klein theory. In Sections 6.3.2 and 7.2.4 of Part I, Pauli & Solomon’s paper was described. Also, in Section 6.3.2 of Part I, we briefly have discussed what O. Veblen & B. Hoffmann called “projective relativity” [671], and the relationship to the Einstein-Mayer theory. Veblen & Hoffmann had introduced projective tensors with components \(T_{\sigma \ldots \tau}^{\alpha \ldots \beta} = {\rm{exp(}}N{x^0})f_{\sigma \ldots \tau}^{\alpha \ldots \beta}({x^1},{x^2},{x^3},{x^4})\) where x1, … x4 are coordinates of space-time, x0 is an additional parameter (a gauge variable) and N a constant named “index”.Footnote 40 x0 transforms as x0 = x0 + log ρ (xα). The auxiliary 5-dimensional space appearing has no physical significance. A projective symmetric metric G αβ of index 2N was given by G αβ = Φ2 γαβ where Φ is an arbitrary projective scalar of index N. In addition, a third symmetric tensor g αβ = γ αβ ϕ α ϕ β , the gravitational metric, appeared. Here, ϕ α := γα0 is a projective vector. Likewise, the Levi-Civita connections \(\Pi _{\alpha \beta}^{\,\,\,\,\,\,\delta}\), \(\Gamma _{\alpha \beta}^{\,\,\,\,\,\,\delta}\) with \(\Pi _{\alpha \beta}^{\,\,\,\,\,\,\,\delta}({G_{\sigma \tau}}) = \Gamma _{\alpha \beta}^{\,\,\,\,\,\,\,\delta}({\gamma _{\sigma \tau}}) + N(\delta _\alpha ^\delta {\Phi _\beta} + \delta _\beta ^\delta {\Phi _\alpha} - {\gamma _{\alpha \beta}}{\Phi ^\delta})\) with \({\Phi _\alpha} = {1 \over N}{{\partial \,{\rm{log}}\,\Phi} \over {\partial {x^\alpha}}}\) and Φδ = γδσ Φ σ were used. For arbitrary index N, the field equations were derived from the curvature scalar P calculated from the connection \(\Pi _{\alpha \beta}^{\,\,\,\,\,\,\delta}({G_{\sigma \tau}})\). One equation could be written in the form of a wave equation:

$${1 \over {\sqrt {- g}}}{\partial \over {\partial {x^\sigma}}}(\sqrt {- g} {\gamma ^{\sigma \tau}}{{\partial \psi} \over {\partial {x^\tau}}}) + {5 \over {27}}(R - 3{N^2})\psi = 0\,,$$
(99)

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

During the 1940s, meson physics became fashionable. Of course, the overwhelming amount of this research happened in connection with nuclear and elementary particle theory, outside of UFT, but sometimes also in classical field theory. Cf. the papers by F. J. Belinfante on the meson field, in which he used the undor-formalismFootnote 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 spinor-field, he used Belinfantes undors [466]. After this paper, he examined which of Kemmer’s five types of meson fields were “in accordance with the requirements of projective relativity” ([467], p. 268).

It is unsurprising that B. Hoffmann in Princeton also applied the projective formalism to a theory intended to unify the gravitational and vector meson fields [278]. The meson field θ α was defined by Hoffmann via: θ α = ϕ α − Φ α with ϕ α = γ0α and Φ α given above. Its space-time components θ j form an affine vector from which the vector meson field tensor \({\theta _{ij}} = {1 \over 2}({{\partial {\theta _i}} \over {\partial {x^j}}} - {{\partial {\theta _j}} \over {\partial {x^i}}})\) follows. The theory again contained three Riemannian curvature tensors (scalars). By skipping all calculations, we arrive at the affine form of Hoffmann’s field equations

$$\begin{array}{*{20}c} {{R^{ij}} - {1 \over 2}{g^{ij}}(R + 12{N^2}) + {1 \over 2}({g^{rs}}\theta _{\;\;r}^i\theta _{\;\;s}^j + {1 \over 4}{g^{ij}}\theta _{\;\;r}^s\theta _{\;\;s}^r) - 12{N^2}({\theta ^i}{\theta ^j} - {1 \over 2}{g^{ij}}{\theta ^s}{\theta _s}) = 0\,,\;\;\;\;\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\theta _{\;\,,j}^{ij} + 12{N^2}{\theta ^i} = 0\,.} \\ \end{array}$$
(100)

In Hoffmann’s words: “except for the term − 6N2gij, these are the classical (i.e. unquantized) field equations for a vector meson and gravitational field in the general theory of relativity” ([278], p. 464). We could name them as well “Einstein-meson” equations in analogy to “Einstein-Maxwell” equations: no unification of both field had been reached. Also, no scalar meson field and the electromagnetic field were present in the theory.

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

$${x^0} \rightarrow {x^0} + {1 \over N}\log k\,,{x^j} \rightarrow x{\prime ^j}({x^k})\,,{x^5} \rightarrow {1 \over {{k^2}}}{x^5}$$
(101)

hold. A symmetric tensor S AB in R6 was given the role of metric; the assumptions S05 = 0, and \({{{S_{55}}} \over {{S_{00}}}}\) independent of xα reduced the number of free functions. The definitions \({S_{AB}}: = {{{S_{AB}}} \over {{S_{00}}}}\) and γ AB := s AB sA5sB5 / s55, γ05 = 0, γ00 = 1 led back to the former vector meson field via γj0 and to a vector in R6 \({\psi _A}: = {{{s_{A5}}} \over {\sqrt {{s_{55}}}}}\) with ψ0 = 0, ψ5 independent of xα and containing the electromagnetic 4-vector ψ j . To abreviate the story, Hoffmann’s final field equations in space-time were:

$$\begin{array}{*{20}c} {{R^{ij}} - {1 \over 2}{g^{ij}}R + 2({g^{rs}}\psi _r^i\psi _s^j + {1 \over 4}{g^{ij}}\psi _{\;\;r}^s\psi _{\;\;s}^r) + {1 \over 2}({g^{rs}}\theta _{\;\;r}^i\theta _{\;\;s}^j + {1 \over 4}{g^{ij}}\theta _{\;\;r}^s\theta _{\;\;s}^r)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- 20{N^2}({\theta ^i}{\theta ^j} - {1 \over 2}{g^{ij}}{\theta ^s}{\theta _s}) = 0\,,\quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {\theta _{\;\;\;,j}^{ij} + 20{N^2}{\theta ^i} = 0\,,\quad \quad \quad \;\;\;} \\ {\psi _{\;\;\;,r}^{ir} = 0\,.} \\ \end{array}$$
(102)

The last equation with \({\psi _{ij}}: = {1 \over 2}({{\partial {\psi _i}} \over {\partial {x^j}}} - {{\partial {\psi _j}} \over {\partial {x^i}}})\) reproduced Maxwell’s equations. In a sequel to this paper, Hoffmann claimed to have derived “the correct trajectories of charged meson testparticles in a combined gravitational, electromagnetic, and vector meson field” ([280], p. 1045).

Geometrical approach

It was Pascual JordanFootnote 43 who in physics re-applied projective geometry (cf. Section 2.1.3.3 of Part I) by showing that the transformation group G5 of the 4-potential A k in electrodynamics, composed of the gauge transformations

$${A_{k\prime}} = {A_k} + {\partial _k}\chi \,,\quad {x^{i\prime}} = {x^i}\,;$$
(103)

and coordinate transformations

$${A_{k\prime}} = {A_s}{{\partial {x^s}} \over {\partial {x^{k\prime}}}}\,,{x^{i\prime}} = {f^i}({x^0}, \ldots ,{x^3})\;(k,s,i = 0,1, \ldots ,3)$$
(104)

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

$${X\prime^{\alpha}} = {X^\alpha}{F^\alpha}({{{X\prime^{0}}} \over {{X\prime^{4}}}},{{{X\prime^{1}}} \over {{X\prime^{4}}}},{{{X\prime^{2}}} \over {{X\prime^{4}}}},{{{X\prime^{3}}} \over {{X\prime^{4}}}})\,,$$
(105)

(no summation over α on the r.h.s.).Footnote 44 Equivalently, the new coordinates X′α are homogeneous functions of degree 1 of the old Xα and transform like a vector:

$${X\prime^{\alpha}} = {X^\rho}{{\partial {X\prime^{\alpha}}} \over {\partial {X^\rho}}}\;\;(\alpha ,\rho = 0,1, \ldots ,4)\;.$$
(106)

for the coordinates xi of space-time, alternatively we may write xi = fi (X0, …, X4) or \({{\partial {x^i}} \over {\partial {x^\rho}}}{X^\rho} = 0\). Jordan defined projector-components \(P_{{\alpha _1}, \ldots, {\alpha _m}}^{{\beta _1}, \ldots, {\beta _n}}\) to transform under (106) like tensor-components \(T_{{\alpha _{1, \ldots,}}{\alpha _m}}^{{\beta _{1, \ldots,}}{\beta _n}}\) which are homogeneous functions of degree (nm) in the Xρ. Thus, Xρ itself is a projector just as the Minkowski (Euclidean) metric g αβ of V5 with the invariant:

$$J = {g_{\alpha \beta}}{X^\alpha}{X^\beta}\,.$$

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

$$L = {1 \over 2}[a(J){\psi _{,\nu}}{\psi _{,\mu}}{g^{\mu \nu}} + b(J){\psi ^2}]$$
(107)

but then became generalized to

$$L = U(J)[\overset {5}{R} + W(J){\psi _{,\nu}}{\psi _{,\mu}}{g^{\mu \nu}} + V(J)]\,.$$
(108)

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

$$L = U(J)[\overset{5}{R} + W(J){J_{,\nu}}{J_{,\mu}}{g^{\mu \nu}} + V(J)]\,.$$
(109)

With

$$^{(5)}R{= ^{(4)}}R + {1 \over 4}J\,{F_{rs}}{F^{rs}} + {2 \over {\sqrt J \sqrt {- g}}}{\partial _r}\;\left({\sqrt {- g} {g^{rs}}{{\partial \sqrt J} \over {\partial {x_s}}}} \right)\;.$$
(110)

we arrive at:

$$L = U(J)\;\left[ {\overset 4 R + {1 \over 4}J{F_{rs}}{F^{rs}} + {J^{- 1}}{g^{rs}}{{\overset 4 \nabla}_r}{\partial _s}J + (W(J) - {1 \over 2}{J^{- 2}}){J_{,\nu}}{J_{,\mu}}{g^{\mu \nu}} + V(J)} \right]\;\,,$$
(111)

where U (J), V(J), W (J) are arbitrary functions. As can be seen from (110), the 5-dimensional curvature scalar used by Jordan and by Thiry (cf. the next Section 3.1.2) follows as the subcase U (J) = J−1/2, W (J) = V (J) = 0 of the general expression (111). Ludwig, at the time of writing the preface to his book, e.g., in May 1951, seemingly did not know of Thiry’s paper of 1948 [604] nor of his PhD thesis published also in 1951: in his bibliography Thiry’s name and paper are missing.

Pauli had browsed in Ludwig’s book and now distanced himself from his own papers on projective relativity of 1933 discussed briefly in Section 7.2.4 of Part I.Footnote 45 He felt deceived:

“The deception consists in the belief that by the projective form, i.e., the homogeneous coordinates, the shortcomings of Kaluza’s formulation have been repaired, and that one has achieved something beyond Kaluza. At the time, in 1933, I did not know explicitly the transition from Kaluza to the projective form (as in [20]); it is too simple and banal to the effect that the factual contents of both equivalent formulations could be somehow different.” (letter of W. P. to P. Jordan, [490], p. 735):Footnote 46

Physical approach: Scalar-tensor theory

Toward the end of the second world war, Kaluza’s five-dimensional theory and projective relativity emerged once again as vehicles for a new physical theory which, much later, came to be known as “scalar-tensor theory of gravitation.”Footnote 47 Cosmological considerations related to the origin of stars seem to have played the major role for the building of a theory by P. Jordan in which the gravitational constant \(\kappa = {{8\pi G} \over {{c^2}}}\) is thought to be varying in (cosmological) time and thus replaced by a scalar function [316]Footnote 48. The theory nicely fit with Dirac’s “large number hypothesis” [122, 123]. The fifteenth field variable in Kaluza’s theory was identified by Jordan with this function, or in projective relativity, with the scalar: J:= g αβ XαXβ by setting \(J = {{2\kappa} \over {{c^2}}}(\alpha, \beta = 0,1, \ldots 4)\) (α, β = 0, 1, …, 4) [321]. In space-time, the field equations for the gravitational field g ij , the electromagnetic 4-potential A k = g4k, and the g44-variable κ were derived by Jordan and MüllerFootnote 49 to be:

$${G_{ik}} + {\kappa \over {{c^2}}}{F_i}{}^s{F_{ks}} = - {1 \over {2\kappa}}\;\left({{\nabla _k}{\nabla _i}\kappa - {1 \over {2\kappa}}{\nabla _i}\kappa {\nabla _k}\kappa} \right)\;\,,$$
(112)
$$\kappa {\nabla _s}{F^{sj}} = - {3 \over 2}{\nabla _s}\kappa \;{F^{sj}}\;,$$
(113)
$$G = - {\kappa \over {2{c^2}}}{F_{rs}}{F^{rs}} + {1 \over {2\kappa}}{g^{rs}}{\nabla _r}\kappa {\nabla _s}\kappa - {1 \over \kappa}{g^{rs}}{\nabla _r}{\nabla _s}\kappa \;.$$
(114)

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

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

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

“The problem of the structure of matter can only be attacked as a problem in quantum mechanics; nevertheless, investigations of the singularities of solutions of the field equations retain considerable importance in this framework. […] the wave functions of matter must be taken into account. Whether this program can be carried through, and to which extent, in the sense of an extension of geometry (to which Schrödinger’s ideas related to the meson field seem to provide an important beginning) is such a widespread question […]”.Footnote 50 ([318], p. 205).

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

$$J\;\left({{}^{(4)}R - {J \over 2}{F_{rs}}{F^{rs}} - \zeta {{{\nabla _r}J{\nabla ^r}J} \over {{J^2}}}} \right)\;.$$
(115)

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

$${\kappa ^\eta}\;\left({{}^{(4)}R - \zeta {{{\nabla _r}\kappa {\nabla ^r}\kappa} \over {{\kappa ^2}}}} \right)\;\,,$$
(116)

but then set one of the two free parameters η = 1. One of those responding to this book was M. Fierz in Basel [195]. Before publication, he had corresponded with W. Pauli, sent him first versions of the paper and eventually received Pauli’s placet; cf. the letter of Pauli to Fierz of 2 June 1956 in [492], p. 578. In the second edition of his book, Jordan also commented on a difficulty of his theory pointed out by W. Pauli: instead of g ik equally well ϕ (x)g ik with arbitrary function ϕ could serve as a metric.Footnote 51 This conformal invariance of the theory is preserved in the case that an electromagnetic field forms the matter tensor. A problem for the interpretation of mathematical objects as physical variables results: by a suitable choice of the conformal factor ϕ, a “constant” gravitational coupling function could be reached, again. In his paper, M. Fierz suggested to either couple Jordan’s gravitational theory to point particles or to a (quantum-) Klein-Gordon field in order to remove the difficulty. Fierz also claimed that Jordan had overlooked a physical effect. According to him, a dielectricity “constant” of the vacuum could be introduced \({\epsilon _0} = {\kappa ^{1 + {1 \over \eta}}} = {J^{1 + \eta}}\) from looking at Jordan’s field equations. His Lagrangian corresponded to \(\kappa (R - \varsigma {{\kappa {,_r}{\kappa ^{,r}}} \over {{\kappa ^2}}}) - {{{\epsilon _0}} \over {{c^2}}}{F_{rs}}{F^{rs}}\) with Jη = κ. On the other hand, in the MKSA system of physical units, the fine-structure constant is \(\alpha = {{{e^2}} \over {\hbar c{\epsilon _0}}}\). Thus, the fine-structure constant would depend on κ.

“Assumed that κ be variable in cosmic spaces, then this variability must show up in the redshift of light radiated from distant stars.” ([195], p. 134)Footnote 52

Because this had not been observed, Fierz concluded that κ = −1. Both, Pauli and Fierz gave a low rating to Jordan’s theoryFootnote 53

Neither Pauli nor Fierz seem to have known that the mathematician Willy Scherrer at the university of Bern had suggested scalar-tensor theory already in 1941 before P. Jordan, and without alluding to Kaluza or Pauli’s projective formulation.Footnote 54 Also, in 1949, Scherrer had suggested a more general Lagrangian [532]:

$${\mathcal L} = (R - 2\Lambda){\psi ^2} + 4\omega {g^{rs}}{{\partial \psi} \over {\partial {x^r}}}{{\partial \psi} \over {\partial {x^s}}}\sqrt {- g} \;.$$
(117)

ψ is considered to be a scalar matter field. He had advised a student, K. Fink, to work on the Lagrangian \({\mathcal L} = (R + 2\omega {g^{rs}}{{\partial \psi} \over {\partial {x^r}}}{{\partial \psi} \over {\partial {x^s}}})\sqrt {- g}\) [198]. The ensuing field equations correspond to those following from Jordan’s Lagrangian if his parameter η = 0 ([319] p. 140). Exact solutions in the static, spherically symmetric case and for a homogeneous and isotropic cosmological model were published in 1951 almost simultaneously by Fink (η = 0) and Heckmann, Jordan & Fricke (η = 1) [243].

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

“Because according to my opinion all “unified field theories” are based on dubious ideas — in particular it is a typically suspect idea of the great masters Einstein and Schrödinger to add up the symmetric and antisymmetric parts of a tensor — I have to pose the question […].” (W. Pauli to M. Fierz, 15 Dec. 1953) ([491], p. 390–391)Footnote 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 Einstein-Schrödinger UFT. His notation for differential forms and tangent vectors living in two reference systems is non-standard. As his most important achievement he regarded “the absolutely invariant and at the same time locally exact conservation laws.” In his correspondence with Fierz, Pauli expressed his lack of understanding: “What he means with this, I do not know, because all generally relativistic field theories abound with energy laws” ([491], p. 403). H. T. Flint wrote a comment in which he claimed to have shown that Scherrer’s theory is kin to Einstein’s teleparallelism theory [213]. For studies of Kaluza’s theory in Paris (Jordan-Thiry theory) cf. Section 11.1.

Continued studies of Kaluza-Klein theory in Princeton, and elsewhere

As described in Section 6.3 of Part I, since 1927 Einstein and again in 1931 Einstein and MayerFootnote 56, within a calculus using 5-component tensorial objects in space-time, had studied Kaluza’s approach to a unification of gravitation and electromagnetism in a formal 5-dimensional space with Lorentz-signature. A decade later, Einstein returned to this topic in collaboration with his assistant Peter Bergmann [167, 166]. The last two chapters of Bergmann’s book on relativity theory are devoted to Kaluza’s theory and its generalization ([20]. Einstein wrote a foreword, in which he did not comment on “Kaluza’s unified field theory” as the theory is listed in the book’s index. He admitted that general relativity “[…] has contributed little to atomic physics and our understanding of quantum phenomena.” He hoped, however, that some of its features as were “general covariance of the laws of nature and their nonlinearity” could contribute to “overcome the difficulties encountered at present in the theory of atomic and nuclear processes” ([20] p. V). Two years before Bergmann’s book appeared, Einstein already had made up his mind against the five-dimensional approach:

“The striving for most possible simplicity of the foundations of the theory has prompted several attempts at joining the gravitational field and the electromagnetic field from a unitary, formal point of view. Here, in particular, the five-dimensional theory of Kaluza and Klein must be mentioned. Yet, after careful consideration of this possibility, I think it more proper to accept the mentioned lack of inner unity, because it seems to me that the embodiment of the hypotheses underlying the five-dimensional theory contains no less arbitrariness than the original theory.”Footnote 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 five-dimensional space was seen by them essentially as a four-dimensional one with a small periodical strip or a tube in the additional spacelike dimension affixed. The 4-dimensional metric then is periodic in the additional coordinate x4.Footnote 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’ gαβ,4 = 0 (cf. Section 4.2 of Part I, Eq. (109)): they set gik,4 = 0, but assumed (gi4,4 = 0 and g44 = 1: the electrodynamic 4-potential remains independent of x4. Due to the restriction of the covariance group (cf. Section 4.2, Part I, Eq. (112)), in space-time many more possibilities for setting up a variational principle than the curvature scalar of 5-dimensional space exist: besides the 4-dimensional curvature scalar R, Einstein & Bergmann list three further quadratic invariants: A rs Ars, grs ,4grs,4, (grsgrs,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. Scalar-tensor theory is excluded due to the restrictions introduced by the authors. Except for the addition of some new technical concepts (p-tensors, p-differentiation) and the inclusion of projective geometry, Bergmann’s treatment of Kaluza’s idea in his book did not advance the field.

The mathematicians K. Yano and G. Vranceanu showed that Einstein’s and Bergmann’s generalization may be treated as part of the non-holonomic UFT proposed by them [713, 681]. Vranceanu considered space-time to be a “non-holonomic” totally geodesic hypersurface in a 5-dimensional space V5, i.e., the hypersurface cannot be generated by the set of tangent spaces in each point. Besides the metric of space-time ds2 = g ab dxa dxb, (a, b = 1, 2, 3, 4), a non-integrable differential form ds5 = dx5ϕ a dxa 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 m0 and 5-vector vA, (A = 1, …, 5) was chosen to be a geodesic tangent to the non-holonomic 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 energy-momentum 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).Footnote 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 m0 be different from zero, a fact which is in accord with the hypothesis of Louis de Broglie (Une nouvelle conception de la lumière; Hermann, Paris 1934).” ([681], p. 524)Footnote 60 More than a decade later, K. Yano and M. Ohgane generalized the non-holonomic UFT to arbitray dimension: n-dimensional space is a non-holonomic hypersurface of (n + 1)-dimensional Riemannian space. It is shown that the theory “[…] seems to contain all the geometries appearing in the five-dimensional unified field theories proposed in the past and to suggest a natural generalization of the six-dimensional unified field theories proposed by B. Hoffmann, J. Podolanski, and one of the present authors” ([714], pp. 318, 325–326). They listed the theories by Kaluza-Klein, Veblen-Hoffmann, Einstein-Mayer, Schouten-Dantzig, Vranceanu and Yano; cf. also Sections 3.1 and 11.2.1.

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

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

“I still work as passionately even though most of my intellectual children, in a very young age, end in the graveyard of disappointed hopes”. ([560], p. 552)Footnote 61

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

“After twenty years of vain searching, this year now I have found a promising field theory which is a quite natural sequel to the relativistic gravitational theory. It is in line with Kaluza’s idea about the essence of the electromagnetic field.” ([163], p. 321)Footnote 62

Non-local fields

Bi-vectors; generalized teleparallel geometry

In 1943, Einstein had come to the conclusion that the failure of “finding a unified theory of the physical field by some generalization of the relativistic theory of gravitation” seemed to require “a decisive modification of the fundamental concepts” ([165], p. 1). He wanted to keep the four-dimensional space-time continuum and the diffeomorphism group as the covariance group, but wished to replace the Riemannian metric by a generalized concept. Together with the assistant at the Institute for Advanced Studies, Valentine BargmannFootnote 63, he set out to develop a new scheme involving “bi-vector fields”. Unlike the concept of a bi-vector used by Schouten in 1924 ([537], p. 17) and ever since in the literature, i.e., for the name of a special antisymmetric tensor, in the definition by Einstein and Bargmann the concept meant a tensor depending on the coordinates of two points in space-time, an object which would be called “bi-local” or “non-local”, nowadays. The two points, alternatively, could be imagined to lie in the same manifold (“single space”), or in two different spaces (“double space”). In the latter case, the coordinate transformations for each point are independent.

In place of the Riemannian metric, a contravariant bi-vector \({g^{\overset {kl} {21}}}\) is defined via

$${g_{\underset 1 i \underset 2 j}}{g^{\underset 2 j \underset 1 k}} = \delta _{\underset 1 i}^{\underset 1 k}\;,$$
(118)

and, similarly,

$${g^{\underset 2 i \underset 1 m}}{g_{\underset 1 m \underset 2 k}} = \delta _{\underset 2 k}^{\underset 2 i}\;.$$
(119)

Here, the numbers refer to the two points, while the Latin indices denote the usual tensor indices. The coordinate transformation \({x^i} \to {\overset * x^i}\) for a simple “bi-vector” \({T^{\overset {ik} {21}}}\) is given by:

$${\overset {\ast} T ^{\underset 2 i \underset 1 k}} = {{\partial {{\overset {\ast} x}^{\underset 2 i}}} \over {\partial {x^{\underset 2 m}}}}{{\partial {{\overset {\ast} x}^{\underset 1 k}}} \over {\partial {x^{\underset 1 l}}}}{T^{\underset 2 m \underset 1 l}}\;.$$
(120)

Already here a problem was mentioned in the paper: there exist too many covariant geometric objects available for deriving field equations. This is due to the possibility to form covariant quantities containing only first order derivatives like the tensorial quantity: \({\gamma _{\underset 1 i \underset 2 j \underset 2 k}}: = {g_{\underset 1 i \underset 2 j,\underset 2 k}} - {g_{\underset 1 i \underset 2 k, \underset 2 j}}\). In order to cut down on this wealth, a new operation called “rimming” was introduced which correlated a new “bi-vector” \({\overset * g_{\underset \alpha i \underset \beta k}}\) with \({g_{\underset \alpha i \underset \beta k}}\) by multiplying it from left and right by tensors of full rank \({\omega _{\underset \alpha i}}^{\overset \kappa \alpha}\), \({\omega _{\underset \beta i}}^{\overset \kappa \beta}\) where each is taken from one of the two manifolds (now Greek indices refer to the two different points)Footnote 64:

$${\overset {\ast} g _{\underset \alpha i \underset \beta k}} = {\omega _{\underset \alpha i}}^{\underset \alpha m}{g_{\underset \alpha m \underset \beta n\prime}}{\omega _{\underset \beta k}}^{\underset \beta n}\;.$$
(121)

All tensors \(\overset * g\) obtained by rimming g were considered as different representations of the same field. The rimming of a contravariant “bi-vector” was done similarly by multiplying from the left with \(\prime{\sigma _{\underset \beta i}}^{\overset \kappa \beta}\) and from the right with tensors \({\sigma _{\underset \alpha i}}^{\overset \kappa \alpha}\). In order that (118), (119) remain invariant under rimming, the relations \(_\alpha ^\sigma = _\alpha ^{{\omega ^{- 1}}},{\,\prime}_\alpha ^\sigma = {{(\prime}\underset \alpha \omega)^{- 1}},\), must hold. A comparison of (120) and (121) shows that a coordinate transformation can be combined with a rimming operation “in such a way that the bi-vector components remain invariant — i.e., behave like scalars under the resultant transformation” ([165], p. 4).

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

$${g_{\underset \alpha i \underset \beta k}}{g^{\underset \beta k \underset \gamma l}}{g_{\underset \gamma l \underset \delta m}}{g^{\underset \delta m \underset \alpha j}} = \delta _i^{j}\;.$$
(122)

if \({g_{\underset \alpha i \underset \beta k}}\) is treated as a matrix, it is easy to see that all higher-order tensorial equations (6, 8, …-point) are dependent on the four-point equation. Next, the authors showed that by a special rimming operation \({g_{\underset \alpha i \underset \beta k}}\) can be transformed into δ ik such that the corresponding space-time is flat. The trace on (i,j) in (122) leads to a scalar equation:

$${g_{\underset \alpha i \underset \beta k}}{g^{\underset \beta k \underset \gamma l}}{g_{\underset \gamma l \underset \delta m}}{g^{\underset \delta m \underset \alpha i}} = \delta _i^{i} = 4\;.$$
(123)

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

In another paragraph, the authors returned to the “single space”-version. Here, a symmetry condition is demanded: \({g_{\underset \alpha i \underset \beta k}}={g_{\underset \beta k \underset \alpha i}}\). Now, the tensorial p-point equations admit the special case that the two points (coordinates α, β) coincide. By a suitable rimming operation \({g_{\underset \alpha i \underset \alpha k}}={\eta _{ik}}\) was reached where η ik is the matrix diag (1, −1, −1, −1). The rimming operations were performed with representations of the Lorentz group. Also, mixed bi-vectors \({g^{\overset i \alpha}}_{\underset \beta k} = {\eta ^{il}}{g_{\underset \alpha l \underset \beta k}}\) were introduced and a tensorial three-point equation had to be satisfied:

$${g^{\underset \alpha i}}_{\underset \beta k}\,{g^{\underset \beta k}}_{\underset \gamma l}\,{g^{\underset \gamma l}}_{\underset \alpha m} = \delta _m^{\;\,i}\,.$$
(124)

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

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

$${A^{\underset \beta k}} = {g^{\underset \beta k}}_{\underset \alpha i}{A^{\underset \alpha i}},\quad {A_{\underset \beta k}} = {A_{\underset \alpha i}}{g^{\underset \alpha i}}_{\underset \beta k}$$
(125)

connects the points with coordinates α and β.Footnote 65 Two conditions are to be fulfilled: the displacement of a vector from α to β and back does not change it, and “the scalar product of a covariant and a contravariant vector is invariant with respect to the affine connection”. This led to

$${g^{\underset \alpha i}}_{\underset \alpha k} = \delta _k^i\,,\;{g^{\underset \alpha i}}_{\underset \beta k}{g^{\underset \beta k}}_{\underset \alpha j} = \delta _j^i\;.$$
(126)

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

$$\underset {\alpha \beta} g \underset {\beta \gamma} g \underset {\gamma \alpha} g \; - 1 = 0\,.$$
(127)

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

$${g_{\underset \alpha i \underset \beta k}} = \pm {g_{\underset \beta k \underset \alpha i}}\,.$$
(128)

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

$$\int\nolimits_G {\;\left({g_{\;\;\,\underset \beta k}^{\underset \alpha i}\,g_{\;\;\;\underset \gamma l}^{\underset \beta k} - g_{\;\;\;\underset \gamma l}^{\underset \alpha i}} \right)\;\,d\underset \alpha \tau = 0} \,.$$
(129)

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

$$\int\nolimits_G {\;g_{\;\;\;\underset \beta k}^{\underset \alpha i}\,g_{\;\;\;\underset \gamma l}^{\underset \beta k}d\underset \alpha \tau \; - g_{\;\;\;\underset \gamma l}^{\underset \alpha i} = 0} \,.$$
(130)

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

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

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

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

$$g_{\;\;\underset \beta k}^{\underset \alpha i} = {(h_\alpha ^i)_{\;at\;\beta}}\;{({h_{k\alpha}})_{\;at\;\alpha}}\,.$$
(131)

Note that Schrödinger denoted the tetrad by \(h_a^\nu\) with Latin indices a, b, … for the number of the leg, and Greek space-time indices such that \({h_{\nu a}}\,h_b^\nu = {\delta _{ab}}\). By comparing the parallel displacement of a 4-vector \(\delta {A^\nu} = - \Delta _{\rho \sigma}^{\,\,\nu}\;{A^\rho}\,d{x^\sigma}\) with \({A^{\overset i \beta}} = {g^{\overset i \beta}}_{\underset \alpha k}{A^{\overset k \alpha}}\) he arrived at his interpretation of \({g^{\overset i \beta}}_{\underset \alpha k}\) as a connection. As the relation between the affine connection — which he called “infinitesimal connection” — and the bi-vector connection, named “distant connection”, he took:

$$\Delta _{ij}^{\;\;k} = {\left\{{{\partial \over {\partial {x_{\underset \alpha j}}}}\left({g_{\;\underset \alpha i}^{\underset \beta k}} \right)} \right\}_{\alpha = \beta}}.$$
(132)

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

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

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

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

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

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

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

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

The laws of nature are invariant not only with regard to the relativistic transformations but also with regard to the substitutions xαp α , p α → −xα […]. All amounts to replace your fundamental invariant xαx α = R by the symmetrical quantity S = R + P where P = pαp α . S is an operator, the integer eigenvalues of which are the distances […].” ([168], p. 242.)Footnote 66

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

Unified Field Theory and Quantum Mechanics

Obviously, Einstein did not trust an investigation like the experimental physicist Osborn’s (1917–2003) trying to show by ideal measurements that the notion of curvature can be applied only “in the large” where “the domain of largeness is fundamentally determined by the momentum of the test particle with which the curvature is measured” — due to limitations from quantum mechanics [465]. Osborn’s feeling obviously was shared by the majority of elementary particle physicists, in particular by F. Dyson:

“The classical field theory of Einstein — electromagnetic and gravitational together — give us a satisfactory explanation of all large-scale physical phenomena. […] But they fail completely to describe the behavior of individual atoms and particles. To understand the small-scale 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.

The impact of Schrödinger’s and Dirac’s equations

In the introduction to Section 7 of Part I, a summary has been given of how Einstein’s hope that quantum mechanics could be included in a classical unified field theory was taken up by other researchers. A common motivation sprang from the concept of “matter wave” in the sense of a wave in configuration space as extracted from Schrödinger’s and Dirac’s equations. Henry Thomas FlintFootnote 67 whom we briefly met in Section 7.1 of Part I, was one of those who wanted to incorporate quantum theory into a relativistic field theory for gravitation and electrodynamics. In Flint’s imagination, the content of quantum mechanics was greatly condensed: it already would have been reproduced by the generation of a suitable relativistic wave equation for the wave function ψ as a geometric object in an appropriate geometry. This might be taken as an unfortunate consequence of the successes of Schrödinger’s wave theory. In the first paper of a series of three, Flint started with a 5-dimensional curved space with metric γ ij , (i, j = 0, 1, …, 4) and an asymmetric connection:

$$\Delta _{ij}^{\;\;k} = \Gamma _{ij}^{\;\,k} + T_{ij}^{\;k}\,,\;T_{ij}^{\;k} = S_{ij}^{\;\,k} - \Theta _{ij}^{\;\,k}\,,$$
(133)

where \(\Gamma _{ij}^{\;k}(\gamma)\) is the Levi-Civita connection of the 5-dimensional space, \(\Delta _{[ij]}^{\;\;\;\;k} = S_{ij}^{\;k}\) the torsion tensor, and \(\Theta _{ij}^{\;k}\) an additional symmetric part built from torsion: \(\Theta _{ij}^k = S_ \cdot ^k\overset \cdot {(ij)}\) [204]. A scalar field ψ was brought in through vector torsion:

$${S_i} = {\partial _i}ln\psi$$
(134)

and was interpreted as a matter-wave function. The metric is demanded to be covariantly constant with regard to \(\Delta _{ij}^{\;k}\), i.e., γij;l = 0. In the next step, the Ricci-scalar of the 5-dimensional space is calculated. Due to (134), it contained the 5-dimensional wave operator. Up to here, an ensuing theory for the scalar field ψ could be imagined; so far nothing points to quantum mechanics nor to particles. By using de Broglie’s idea that the paths of massive and massless particles be given by geodesics in 5-dimensional space and O. Klein’s relation between the 5th component of momentum and electric charge, Flint was led to equate the curvature scalar to a constant containing charge, mass and Planck’s constant:

$${1 \over 4}{\gamma ^{rs}}{R_{rs}}(\Delta) = {1 \over {{\hbar ^2}}}\;\left({{m^2}{c^2} + {{{e^2}} \over {{\alpha ^2}{\gamma _{55}}}}} \right)\;.$$
(135)

Following Kaluza and Klein, \({\alpha ^2}\gamma 55 = {{16\pi G} \over {{c^2}}}\) was set with G the Newtonian gravitational constant. The linear one-particle wave equation thus obtained contains torsion, curvature, the electromagnetic field F ab , (a, b = 0, 1, …, 3) and a classical spin tensor Alm. It is:

$${(\psi)^{- 1}}{\gamma ^{rs}}({{{\partial ^2}\psi} \over {\partial {x^r}\partial {x^s}}} - \Gamma _{rs}^{\;\;l}{\partial _l}\psi) - {{2\pi ie} \over h}{F_{ab}}{A^{ab}} + {1 \over 4}(R(\gamma) - {{4\pi G} \over {{c^2}}}H) + {{4{\pi ^2}} \over {{h^2}}}({m^2}{c^2} - {{{e^2}} \over {{c^2}}}{\phi ^a}{\phi _a}) = 0\,,$$
(136)

where R is the 4-dimensional curvature scalar, H = F ab Fab, and ϕ a the electromagnetic 4-potential. The particle carries charge e and mass m. While Flint stated that “[…] the generalized curvature, is determined by the mass and charge of the particle situated at the point where the curvature is measured” ([204], p. 420), the meaning of (136) by no means is as trivial as claimed. A solution ψ determines part of torsion (cf. (134), but torsion is needed to solve (136); we could write R = R (ψ), \(\Gamma _{r\;s}^{\;\;\;l} = \Gamma _{r\;s}^{\;\;\;l}(\psi)\). Hence, (136) is a highly complicated equation.

As a preparation for the second paper in the series mentioned [205], a link to matrix theory as developed by Schrödinger was given through replacement of the metric “by more fundamental quantities”, the 5 by 5 matrices α i :

$${\alpha _i}{\alpha _j} + {\alpha _j}{\alpha _i} = 2{\gamma _{ij}},\;\,{\alpha _{i;k}} = T_{ik}^{\;\;r}{\alpha _r},$$
(137)

where the covariant derivative refers to the Levi-Civita connection of γ ij . Both formulations, with and without matrices were said “to be in harmony”. In this second paper, Dirac’s equation is given the expression αs Π s ψ = 0 with \({\rm{I}}{{\rm{I}}_j} = {p_j} + {e \over c}{\phi _j}\), \({p_j} = {\hbar \over i}{\partial \over {\partial {x^j}}}\), ϕ5 = 0. \({\partial \over {\partial {x^5}}}\) is replaced by \({{imc} \over \hbar}\). In place of (134) now

$${S_r}{\alpha ^r}\psi = {\alpha ^r}{{\partial \psi} \over {\partial {x^r}}}$$
(138)

is substituted.Footnote 68 Dirac’s equation then is generalized to \({\alpha ^r}{{\partial \psi} \over {\partial {x^r}}} + {i \over \hbar}{\rm{I}}{{\rm{I}}_s}{\alpha ^s}\psi = 0\). The resulting wave equation of second order contains terms which could not be given a physical interpretation by Flint.

In the third paper of 1935 [206], Flint took up the idea of “matrix length” = α j dxj Fock and Ivanenko had presented six years before without referring to them [215]. ψ now is taken to be a column (ψ1, ψ2, …,ψ5)t and the matrix length of a vector Ak defined to be L:= Arα r ψ such that L2 = (ψ*ψ)γrsA r A s with ψ* being the conjugate to ψ. Flint seemed undecided about how to interpret ψ*ψ. On the one hand, he said that “[…] ψ*ψ has been interpreted as the density of matter” ([206], p. 439), on the other he apparently had taken note of the Kopenhagen interpretation of quantum mechanics (without sharing it) when writing:

“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 re-direction 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 gauge-equations” ([214], p. 260). Under parallel transport, the matrix length L of a vector A k is assumed to change by dL = ΘR r αsA s ψ dxr, where R r is an operator (a matrix) corresponding to the 5-vector 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 wave-equations [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.

Other approaches

We come back to a paper by M. Born which was referred to already in Section 3.3.2, but under a different perspective. In view of the problems of quantum field theory at the time with infinite self-energy of the electron, the zero-point energies of radiation fields adding up to infinity etc., Max Born preferred to unify quantum theory and “the principle of general invariance”, i.e., inertial fields rather than include the gravitational field. The uncertainty relations between coordinates and momenta served as a motivation for him to assume independent and unrelated metrics g ij (xl) in configuration and γij (pn) in momentum space [39]. As field equations in momentum space he postulated the Einstein field equations for a correspondingly calculated Ricci-tensor (as a function of momenta) Pkl (pr):

$${P^{kl}} - \left({{1 \over 2}P + \lambda \prime} \right)\;{\gamma ^{kl}} = - \kappa \prime{T^{kl}}$$
(139)

The “nuclear constants” λ′, κ′ remained undetermined. Born was silent on the matter tensor. His applications of the formalism turned toward quantum electrodynamics, black body radiation and the kinetic theory of gases (of atoms). By choosing, in momentum space, the analogue to the Friedman cosmological solution with space sections of constant curvature, an upper limit b for momentum ensued. The number of quantum states in volume elements V of configuration space and in a volume element of momentum space turned out to be \(g{{{\pi ^2}V} \over {{h^3}}}b\) and had many consequences e.g., for Planck’s and Coulomb’s laws and for nuclear structure. The parameter b determined all deviations from previous laws: the Coulomb law for two particles became \({{{e_1}{e_2}} \over {r_{12}^2}}f(2{{{r_{12}}} \over \hbar}b)\) with the function \(f(x) = \int\nolimits_0^x {{J_0}(y)} \;dy\) and the Bessel function J0; the Planck law for the energy density of black body radiation \(u(\nu, T) = {{8\pi h} \over {{c^3}}} - {{{\nu ^3}} \over {(\exp ({{h\nu} \over {kT}} - 1)(1 - {{(\nu \tau)}^2})}}\) with \(\tau = {h \over {bc}}\). Born fixed b such that the classical electron radius \({r_0} = {\hbar \over b} = {{{e^2}} \over {m{c^2}}}\). The paper’s main result was a geometric foundation for the assumption of an upper limit for momentum — not a unification of quantum mechanics with anything else. Perhaps, Born had recycled an idea from his paper with Infeld, in which they had introduced an upper limit for the electrical field (cf. Section 5).

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 hypo-center of the explosion lying 1.5 km away from the Research Institute for Theoretical Physics.Footnote 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 space-geometry 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 space-time 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).Footnote 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.

The physical system, “the space-time-matter” manifold, was to be seen as a (quantum mechanical) state ψ, a 4-component (Dirac) spinor; “distance” in microscopic space became defined as an eigenvalue of a linear distance operator. In order to find this operator, by following Dirac, a principle of linearization was applied:

$$ds\,\psi : = {\gamma _i}\,d{x^i}\psi ,$$
(140)

with \({\gamma _i} = h_i^a\overset 0 {{\gamma _a}}\) where \(h_i^a\) is an arbitrary tetrad (with the tetrad index a = 1, 2, 3, 4), and \(\overset 0 {{\gamma _i}}\) denoting the Dirac matrices. If

$${\gamma _{\left(i \right.}}{\gamma _{\left. j \right)}} = {g_{ij}}$$
(141)

is demanded, i.e.,

$${\Sigma _a}h_i^ah_j^a = {g_{ij}}\,,$$
(142)

the eigenvalues of the distance operator are \(\pm \sqrt {{g_{ij}}d{x^i}d{x^j}}\). If Riemannian covariant differentiation is used, then

$${\nabla _i}{\gamma _j}: = {{\partial {\gamma _j}} \over {\partial {x^i}}} - \{_{ij}^{\;\,k}\} {\gamma _k} - {\Gamma _i}{\gamma _j} + {\Gamma _j}{\gamma _i}\,.$$
(143)

Here, Γ denotes the spin connection. As the fundamental equation of the theory

$${\nabla _i}\;\psi = {\Sigma _i}\;\psi \,,$$
(144)

was written down where Σ i is an as yet undetermined 4-vector with matrix entries. It was expected that (144) describe the gravitational, electromagnetic and the matter field “in unified form not discriminating macroscopic and microscopic phenomena” ([427], p. 11). In 1929, (140) had also been suggested by Fock and Ivanenko [215], a paper mentioned briefly in Section 7.2 of Part I. As we have seen, at around the same year 1935, H. T. Flint had set up a similar unified theory as Mimura [206]. The theory of Mimura and Takeno was to be applied to the universe, to local irregularities (galaxies) in the universe and to the atom. Only the Einstein cosmos and de Sitter space-time were allowed as cosmological metrics. For the atom, a solution in a space-time with metric \(d{s^2} = {R^2}({(d{x^1})^2} + {\sin ^2}{x^1}{(d{x^2})^2}) + {{{R^2}} \over {{{({x^3})}^2}}}({(d{x^3})^2} + {(d{x^0})^2})\) was obtained and a wave function “which can be identified with the Dirac level of the hydrogen atom if the arbitrary functions and constants in the equation are chosen suitably” ([427], p. 66). With a particular choice of Σ i , (cf. [428], Eq. 4.15 on p. 4) the fundamental equation for ψ was then determined to be the matrix equation:

$$({\partial _i} - {\Gamma _i})\psi = (T_i^5{\gamma _5} - {L_i}I)\psi \,,$$
(145)

with \({\Gamma _i}\;: = {\textstyle{1 \over 4}}[h_r^s{\partial _i}h_t^r - {\{_{ti}^s\} _g}]{\gamma ^t}{\gamma _s}\). \(T_i^5\) and L i are arbitrary vectors. In [428], a second fundamental equation was added:

$${\gamma ^i}({\partial _i} - {\Gamma _i})\psi = \mu \psi$$
(146)

with a scalar μ. For complete integrability of (145), the Riemannian curvature tensor \(K_{klm}^i\) must vanish. Equation (145) reduces to i ψ = 0, g ij = η ij with the solution ψ a = const., (a = 0, 1, …, 3). This being too restrictive, (145) was weakened to ( i − Γ i )ψ = 0 for either ψ0 = ψ3 = 0 or ψ1= ψ2 = 0 with the integrability conditions \({{\sqrt g} \over 2}{\epsilon _{lmrs}}K_{ij}^{\;\;\;rs} = \pm {K_{ijlm}}\), respectively. T. Sibata gave a solution of this equation expressing self-duality for weak fields [576]. He also set out to show that Born-Infeld theory follows from his approach to wave geometry in the case of vacuum electrodynamics [577]. In this paper, the condition of complete integrability for his version of (144) read as \({g \over 4}{\epsilon _{lmrs}}{\epsilon _{ijpq}}{K^{pqrs}} = \pm {K_{ijlm}}\).

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” space-time, 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 space-time.” ([428], p. 41.) Clearly, the algebra of γ-matrices which is all what is behind the distance operator, was an insufficient substitute for the algebra of non-commuting observables in quantum field theory.

Born-Infeld Theory

In 1934, M. Born and L. Infeld published a paper on “The Foundations of the New Field Theory” [42]. Its somewhat vague title hid a non-linear theory of the electromagnetic field using a non-symmetric metric but denying a relationship with “‘unitary’ field due to Einstein, Weyl, Eddington, and others […]”. In fact, the original idea for the new theory originated in July 1933 while Born was still a member of the University of Göttingen but already on the move from Germany for vacations in South Tyrol to only return after World War II. Born’s next publication, submitted in August 1933 without institutional address, dealt with the quantization of the electromagnetic field; in it the new Lagrangian was also shown [37]. In view of the problems with divergent terms in quantum (field) electrodynamics at the time, he set out to modify Maxwell’s equations in such a way that an electron with finite radius r0 could be described; its electric potential remained finite for \({r \over {{r_0}}} \ll 1\) [36]. The Lagrangian for the new electrodynamics was \(L = {1 \over {{a^2}}}\sqrt {1 + {a^2}({H^2} - {E^2})}\) with the constant a of dimension \({{r_0^2} \over e}\), where e is the elementary electric charge and r0 the electron radius. In the limit a → 0 the Lagrangian of Maxwell’s theory reappeared: \({1 \over {{a^2}}} + {1 \over 2}({H^2} - {E^2})\). In the paper with Infeld, the Lagrangian is generalized in order to include the gravitational field:

$$L = \sqrt {- \det ({g_{(ij)}} + {f_{ij}})} - \sqrt {- \det ({g_{(ij)}})}$$
(147)

where g(ij) is the (Riemannian) metric and f ij = f[ij ] the electromagnetic field tensor; g ij = g(ij) + f ij , formally is an asymmetric metric. The Lagrangian (147) can be expressed by the two invariants of Maxwell’s theory \(F\;: = {1 \over 2}{f_{mr}}{f_{ns}}{g^{mn}}{g^{rs}}\) and \(G\;: = {1 \over 4}{\epsilon ^{mnrs}}{f_{mn}}{f_{rs}} = = {f^{*rs}}{f_{rs}}\) as

$${\mathcal L} = \sqrt {- g} L = \sqrt {- g} (\sqrt {1 + F - {G^2}} - 1)\,.$$
(148)

The new field equations become:

$${{\partial \sqrt {- g} {f^{{\ast}is}}} \over {\partial {x^s}}} = 0\,,\;\,{{\partial \sqrt {- g} \;{p^{is}}} \over {\partial {x^s}}} = 0\,,$$
(149)

with the definition \(\sqrt {- g} \;{p^{ik}}\;: = {{\partial {\mathcal L}} \over {\partial {f_{ik}}}}\). Insertion of \(F = 1/{b^2}({\overrightarrow {\bf{B}} ^2} - {\overrightarrow {\bf{E}} ^2})\), \(G = 1/{b^2}\overrightarrow {\bf{B}}. \overrightarrow {\bf{E}}\) led to Maxwell’s equations plus the relations between fields and inductions:

$$\overset \rightarrow {\bf{H}} = {b^2}{{\partial L} \over {\partial \overset \rightarrow {\bf{B}}}} = {{\overset \rightarrow {\bf{B}} \, - G\overset \rightarrow {\bf{E}}} \over {\sqrt {1 + F - {G^2}} - 1}}\,,\;\,\overset \rightarrow {\bf{D}} = - {b^2}{{\partial L} \over {\partial \overset \rightarrow {\bf{E}}}} = {{\overset \rightarrow {\bf{E}} \, + G\overset \rightarrow {\bf{B}}} \over {\sqrt {1 + F - {G^2}} - 1}}\,.$$
(150)

“The quotient of the field strength expressed in the conventional units divided by the field strength in the natural units” was denoted by b and named the “absolute field”. As was well known, many asymmetric energy-momentum tensors for the electromagnetic field could be formulated. Years later, St. Mavridès took up this problem and derived identities for the symmetric Minkowski tensor, the fields and inductions, independent of whether the relations between fields and inductions were linear or more general [410]. Xinh Nguyen Xua then showed that with the relations (150), all the various energy-momentum tensors can be derived from one such symmetric tensor [711]. Born & Infeld chose

$$T_k^l = L\delta _k^l - {{{f_{rk}}{f^{rl}} - \delta _k^l{G^2}} \over {\sqrt {1 + {F^2} - {G^2}}}}\,.$$
(151)

The static solution of the new equations for the potential of a point charge was determined to be

$$\phi (r) = {e \over {{r_0}}}\int\nolimits_r^\infty {{{dv} \over {\sqrt {1 + {v^4}}}}} \,,\;\,{r_0}: = \sqrt {{e \over b}} \,.$$
(152)

It turned out that, from \(E = {m_0}{c^2}: = {{{b^2}} \over {4\pi}}\int {L{d^3}x}\), r0 could be calculated numerically via \(2{{{e^2}} \over {{r_0}}}(\int\nolimits_0^\infty {(1 - {\textstyle{{{x^2}} \over {\sqrt {1 + {x^4}}}}})}\) to take the value \({r_0} = 1,2361{{{e^2}} \over {{m_0}{c^2}}}\) and thus b could also be determined. According to Born and Infeld: “The new field theory can be considered as a revival of the old idea of the electromagnetic origin of mass” Also, the existence of an absolute field as a “natural unit for all field components and the upper limit for a purely electric field” ([42], p. 451) had been assumed.

Unsurprisingly, Pauli was unhappy with the paper by Born and Infeld as far as its inclusion of the gravitational field via g(ij) + f ij was concerned. Instead, in his letter of 21 December 1933 to Max Born ([488], p. 241), he suggested to take as a Lagrangian density \({\mathcal L} = \sqrt {- g} F(P)\) with Kaluza’s curvature scalar in 5 dimensions \(P = R + {k \over {2{c^2}}}{f_{rs}}{f^{rs}}\). “In particular, it is possible to set \(F(P) = \sqrt {1 + {1 \over {{\rm{const}}{\rm{.}}}}P}\), and therefore your electrodynamics is compatible with the projective view on the electromagnetic and gravitational field.”Footnote 71 But for Born, electrodynamics was in the focus. Three months after Pauli’s criticism, he wrote to Einstein in connection with his paper with Infeld:

“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).Footnote 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].Footnote 73 He suggested the Lagrangian \({\mathcal L} = (\sqrt {1 + B} - 1)\sqrt {- \gamma}\) where γ ij is the 5-dimensional projective metric and B the projective curvature scalar. Due to B = Rgpr gqs 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 non-vanishing electromagnetic field. In the augmented Born-Infeld 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.

In connection with the work of Euler and Kockel on the scattering of light by light under his guidance, W. Heisenberg wrote Pauli on 4 November 1934: “The terms to be added to the Lagrangian look like in the theory of Born and Infeld, but they are twenty times larger than those of Born and Infeld” ([488], p. 358).Footnote 74 But Pauli had not changed his opinion; in connection with the scattering of light by light, he answered Heisenberg curtly: “I do not care about Born’s theory” ([488], p. 372). Ten years later, in his letter to Einstein of 10 October 1944 Born assessed his theory with some reservation ([168], p. 212):

“[…] 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 non-linear electrodynamics, and this is no particular success […].”Footnote 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 non-linear 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.Footnote 76

One who became attracted by the Born-Infeld theory was E. Schrödinger. He had come “across a further representation, which is so entirely different from all the aforementioned, and presents such curious analytical aspects, that I desired to have it communicated” ([542], p. 465). He used a pair \({\mathcal F},{\mathcal G}\) of complex combinations of the 3-vector fields \(\vec B,\;\vec E,\;\vec H,\;\vec D\) such that \({\mathcal F} = \vec B - i\vec D,\;{\mathcal G} = \vec E + i\vec H\). The Lagrangian \({\mathcal L}\) was to be determined such that its partial derivatives with respect to \({\mathcal F}\) and \({\mathcal G}\) coincided with the complex conjugates: \(\bar {\mathcal F} = {{\partial {\mathcal L}} \over {\partial {\mathcal G}}}\) and \(\bar {\mathcal G} = {{\partial {\mathcal L}} \over {\partial {\mathcal F}}}\). The result is

$${\mathcal L} = {{{{\mathcal F}^2} - {{\mathcal G}^2}} \over {{\mathcal F}\,\cdot\,{\mathcal G}}}\,.$$

Born’s constant b was set equal to one. Schrödinger showed that his formulation was “entirely equivalent to Born’s theory” and did not provide any further physical insight. Thus, Schrödinger’s paper gave a witty formal comment on the Born-Infeld theory. Ironically, it had been financed by Imperial Chemical Industries, Limited.

S. KichenassamyFootnote 77 studied the subcase of an electromagnetic null field with matter tensor: T ij = A2k i k j , k i ki = 0 and showed that in this case the Born-Infeld theory leads to the same results as Maxwell’s electrodynamics [328, 340].

Affine Geometry: Schrödinger as an Ardent Player

A unitary theory of physical fields

When in peaceful Dublin in the early 1940s E. SchrödingerFootnote 78 started to think about UFT, he had in mind a theory which eventually would give a unitary description of the gravitational, electromagnetic and mesonic field. Mesons formed a fashionable subject of research at the time; they were thought to mediate nuclear interactions. They constituted the only other field of integral spin then known besides the gravitational and electromagnetic fields. Schrödinger had written about their matrix representations [546]. In his new paper, he deemed it “probable that the fields of the Dirac-type can also be accounted for. […] It is pretty obvious that they must result from the self-dual and self-antidual constituents into which the anti-symmetric part of R kl can be split.” ([545], p. 44, 57.) This is quickly constrained by another remark: “I do not mean that the new affine connection will be needed to account for the well-known Dirac fields”. ([545], p. 58.) He followed the tradition of H. Weyl and A. Eddington who had made the concept of affine connection play an essential role in their geometries — beside the metric or without any. He laid out his theory in close contact with Einstein’s papers of 1923 on affine geometry (cf. Section 4.3.2 of Part I) and the nonlinear electrodynamics of M. Born & L. Infeld [42] (cf. Section 5). On 10 May 1943, M. Born reported to Einstein about Schrödinger’s work: “[…] He has taken up an old paper of yours, from 1923, and filled it with new life, developing a unified field theory for gravitation, electrodynamics and mesons, which seems promising to me. […]” ([168], p. 194.) Einstein’s answer, on 2 June 1943, was less than excited:

“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.).Footnote 79

As I suppose, the “At the time” refers to 1923. With “skew symmetric curvature”, the antisymmetric part of the Ricci-tensor is meant. Schrödinger believed that Einstein had left affine theory because of “aesthetic displeasure” resulting from a mistake in his interpretation of the theory.

Symmetric affine connection

In his first papers on affine geometry, Schrödinger kept to a symmetric connection.Footnote 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 metric-affine or mixed geometry. By contraction of the curvature tensor, second-rank 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.

In his paper, Schrödinger took as such a variational principle:

$$\delta \;\int {{\mathcal L}({K_{ij}})\,d\tau} = 0\,,$$
(153)

thus neglecting homothetic curvature as a further possible ingredient.Footnote 81 Kij(Γ) is the Ricci-tensor introduced in (55) or (56) due to Γ being symmetric. Enthusiastically, he started at the point at which Einstein had given up and defined the symmetric and skew-symmetric quantities:

$${\hat g^{ik}}: = {1 \over 2}({{\partial {\mathcal L}} \over {\partial {K_{ik}}}} + {{\partial {\mathcal L}} \over {\partial {K_{ki}}}})\,,\;\;{\hat f^{ik}}: = {1 \over 2}({{\partial {\mathcal L}} \over {\partial {K_{ik}}}} - {{\partial {\mathcal L}} \over {\partial {K_{ki}}}})\,.$$
(154)

The variation of (153) with respect to the components of the connection \(\Gamma _{ij}^{\;\;k}\) now can be written as:

$${\hat g^{ik}}_{\;\;\parallel l} + {1 \over 3}\delta _l^k\hat f_{\;\;\;\parallel s}^{is} + {1 \over 3}\delta _l^i\hat f_{\quad \;\parallel s}^{ks} = 0\,.$$
(155)

Note that (155) is formally the same equation which Einstein had found in his paper in 1923 when taking up Eddington’s affine geometry [141]. A vector density \({\hat j^k}\) is introduced via

$${\hat j^k}: = \hat f_{\quad \;\parallel s}^{is} = {{\partial {{\hat f}^{is}}} \over {\partial {x^s}}}$$
(156)

with \({{\hat j}^k}\) being interpreted as the (electric) current density. By help of ĝik a (symmetric) metric tensor g ik is introduced by the usual relations:

$${\hat g^{ik}} = \sqrt {- \det ({g_{lm}})} \;{g^{ik}}\,,\;\,{g_{is}}{g^{ks}} = \delta _i^k\; \rightarrow {g^{ik}} = \sqrt {- \det ({{\hat g}_{lm}})} \;{\hat g^{ik}}.$$
(157)

(155) can be formally solved for the components of the symmetric connection to give:

$$\Gamma _{ij}^{\;\;\,k} = \{_{ij}^k\} - {1 \over 2}{g_{ij}}\;{j^k} + {1 \over 6}\delta _i^{\;k}\;{j_j} + {1 \over 6}\delta _j^k\;{j_i}$$
(158)

with \(j_k= {(- g)^{- {1 \over 2}}}{g_{ks}}{\hat j^s}\),g = det(g lm ).Footnote 82 This expression is similar but unequal to the connection in Weyl’s theory (cf. Section 4.1.1 of Part I, Eq. (100)). The intention is to express \(\Gamma _{ij}^{\;\;k}\) as a functional of the components of K ik , insert the expression into (56) (for L = Γ), and finally solve for K ik . What functions here as a metric tensor, is only an auxiliary quantity and depends on the connection (cf. (154), (157)).

In order to arrive at a consistent physical interpretation of his approach, Schrödinger introduced two variables conjugate to ĝik, \({\hat g^{ik}},{\hat f^{ik}}\) by:

$${\gamma _{ik}}: = {K_{(ik)}}\,,{\phi _{ik}}: = - {K_{[ik]}}\,,$$
(159)

and carried out what he called a contact transformation:

$$\overline {\mathcal L} ({\hat g^{lm}},\;\,{\phi _{lm}}) = {\hat g^{lm}}{\gamma _{lm}} - {\mathcal L}\,.$$
(160)

From (160) we get:

$${\gamma _{ik}} = {{\partial \overline {\mathcal L}} \over {\partial {{\hat g}^{ik}}}}\,,\quad {\hat f^{ik}} = {{\partial \overline {\mathcal L}} \over {\partial {\phi _{ik}}}}\,.$$
(161)

With the new variables, Eqs. (155) and (156) may be brought into the form of the Einstein-Maxwell equations:

$$- \,({G_{ik}} - {1 \over 2}{g_{ik}}G) = {T_{ik}} + {1 \over 6}({j_i}{j_k} - {1 \over 2}{g_{ik}}{j^s}{j_s})\,,$$
(162)
$${\phi _{ik}} = {1 \over 6}({{\partial {j_k}} \over {\partial {x^i}}} - {{\partial {j_i}} \over {\partial {x^k}}})\,,$$
(163)
$${{\partial {{\hat f}^{ks}}} \over {\partial {x^s}}} = {\hat j^k}\,.$$
(164)

In (162), the (Riemannian) Ricci tensor G ik and Ricci scalar G are formed from the auxiliary metric g ik ; the same holds for the tensor T ik := −(γ ik − ½ g ik grsγ rs ). Of course, in the end, g ik and all quantities formed from it will have to be expresses by the affine connection Γ.

Schrödinger’s assignment of mathematical quantities to physical observables is as follows:

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

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

  • jk corresponds to the electric 4-current density,

  • T ik corresponds to the “field-energy-tensor of the electromagnetic field”.

We note from (163) that the electric current density is the negative of the electromagnetic 4-potential. The meson field is not yet included in the theory.

Up to here, Schrödinger did not specify the Lagrangian in (153). He then assumed:

$$\overline {\mathcal L} = 2\alpha \{\sqrt {- \det ({g_{ik}} + {\phi _{ik}})} - \sqrt {- \det ({g_{ik}})} \} \,,$$
(165)

with a numerical constant α (his Eq. (4,1) on p. 51). According to Schrödinger:

\({\mathcal L}\) 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 tensor-density (\({\bf{\vec H}},\; - {\bf{\vec D}}\)) […].” ([545], p. 52.)

This refers to the paper by Born and Infeld on a non-linear electrodynamics;Footnote 83 cf. Section 5. At the end of the paper, Schrödinger speculated about taking into account a cosmological constant, and about including a meson field of spin 1 described by a symmetrical rank 2 tensor ψ ik in a more complicated LagrangianFootnote 84:

$$\overline {\mathcal L} = 2\alpha \{\sqrt {- \det ({g_{ik}} + {\phi _{ik}})} + {{\alpha \prime} \over \alpha}\sqrt {- \det ({g_{ik}} + {\phi _{ik}} + {\psi _{ik}})} - \sqrt {- \det ({g_{ik}})} \} \,.$$
(166)

As field equations, he obtained the following system:

$${G_{ik}} = {\alpha \over w}(\phi _i^{\;\,r}{\phi _{kr}} + {g_{ik}}(w - 1) - {1 \over 6}\;{j_i}{j_k})\,,$$
(167)
$${\phi _{ik}} = {1 \over 6}({{\partial {j_k}} \over {\partial {x^i}}} - {{\partial {j_i}} \over {\partial {x^k}}})\,,$$
(168)
$${1 \over {\sqrt {- g}}}{\partial \over {\partial {x^r}}}\;\left[ {{{\sqrt {- g}} \over w}({\phi ^{kr}} - {I_2}\;{\phi ^{{\ast}\;\,kr}})} \right] = {g^{kr}}{j_k}\,,\quad w = \sqrt {1 + {1 \over 2}\;{\phi ^{rs}}{\phi _{rs}} - {{({I_2})}^2}} \,.$$
(169)

Here, \({\phi ^{*\;ik}}: = {1 \over 2}{1 \over {\sqrt {- g}}}{\epsilon ^{ikrs}}{\phi _{rs}}\), \({I_2}: = {1 \over 4}{\phi ^{*\;rs}}{\phi _{rs}}\) as in [545], p. 51, Eq. (4,3). For a physical interpretation, Schrödinger re-defined all quantities g ik , A k , ϕ ik , j k by multiplying them with constants having physical dimensions. This is to be kept in mind when his papers in which applications were discussed, are compared with this basic publication.

Application: Geomagnetic field

Schrödinger quickly tried to draw empirically testable consequences from his theory. At first he neglected gravity in his UFT and obtained the equations “for not excessively strong electromagnetic fields”:

$$\overset \rightarrow H = \overset \rightarrow {{\rm{rot}}} \overset \rightarrow A \,,\quad \overset \rightarrow E = - \overset \rightarrow \nabla V - {{\partial \overset \rightarrow A} \over {\partial t}}\,,$$
(170)
$$\overset \rightarrow {{\rm{rot}}} \overset \rightarrow H - {{\partial \overset \rightarrow E} \over {\partial t}} = - {\mu ^2}\overset \rightarrow A \,,\quad {\rm{div}}\overset \rightarrow E = - {\mu ^2}V\,,$$
(171)

in which the electric current 4-density is replaced by the 4-potential; cf. (163). The equations then were applied to the permanent magnetic field of the Earth and the Sun [544]. Deviations from the dipole field as described by Maxwell’s theory are predicted by (171). Schrödinger’s careful comparison with available data did not show a contradiction between theory and observation, but remained inconclusive. This was confirmed in a paper with the Reverend J. McConnelFootnote 85 [419] in which they investigated a possible (shielding) influence of the earth’s altered magnetic field on cosmic rays (as in the aurora).

After the second world war, the later Nobel-prize winner Maynard S. Blackett (1897–1974) suggested an empirical formula relating magnetic moment M and angular momentum L of large bodies like the Earth, the Sun, and the stars:

$$M = \beta {{\sqrt G} \over {2c}}L\,,$$
(172)

with G Newton’s gravitational constant, c the velocity of light in vacuum, and β a numerical constant near \({1 \over 4}\) [28]. The charge of the bodies was unimportant; the hypothetical effect seemed to depend only on their rotation. Blackett’s idea raised some interest among experimental physicists and workers in UFT eager to get a testable result. One of them, the Portuguese theoretical physicist Antonio Gião, derived a formula generalizing (172) from his own unified field theory [225, 226]:

$$M = 2{\xi \over \chi}{{{m_0}} \over {2e}}GL\,,$$
(173)

where e, m0 are charge and rest mass of the electron, ξ a constant, and χ the “average curvature of space-time”.

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.

Theoreticians outside the circle of those working on unified field theory were not so much attracted by Blackett’s idea. One of them was Pauli who, in a letter to P. Jordan of July 13, 1948, wrote:

“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.”Footnote 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).

In 1949, the Royal Astronomical Society of England held a “Geophysical Discussion” on “Rotation and Terrestrial Magnetism”[519]. Here, Blackett tried to avoid Pauli’s criticism by retaining his formula in differential form:

$$d\underset - H \sim {G^{{1 \over 2}}}dm\;[\underset - \omega \times \underset - R ] \times {{\underset - r} \over {{r^3}}}\,.$$
(174)

For a translation \(\underset - \omega = 0\), and no effect obtains. T. Gold questioned Blackett’s formula as being dependent on the inertial system and asked for a radial dependence of the angular velocity ω. A. Papapetrou claimed that Blackett’s postulate “could be reconciled with the relativistic invariance requirements of Maxwell’s equations” and showed this in a publication containing Eq. (174), if only forcedly so: he needed a bi-metric gravitational theory to prove it [476]. In the end, the empirical data taken from the earth did not support Blackett’s hypothesis and thus also were not backing UFT in its various forms; cf. ([19], p. 295).

Application: Point charge

A second application pertains to the field of an electrical point charge at rest [548]. Schrödinger introduced two “universal constants” which both appear in the equations for the electric field. The first is his “natural unit” of the electromagnetic field strength \(b: = {e \over {r_0^2}}\) called Born’s constant by him, where e is the elementary charge and \({r_0}: = {{{e^2}} \over {{m_0}{c^2}}}{2 \over 3}\int\nolimits_0^\infty {{{dx} \over {\sqrt {1 + {x^4}}}}}\) the electron radius (mass m0 of the electron). The second is the reciprocal length introduced in a previous publication \(f: = {{b\;\sqrt {2G}} \over {{c^2}}}\) with Newton’s gravitational constant G and the velocity of light c. Interestingly, the affine connection has been removed from the field equations; they are written as generalized Einstein-Maxwell equations as in Born-Infeld theoryFootnote 87 (cf. Section 5):

$${G_{ik}} = {f^2}\left({{{{c^2}} \over {{b^2}w}}\phi _i^{\;r}{\phi _{kr}} - {g_{ik}}{{w - 1} \over w} - {{{\mu ^2}{c^2}} \over {{b^2}}}{A_i}{A_k}} \right)\,\,,$$
(175)
$${\phi _{ik}} = {{\partial {A_k}} \over {\partial {x^i}}} - {{\partial {A_i}} \over {\partial {x^k}}}\,,$$
(176)
$${1 \over {\sqrt {- g}}}{\partial \over {\partial {x^r}}}\;\left[ {{{\sqrt {- g}} \over w}\;\left({{\phi ^{kr}} - {{{c^2}} \over {{b^2}}}{I_2}\;{\phi ^{{\ast}\;kr}}} \right)} \right] = {\mu ^2}{A^k}\,,w = \sqrt {1 + {1 \over 2}{{{c^2}} \over {{b^2}}}\;{\phi ^{rs}}{\phi _{rs}} - {{{c^4}} \over {{b^4}}}{{({I_2})}^2}} \,.$$
(177)

If only a static electric field is present, ϕ*ik = 0 and thus I 2 = ϕ*ikϕ ik = 0.

An ansatz for an uncharged static, spherically symmetric line element is made like the one for Schwarzschild’s solution in general relativity, i.e.,

$$d{s^2} = \exp (\nu)\,d{t^2} - \exp (\lambda)\,d{r^2} - {r^2}{(d\theta)^2} - {r^2}{\sin ^2}\theta {(d\phi)^2}\,.$$
(178)

The solution obtained was:

$$\exp (\nu) = \exp (- \lambda) = 1 - {f^2}{({r_0})^2}{1 \over x}\int\nolimits_0^x \, dx\left({\sqrt {1 + {x^4}} - {x^2}} \right)\,,$$
(179)

with \(x: = {r \over {{r_0}}}\). The integral is steadily decreasing from x = 0 to x → ∞, where it tends to zero. Like the Schwarzschild solution, (178) with (179) has a singularity of the Ricci scalar at x = 0 = r. For r → ∞ the Schwarzschild (external) solution is reached. According to Schrödinger, “[…] we have here, for the first time, the model of a point source whose gravitational field is accounted for by its electric field energy. The singularity itself contributes nothing” ([548], p. 232).

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 gauge-invariant.Footnote 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.

Semi-symmetric connection

Schrödinger’s next paper on UFT continued this line of thought: in order to be able to include the mesonic field he dropped the symmetry-condition on the affine connection ([549], p. 275). This brings homothetic \({\underset - V _{ik}}\) curvature into the game (cf. Section 2.3.1, Eq. (65)). Although covariant differentiation was introduced through \({\bar \nabla _k}{X^i}\) and \({\bar \nabla _k}{\omega _i}\), in the sequel Schrödinger split the connection according to \({L_{ij}}^k = {\Gamma _{ij}}^k + {S_{ij}}^k\) and used the covariant derivative \(\overset 0 \nabla\) (cf. Section 2.1.2) with regard to the symmetric part Γ of the connection.Footnote 89 In his first attempt, Schrödinger restricted torsion to non-vanishing vector torsion by assuming:

$$S_{ij}^{\;\;\,k} = \delta _i^k{Y_j} - \delta _j^k{Y_i}$$
(180)

with arbitrary 1-form Y = Y i dxi. Perhaps this is the reason why he speaks of “weakly asymmetric affinity” without giving a precise definition. Schouten called this type of connection “semi-symmetric” (cf. Section 5 of Part I, Eq. (132)). Obviously, vector torsion S i = −3Y i . The two contractions of the curvature tensor \({\underset - K ^i}_{jkl}\), i.e., \({\underset - K _{ik}}\) and \({\underset - V _{ik}}\). are brought into the form [cf. (63)]:

$${\underset - K _{ik(L)}} = {\underset - K _{ik(\Gamma)}} - {Y_{\underset 0 k \parallel i}} - 2{Y_{\underset 0 i \parallel k}} - 3{Y_i}{Y_k}\,,$$
(181)
$${\underset - V _{ik(L)}} = {\underset - V _{ik(\Gamma)}} - 3{Y_{\underset 0 i \parallel k}} + 3{Y_{\underset 0 k \parallel i}}\,.$$
(182)

Here, \(_0^i\Vert k\) denotes covariant differentiation with respect to the symmetric connection Γ. Two linear combinations of these tensors are introduced:

$${M_{ik}}: = {1 \over 4}({\underset - V _{ik}} + {\underset - K _{[ik]}})\,,\qquad {P_{ik}}: = {3 \over 2}{\underset - K _{ik}} - {1 \over 2}{\underset - K _{ki}} + {1 \over 2}{\underset - V _{ik}}\,.$$
(183)

A calculation shows that:

$${P_{ik}} = {({K_{ik}})_\Gamma} + {Y_{k\parallel i}} - 4{Y_{i\parallel k}} - 3{Y_i}{Y_k}$$
(184)

with (K ik )Γ being the Ricci-tensor formed with the symmetrized connection, and

$${M_{ik}} = {Y_{k\parallel i}} - {Y_{i\parallel k}} = {Y_{k,i}} - {Y_{i,k}}\,.$$
(185)

In the same way as in (154) two tensor densities are introduced:

$${\hat g^{ik}}: = {1 \over 2}({{\partial {\mathcal L}} \over {\partial {P_{ik}}}} + {{\partial {\mathcal L}} \over {\partial {P_{ki}}}})\,,\qquad {\hat f^{ik}}: = {1 \over 2}({{\partial {\mathcal L}} \over {\partial {P_{ik}}}} - {{\partial {\mathcal L}} \over {\partial {P_{ki}}}})\,,$$
(186)

and a third one according to

$${\hat m^{ik}} = {{\partial {\mathcal L}} \over {\partial {M_{ik}}}}\,.$$
(187)

Conjugated variables to \({\hat g^{ik}},{\hat f^{ik}}\) are [cf. (159]:

$${\gamma _{ik}}: = {P_{(ik)}}\,,\quad {\phi _{ik}}: = {P_{[ik]}}\,.$$
(188)

The Lagrangian is to depend only on P ik and Mik. A (symmetric) metric tensor g ik . is introduced as in (157). Variation of the Lagrangian with respect to P ik and Mik leads to field equations now containing terms from homothetic curvature:

$$\hat g_{\;\;\;\parallel l}^{ik} + \delta _l^k({1 \over 3}\hat f_{\;\;\;\,\;\parallel s}^{is} - {\hat g^{is}}{Y_s}) + \delta _l^i({1 \over 3}\hat f_{\;\;\;\;\;\;\parallel s}^{ks} - {\hat g^{ks}}{Y_s}) + 3{\hat g^{ik}}{Y_l}$$
(189)

and to the same equation (156) as before, as well as to an additional equation:

$$\hat m_{\;\;\;\;\;\parallel s}^{is} = {{\partial {{\hat m}^{is}}} \over {\partial {x^s}}} = 0\,.$$
(190)

According to Schrödinger, (185) and (190) “form a self-contained Maxwellian set”. The formal solution for the symmetric part of the connection replacing (158) now becomes:

$$\Gamma _{ij}^{\;\;\;k} = \{_{ij}^k\} - {1 \over 2}{g_{ij}}\;{j^k} + \delta _i^{\;k}\;\left({{1 \over 6}\;{j_j} + {Y_j}} \right) + \delta _j^k\;\left({{1 \over 6}\;{j_i} + {Y_i}} \right)\;\,.$$
(191)

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.

His main argument was:

“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 non-symmetrical case gives me the impression that the exchange of roles will very probably be imperative, […]” ([549], p. 282).

As to the field equations, they still were considered as preliminary because: “the investigation of the fully non-symmetric case is imperative and may have surprises in store.” ([549], p. 282.) The application of Weyl’s gauge transformations g ik λg ik in combination with

$${Y_k} \rightarrow {Y_k} - {1 \over {3\lambda}}{{\partial \lambda} \over {\partial {x^k}}}\,,\qquad {j_k} \rightarrow {j_k} - {1 \over \lambda}{{\partial \lambda} \over {\partial {x^k}}}$$
(192)

leaves invariant \({\Gamma _{ij}}^k,{\phi _{ik}}\), and M ik , but changes γ ik and destroys the vanishing of the divergence \({\hat j^s}_{,s}\). Schrödinger thought it to be “imperative to distinguish between the potential j k . and the current-and-charge j k the two coinciding only in the original gauge.” ([549], p. 284.) He also claimed that only the gravitational and mesonic fields had an influence on the auto-parallels (cf. Section 2.1.1, Eq. (22)).

These first two papers of Schrödinger were published in the Proceedings of the Royal Irish Academy, a journal only very few people would have had a chance to read, particularly during World War II, although Ireland had stayed neutral. Schrödinger apparently believed that, by then, he had made enough progress in comparison with Eddington’s and Einstein’s publications.Footnote 90 Hence, he wrote a summary in Nature for the wider physics community [547]. At the start, he very nicely laid out the conceptual and mathematical foundations of affine geometry and gave a brief historical account of its use within unified field theory. After supporting “the superiority of the affine point of view” he discussed the ambiguities in the relation between mathematical objects and physical observables. An argument most important to him came from the existence of

“a third field […], of equally fundamental standing with gravitation and electromagnetism: the mesonic field responsible for nuclear binding. Today no field-theory which does not embrace at least this triad can be deemed satisfactory at all.” ([549], p. 574.)Footnote 91

He believed to have reached “a fully satisfactory unified description of gravitation, electromagnetism and a 6-vectorial 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.)

In a letter to Einstein of 10 October 1944, in a remark about an essay of his about Eddington and Milne, M. Born made a bow to Einstein:

“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)Footnote 92

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 5-dimensional 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 bi-vector (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 tele-parallel, 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 skew-symmetric 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.Footnote 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.

As we shall note in Section 10.5.1, in 1942–1944 Einstein’s interest was also directed to the question to what extent singularities occur in the solutions of the field equations of general relativity and of Kaluza-Klein. It is during this time that he wrote to Hans Mühsam in spring of 1942 (as quoted by Seelig [570], p. 412):

“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.” Footnote 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 4-current, 4 components of the magnetic 4-current, 5 components for an ideal fluid, more for the unspecified energy-momentum-tensor 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 metric-affine geometry a two-parameter 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.Footnote 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 skew-symmetric, 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 complex-valued fields on real space-time 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.

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 space-time, 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″.

In his paper of 1945 Einstein gave two formal criteria as to when a theory could be called a “unified” field theory. The first was that “the field appear as a unified entity”, i.e., that it must not be decomposable into irreducible parts. The second was that “neither the field equations nor the Hamiltonian function can be expressed as the sum of several invariant parts, but are formally united entities”. He readily admitted that for the theory presented in the paper, the first criterion was not fulfilled ([147], p. 578). As remarked above, the equations of general relativity as well as Maxwell’s equations ought to be contained within the field equations of UFT; by some sort of limiting process it should be possible to regain them. A further requirement was the inclusion of matter into geometry: some of the mathematical objects ought to be identified with physical quantities describing the material sources for the fields. With UFT being a field theory, the concept of “particle” had no place in it. Already in 1929, in his correspondence with Elie Cartan, Einstein had been firm on this point:

“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.)Footnote 96

Thus, the discussion of “equations of motion” would have to use the concept of thin timelike tubes, and integrals over their surface, or some other technique. Einstein kept adamantly to this negatory position concerning point particles as shown by the following quote from his paper with N. Rosen of 1935 [178]:

“[…] 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, λ-transformations; 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].

Einstein 1945

As early as in 1942, in his attempts at unifying the gravitational and electromagnetic fields, Einstein had considered using both a complex valued tensor field and a 4-dimensional complex space as a new framework. About this, he reported to his friend M. Besso in August 1942 ([163], p. 367):

“What I now do will seem a bit crazy to you, and perhaps it is crazy. […] I consider a space the 4 coordinates x1, … x4 which are complex such that in fact it is an 8-dimensional space. To each coordinate xi belongs its complex conjugate x. […] In place of the Riemannian metric another one of the form g ik obtains. We ask it to be real, i.e., g ik = ki must hold (Hermitian metric). The g ik are analytical functions of the xi, and x. […]”Footnote 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 space-time 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. SohFootnote 98, during his stay at the Massachusetts Institute of Technology, had published a paper on a theory of gravitation and electromagnetism within complex four-dimensional 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).

Einstein derived the field equations from the LagrangianFootnote 99

$${\mathcal H} = K_{ik}^{\;\;\; {\ast}}\;{\hat g^{ik}}$$
(193)

withFootnote 100

$$K_{ik}^{\;\;\,{\ast}} = :{\overset {{\rm{Her}}} K _{ik}} - {X_{\underset - i \Vert k}}$$
(194)

and

$${X_i} = {{\partial (\log \sqrt {- g})} \over {\partial {x_i}}} - {1 \over 2}(L_{im}^{\;\;\;m} + L_{mi}^{\;\;\;m})\,.$$
(195)

Here, \({\overset {{\rm{Her}}} K_{ik}}\) is the Hermitian-symmetrized Ricci-tensor obtained from the curvature tensor (54) as: \(\overset {\rm{Her}} {{{\underset - K}_{ik}}} = {1 \over 2}({\underset - K ^m}_{imk} + \overset - {\underset - K} {\, ^m}_{kmi})\) (cf. (73) of Section 2.3.2):

$${\underset - {\overset {{\rm{Her}}} K} _{ik}} = L_{ik\;\,,l}^{\;\;\;\;l} - L_{im}^{\;\;\;\;l}L_{lk}^{\;\;\;\;\;m} - {1 \over 2}(L_{il\;\,,k}^{\;\;\;\;\,l} + L_{lk\;\,,i}^{\;\;\;\;l}) + {1 \over 2}L_{ik}^{\;\;\;m}(L_{ml}^{\;\;\;\,l} + L_{lm}^{\;\;\;\,l})\,,$$
(196)

with L being a connection with Hermitian symmetry. Note that K ik * is not Hermitian thus implying a non-Hermitian Lagrangian \({\mathcal H}\).

The quantities to be varied are ĝik, \({L_{ik}}^l\). From \(\delta {\mathcal H} = - \hat U_m^{ik}\delta L_{ik}^m + {G_{ik}}\delta {\hat g^{ik}}\), Einstein then showed that the first equation of the field equations:

$$\hat U_m^{\;\,ik} = 0,\;\;{G_{ik}} = 0$$
(197)

is equivalent to the compatibility equation (30) while the second may be rewritten as

$${G_{ik}} = K_{ik}^{\;\;\,{\ast}} - {1 \over {2\sqrt {- g}}}{({\hat g^{lm}}{L_l})_{,m}}\;{g_{ik}}\,.$$
(198)

The proofs are somewhat circular, however, because he assumed (30) beforehand. He also claimed that a proof could be given that the equation

$${L_i} = 0 = {1 \over 2}(L_{im}^{\;\;\,\,\,m} - L_{mi}^{\;\;\,\,m})\,,$$
(199)

expressing the vanishing of vector torsion, could be added to the field equations (197). Its second Eq. (198) would then bear a striking formal resemblance to the field equations of general relativity. The set

$$0 = {g_{\underset + i \underset - k \parallel l}}: = {g_{ik,l}} - {g_{rk}}L_{il}^{\;\;\;r} - {g_{ir}}L_{lk}^{\;\;\;r}\,,$$
(200)
$${S_j}(L): = {L_{[im]}}^m = 0\,,$$
(201)
$${K_{jk}}(L) = 0$$
(202)

with a more general connection L later was named the strong field equations.

Equation (200) replaced the covariant constancy of the metric in general relativity, although, in general, it does not preserve inner products of vectors propagated parallelly with the same connection. The problem was already touched in Section 2.1.2: how should we define \({({g_{ik}}{A^i}{B^k})_{\underset \pm {\Vert} l?}}\)? M. Pastori has shown that ([485], p. 112):

$${({g_{ik}}{A^i}{B^k})_{,s}}\,d{x^s} = \left({{g_{\underset + i \underset - k \parallel l}}{A^i}{B^k} + {g_{ik}}A_{\parallel l}^{\underset + i}{B^k} + {g_{ik}}{A^i}B_{\parallel l}^{\underset - k}} \right)\,.$$
(203)

With the help of (20) and (31), we can re-write (203) in the form:

$${({g_{ik}}{A^i}{B^k})_{,s}}\,d{x^s} = \left({{g_{\underset + i \underset - k \parallel l}}{A^i}{B^k} + {g_{ik}}A_{\parallel l}^{\underset 0 i}{B^k} + {g_{ik}}{A^i}B_{\parallel l}^{\underset 0 k}} \right)\,.$$
(204)

Hence, besides the connection \({L_{ij}}^k\) a second one \({L_{(ij)}}^k\) enters.

Einstein-Straus 1946 and the weak field equations

It turned out that the result announced, i.e., L i = 0, could not be derived within the formalism given in the previous paper ([147]). Together with his assistant Ernst StrausFootnote 101, Einstein wrote a follow-up in which the metric field did not need to be complex [179]. Equation (199) now is introduced by a Lagrangian multiplier Âi. The new Lagrangian is given by:

$$H = {P_{ik}}{\hat g^{ik}} + {\hat A^i}{L_i} + {b_k}\hat g_{\,,m}^{[km]}\,,$$
(205)

where P ik is the same quantity as \({\overset {\rm{Her}} K _{ik}}\) of [147]; cf. also Eq. (74) of Section 2.3.2.Footnote 102 The variables to be varied independently are ĝik, \({L_{ik}}^l\), and the multipliers Âi, b k . After some calculation, the following field equations arose:

$${L_i} = 0\,,$$
$$\hat g^{\underset + i \underset - k}\,_{\Vert l} = 0\,,$$
(206)
$${P_{(ik)}} = 0,$$
(207)
$${P_{[ik],l}} + {P_{[kl],i}} + {P_{[li],k}} = 0.$$
(208)

The last Eq. (208) is weaker than P[ik ]= 0; therefore this system of equations is named the weak field equations of UFT. However, cf. Section 9.2.2 for a change in Einstein’s wording. From the calculations involved, it can be seen that (206) is equivalent to (30), and that Âi vanishes. The second multiplier satisfies:

$${b_{i,k}} - {b_{k,i}} = 2{P_{[ik]}}\,.$$
(209)

On first sight, according to Eqs. (206) and (208), either the skew-symmetric part of \({\hat g^{\underset +i \underset - k}}\), or the skew-symmetric part of P ik might be related to the electromagnetic field tensor. In the paper, homothetic curvature V ik is introduced but not included in the Lagrangian. It will vanish as a consequence of the field equations given

In linear approximation, the ansatz g ij = η ij + γ ij is used with γ(ij) related to the gravitational and γ[ij ] to the electromagnetic field. For the skew-symmetric part γ[ij ] of the metric, the field equations reduce to

$${\eta ^{jk}}{\gamma _{[ij],k}} = 0\,,$$
(210)
$${\eta ^{mn}}{\partial _m}{\partial _n}({\gamma _{[ik],l}} + {\gamma _{[kl],i}} + {\gamma _{[li],k}}) = 0\,.$$
(211)

The system (210), (211) is weaker than the corresponding Maxwell’s equations in vacuum. According to the authors, this is no valid objection to the theory “since we do not know to which solutions of the linearized equations there correspond rigorous solutions which are regular in the entire space.” Only such solutions are acceptable but: “Whether such (non-trivial) solutions exist is as yet unknown” ([179], p. 737).

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).

In the long last paragraph of the paper, the authors derive necessary and sufficient (algebraic) conditions for g ik in order that the Eqs. (30) or (206) determine the connection (in terms of the metric) “uniquely and without singularities”. If we set \({I_1}: = {\rm{det(}}{h_{ik}}{\rm{);}}\,{I_2}: = {1 \over 4}{\epsilon ^{ijkl}}{\epsilon ^{abcd}}{h_{ia}}{h_{jb}}{k_{kc}}{k_{kd}}\), and I3:= det(k ik ), then they are given byFootnote 103:

$$\begin{array}{*{20}c} {\quad \quad \quad \quad \quad {I_1} \neq 0\,,} \\ {g = {I_1} + {I_2} + {I_3} \neq 0\,,} \\ {\,\,\,{{({I_1} - {I_2})}^2} + {I_3} \neq 0\,.} \\ \end{array}$$
(212)

The second equation in (212) is equivalent to (8) in Section 2.1. We will have to compare this result with those to be given in Section 10, and in Section 12.2.

Einstein wrote on 6 March 1947 to Schrödinger that he:

“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.”Footnote 104

And four months later (16 July 1946), Einstein confessed to Schrödinger:

“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.”Footnote 105

Both quotes are taken from the annotations of K. von Meyenn in ([489], p. 383). In fall, Pauli who had returned to Zurich wrote to Einstein:

“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)

Unimpressed, Einstein went on squeezing the lemon for the next nine years until his death. On 9 April 1947, he wrote to his friend from student days, Maurice Solovine (1875–1958):

“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)Footnote 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, well-known 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 non-linear equations” (letter to H. Zangger of 28 July 1947 in [560], p. 579).

How appropriate Pauli’s remark was, is made clear by a contemporary paper on “non-symmetric gravity theories”. Damour, Deser & McCarthy show that the theories of Einstein and Einstein & Straus (together with further geometrical theories) “violate standard physical requirements” such as to be free of ghostsFootnote 107 and with absence of algebraic inconsistencies [101, 102]. On the other hand, the authors showed that the following Lagrangian, closely related to an expansion in powers of k ij = g[ij ] of the Einstein-Straus Lagrangian, would be acceptable:

$${{\mathcal L}^I} = \sqrt h \left[ {R(h) - {1 \over {12}}{H_{rst}}{H^{rst}} - {2 \over 3}{k^{lm}}({\partial _l}{S_m} - {\partial _m}{S_l})} \right]\,,$$
(213)

with H rst := r k st + d t k rs + s k tr . Indices are moved with the symmetric part of the asymmetric metric [100].

Another link from Einstein’s Hermitian theory to modern research leads to “massive gravity” theories, i.e., speculative theories describing an empirically unknown spin-2 particle (graviton) with mass [76, 255]. However, it is not clear whether these theories are free of ghosts.

Schrödinger II: Arbitrary Affine Connection

After an interruption of more than two years, in January 1946, the correspondence between Schrödinger and Einstein resumed; he sent Schrödinger two unpublished papers, among them his paper with E. Straus [179].

“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 non-symmetric 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 non-symmetric 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 second-order equations.” ([446], p. 424.)Footnote 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.

In April, Schrödinger had progressed with his research such that he could present its essence before the Irish Academy. In hindsight, he confessed to Einstein in a letter of 1 May 1946:

“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.)

M.-A. Tonnelat from Paris visited Schrödinger in Dublin during 1946 and, perhaps, discussions with her had influenced him. Unlike what he had stated in his earlier paper when using a semi-metric connection two years before [549], Schrödinger now thought that: “[…] only the general case, dealt with here, is completely satisfactory and gives new information” ([551], 41). The “general case” meant: a general affine connection, named \({\Delta _{ik}}^l\) by him. After splitting it up into a symmetric part, the trace-free part \({W_{ik}}^l\) of the torsion tensor \({S_{ik}}^l\) and vector torsion:

$$\Delta _{ik}^l = \Gamma _{ik}^l + W_{ik}^l + 2\delta _{\left[ i \right.}^lY{}_{\left. k \right]}\,,$$
(214)

with \({\Gamma _{ik}}^l: = {\Delta _{(ik)}}^l,\,{Y_i}: = {1 \over 3}{S_{il}}^l\) and \({W_{il}}^l = 0\). Instead of the contractions of the curvature tensor (181), (182) now

$$\underset - K {}_{ik}\,(L) = \underset - K {}_{ik}(\Gamma) - {Y_{k\underset 0 \parallel i}} - 2{Y_{i\underset 0 \parallel k}} - 3{Y_i}{Y_k} - W_{ik\underset 0 \parallel r}^r + W_{ir}^sW_{sk}^r + 3{Y_s}W_{ik}^s,$$
(215)
$${\underset - V _{ik}}(L) = {\underset - V_{ik}}(\Gamma) - 3{Y_{i,k}} + 3{Y_{k,i}}$$
(216)

obtain. Again, \(\underset 0 {\Vert} \,k\) denotes covariant differentiation with respect to the symmetric connection Γ. The quantities Γ, W, and Y are varied independently. That \({W_{ik}}^l\) is tracefree is taken into account by help of a multiplier term in the variational principle:

$$\delta \int d \tau [\hat {\mathcal L}({\underset - K _{ik}}(L),{\underset - V _{lm}}(L)) + 2{\hat p^r}W_{rs}^{s}] = 0\,.$$
(217)

The Lagrangian density \(\hat {\mathcal L}\) is demanded to be a functional of the two contractions (215), (216) of the affine curvature tensor. The field variables ĝik, \({\hat f^{ik}}\), γ ik and ϕ ik are introduced as they were in Section 6.1.1 by (154) and (159). A decomposition of (215) then leads to:

$${\gamma _{ik}}: = {\underset - K _{(ik)}}(L) = {\underset - K _{(ik)}}(\Gamma) - 3{Y_{\underset 0 {(i} \parallel k)}} - 3{Y_i}{Y_k} + W_{ir}^{s}W_{sk}^{r}\,,$$
(218)
$${\phi _{ik}}: = {\underset - K _{[ik]}}(L) = {1 \over 2}[{{\partial \Gamma _{ri}^{r}} \over {\partial {x^k}}} - {{\partial \Gamma _{rk}^{r}} \over {\partial {x^i}}}] - {Y_{[i,k]}} - W_{ik\underset 0 \parallel r}^{r} + 3{Y_s}W_{ik}^{s}\,.$$
(219)

The field equations following from the variational principle (217) with respect to variation of Γ and W are similar to (189):

$${\hat g^{ik}}_{\,\,\,\,\parallel l} + \delta _l^k[{1 \over 3}\hat f_{\quad \parallel s}^{is} - {\hat g^{is}}{Y_s} - {2 \over 3}{\hat r^i}] + \delta _l^i[{1 \over 3}\hat f_{\quad \,\parallel s}^{ks} - {\hat g^{ks}}{Y_s} - {2 \over 3}{\hat r^k}] + 3{\hat g^{ik}}{Y_l} = 0\,,$$
(220)
$$\hat f_{\quad \parallel l}^{ik} - \delta _l^k({1 \over 3}\hat f_{\quad \parallel s}^{is} + {\hat f^{is}}{Y_s}) + \delta _l^i({1 \over 3}\hat f_{\quad \,\parallel s}^{ks} + {\hat f^{ks}}{Y_s}) + 3{\hat f^{ik}}{Y_l} + {\hat g^{is}}W_{sl}^{\;\;\;k} - {\hat g^{ks}}W_{sl}^{\;\;\;i} = 0\,.$$
(221)

The abbreviation, or rather definition of the vector density \({\hat j^k}\) is as before \((= {{\partial {{\hat f}^{ks}}} \over {\partial {x^s}}})\) while

$${\hat r^k}: = \hat s_{\,\,\,\,\, \parallel r}^{kr} = \hat s_{\,\,\,\,\,,r}^{kr}$$
(222)

withFootnote 109 \({\hat s^{ik}}: = {{\partial {\mathcal L}} \over {\partial {V_{ik}}}}\). The variation with respect to Yk, after some calculation, led to the simple relation between the current densities:

$${\hat r^k} = {1 \over 4}{\hat j^k}\,.$$
(223)

By the field equations, the dynamics of three fields were to be determined, gravitational, electromagnetic, and mesonic field:

“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 order to arrive at equations better separated in the new fields, Schrödinger redefined the field variables by forming the linear combinations:

$$\prime{\phi _{ik}}: = {\phi _{ik}} + {1 \over 4}{V_{ik}}\,,\qquad \prime{\hat f^{ik}} = {\hat f^{ik}}\,,$$
(224)
$$\prime {V_{ik}} = {V_{ik}}\,,{\qquad \prime}{\hat s^{ik}} = {\hat s^{ik}} - {1 \over 4}{\hat f^{ik}}\,.$$
(225)

The equations for the Maxwellian field ′ŝ, V are claimed to then be “kept entirely aloof from the rest by the remarkable fact that the Y-vector drops out rigorously from all the other equations except the last eqn. (216)”. The fields \(\hat f\), ′ϕ with the current density \(\hat j\) are related to the meson field.

As in his earlier papers, a metric was introduced by (157), i.e., via ĝik. Schrödinger then calculated the expression for the symmetric part of the connection as in (191) but now only in first approximation in \(\hat f\), W:

$$\Gamma _{ij}^{\,\,\,\,k} = \{_{ij}^k\} - {1 \over 4}{g_{ij}}{j^k} + \delta _i^{k}({1 \over {12}}{j_j} + {Y_j}) + \delta _j^k({1 \over {12}}{j_i} + {Y_i}) + \ldots \,.$$
(226)

The trace-free part of torsion was given by:

$$W_{ik}^{l} = - {1 \over 2}{g^{ls}}({f_{ks;i}} + {f_{si;k}} - {f_{ik;s}}) + {2 \over 3}{\delta ^l}_{\left[ i \right.}j{}_{\left. k \right]} + \ldots \,,$$
(227)

and the field equations for the meson field, again in linear approximation, were:

$${\phi _{ik}} = - {3 \over 8}({{\partial {j_k}} \over {\partial {x^i}}} - {{\partial {j_i}} \over {\partial {x^k}}}) + {1 \over 2}(f_{i;k;s}^s - f_{k;i;s}^s - {g^{rs}}{f_{ik;r;s}} + \ldots)\,,{\hat j^k} = {{\partial {{\hat f}^{ks}}} \over {\partial {x^s}}}\,.$$
(228)

The covariant derivative is the one formed with the Christoffel-symbol (Levi-Civita connection) built from (the symmetric) g ik . Schrödinger did interpret (228) as Proca equationFootnote 110 for the meson, “except for the term which contains explicitly second derivatives”. According to him, the additional term was taking into account a slight direct influence of gravitation on the meson field ([551], 47).

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.

Schrödinger’s conclusion was cautious:

“This encourages one to regard an affine connection of space-time 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 non-linear.

In his correspondence with Schrödinger, Einstein doubted that a theory using only the connection (i.e., without additional metric) be feasible. He reported about difficulties in his theory to solve for the connection as a function of the metric and its first derivative:

“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.)

In another letter of 16 July 1946 to Schrödinger, Einstein did explain the progress achieved “thanks to the truly great skill and persistence of my assistant Straus”, then published in [179]. He also commented on conceptual differences. Schrödinger used the wave model for the transport of electromagnetic energy while Einstein thought this to be “really false on account of quantum actualities” ([446], p. 427). Another of his correspondents, W. Pauli, also did not believe in purely affine theory. He wrote to him on the same day at which Schrödinger’s paper finally had been issued, i.e., on 21 November 1946:

“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)”Footnote 111

Schrödinger’s debacle

Schrödinger kept Einstein informed about his continuing work on UFT within affine geometry “[…] by reports at about fortnightly intervals” ([446], p. 429). He had read the papers of Einstein, and Einstein & Straus [179, 147] from the previous two years. (Cf. Sections 7.2 and 7.3 above.) Now he again presented his newest development of the theory to the Royal Irish Academy on 27 January 1947. Believing that he had made a break-through, he had written to Einstein a day earlier:

“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.)

Well, this might have tasted a bit stale to Einstein because he had used this same Lagrangian about twenty five years ago [141] and abandoned the theory, nonetheless! (cf. Section 4.3.2 of Part I.) To the exuberant Schrödinger a modest statement to the Academy would not do: the press was also invited. So, Schrödinger began:

“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)
$${R_{ik}} \equiv - \underset - K {\,_{ik}} = {{\partial \Gamma _{il}^{l}} \over {\partial {x_k}}} - {{\partial \Gamma _{ik}^{l}} \over {\partial {x_l}}} + \Gamma _{mk}^{l}\Gamma _{il}^{m} - \Gamma _{ml}^{l}\Gamma _{ik}^{m}\,.$$
(230)

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.)

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.)

The presentation to the Academy and the press did not contain the finer details. For this kind of public, he wrote down the field equations in the reduced form:Footnote 112

$${{\partial {R_{ik}}} \over {\partial {x^l}}} - {R_{sk}}^{\ast}{\Gamma _{il}}^s - {R_{is}}^{\ast}{\Gamma _{il}}^s = 0\,,$$
(231)

with his “star”-connection (cf. (27))

$${}^{\ast}{\Gamma _{ik}}^l: = {\Gamma _{ik}}^l + {1 \over 3}\delta _i^{l}({\Gamma _{ks}}^s - {\Gamma _{sk}}^s)\,,$$
(232)

while the complete equations in his paper are:

$${{\partial [R_{ik}^{\ast} + {{\mathcal F}_{ik}}]} \over {\partial {x^l}}} - [R_{sk}^{\ast} + {{\mathcal F}_{sk}}]{}^{\ast}{\Gamma _{il}}^s - [R_{is}^{\ast} + {{\mathcal F}_{is}}]{}^{\ast}{\Gamma _{lk}}^s = 0\,,$$
(233)

where

$${{\mathcal F}_{ik}}: = {2 \over 3}\left({{{\partial {\Gamma _k}} \over {\partial {x^i}}} - {{\partial {\Gamma _i}} \over {\partial {x^k}}}} \right) \equiv - {4 \over 3}{S_{[i,k]}}\,,$$
(234)

if vector torsion S i is used. Equation (233) expresses nothing but the vanishing of the ±-derivative of \(R_{ik}^* + {{\mathcal F}_{ik}} \equiv - \underset - K\ _{ik}^* + {{\mathcal F}_{ik}} \cdot \,R_{ik}^*(\underset - K\ _{ik}^*)\) is formed with the “star”-connection. The (asymmetric) metric again was defined as a variational derivative with respect to R ik . By some manipulation of the formalism, Schrödinger was able to show that (233) is equivalent to the slightly generalized weak field equations of Einstein & Straus (206)(208) (cf. below, Section 7.3):

$${\hat g^{[is]}}_{\quad \,\,,s} = 0\,,$$
(235)
$$R_{(ik)}^{\ast} - \lambda {g_{ik}} = 0\,,$$
(236)
$${(R_{[ik]}^{\ast} - \lambda {g_{[ik]}})_{,l}} + {(R_{[kl]}^{\ast} - \lambda {g_{[kl]}})_{,i}} + {(R_{[li]}^{\ast} - \lambda {g_{[li]}})_{,k}} = 0\,.$$
(237)

Schrödinger was well aware of this:

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 λ-terms.Footnote 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.)

Equations (235) to (237) were also called the “para-form” of his field equations [526]. They may be seen as 18 equations for the 16 field variables g ik . According to Schrödinger: “The surplus of 2 equations is vindicated by two trivial identities, one between the first members of (237), and one between those of (236).” (236) is not a definition, such as it was used by Eddington (cf. Section 4.3.1 of Part I) but derived from the Lagrangian (229), the definition in (157) and:

$${\hat g^{ik}}: = {{\partial {\mathcal L}} \over {\partial {R_{ik}}}}\,.$$
(238)

Schrödinger also confessed that “it may turn out that I have overrated the practical advantage of (233) over (235)(237).”

The Irish Press caught the bait: “Twenty persons heard and saw history being made in the world of physics yesterday as they sat in the lecture hall of the Royal Irish Academy, Dublin, and heard Dr. Erwin Schrödinger. […] It was later told me that the theory should express everything in field physics.” ([446], p. 432.) The news spread quickly; the science editor of The New York Times sent Schrödinger’s statement and a copy of his paper to Einstein, and asked him for a comment. In the text supplied by Einstein which became also widely distributed, he said:

“Schrödinger’s latest effort […] can be judged only on the basis of mathematical-formal 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.)

Schrödinger must have had second thoughts about his going public; he tried to justify himself vis-à-vis Einstein — although he had not yet seen Einstein’s rebuff. In his letter, he admitted to have indulged “in a little hot air […] I blew myself up quite a bit. […] This thing is being done for the purpose of obtaining cheap and fraudulent publicity for a discredited administration.” His excuse was that he had tried by this “commotion” to increase his salary and to bring the authorities to reach a decision whether his wife, as a widow, could get a pension or not ([446], p. 433). Einstein replied coolly and curtly:

“I was not correct in my objection to your Hamilton-function. But your theory does not really differ from mine, only in the presentation and in the ‘cosmological term’ which mine lacks. […] Not your starting-point but your equations permit a transition to vanishing cosmological constant, then the content of your theory becomes identical with mine” ([446], p. 434),

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:

“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 non-irreducible object as a basis continue unabatedly. […]” ([489], p. 415).Footnote 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).

As to the physical interpretation of the geometrical objects in his “ultimate” theory, Schrödinger associated the two skew-symmetric fields ĝ[ik ], Ƒ ik . as contravariant density and covariant field tensor, so that (234), and (233), or (235) amount to (modified) Maxwell equations. The quantities \(g[ik],\,\sqrt {- g} {g^{ir}}{g^{ks}}\,{{\mathcal F}_{rs}},\,\,\sqrt {- g} {g^{ri}}{g^{sk}}{{\mathcal F}_{rs}},\,{1 \over 2}\sqrt {- g} ({g^{ri}}{g^{ks}} - {g^{rk}}{g^{is}}){{\mathcal F}_{rs}})\) were assumed to be linked to “electric charge, mesonic charge and matter”.

“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.

Recovery

Schrödinger must have been depressed after so much self-confidence! To a friend he had written:

“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)Footnote 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 ĝrsR 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 non-symmetric metric and symmetric connection “since there is no simple and natural clue” by which the Levi-Civita 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 auto-parallels (cf. Section 2.1.1, after (22)) and arrived, without noting it, at Einstein’s λ-transformations (52) introduced before.

In commenting Schrödinger’s work on affine field theories, Pauli also contributed to UFT, if only in a letter to Pascual Jordan of 13 July 1948. Once more he criticized the use of reducible tensors by Schrödinger, notably of the Ricci tensor K ik instead of its symmetric K(ik) and skew symmetric K[ik ] parts, separately. He then derived the “mathematically simplest scalar densities” as building elements of the Lagrangian. If a symmetric connection is used, they are given byFootnote 116

$${H_0} = \sqrt {\det {K_{(ik)}}} \,,\quad {H_1} = \sqrt {{K_{(ip)}}{K_{(jq)}}{K_{[kr]}}{K_{[ls]}}{\epsilon ^{ijkl}}{\epsilon ^{pqrs}}} \,,\quad {H_2} = {K_{[ij]}}{K_{[kl]}}{\epsilon ^{ijkl}}\,.$$
(239)

Possible Lagrangians then would be

$${\mathcal L} = {H_0}\left\{{1 + {f_1}\left({{{{H_1}} \over {{H_0}}}} \right) + {f_2}\left({{{{H_2}} \over {{H_0}}}} \right)} \right\}\,,\qquad {\mathcal L}\prime = {H_0}f\left({{{{H_1}} \over {{H_0}}},{{{H_2}} \over {{H_0}}}} \right),$$
(240)

with arbitrary functions f1, f2, f. Pauli’s conclusion was:

“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).Footnote 117

Schrödinger was still convinced of his approach to unified field theory. In a paper of 1951, he showed a pragmatic attitude: He set out to solve approximately the field equations with asymmetric metric and asymmetric connectionFootnote 118:

$${g_{\underset + i \underset - k \parallel l}} = 0\,,\,\,L_{[is]}^{s} = 0\,,\,\,\underset - K {\,_{ik}} = \lambda {g_{ik}}\,.$$
(241)

He argued that:

“[…] 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 pseudo-classical field theories of independent origin, cemented together by interaction terms’.” ([558], p. 555.)

As a result he claimed that “a pure charge-free 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 non-vanishing of the trace of a correspondingly defined energy-momentum 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 space-space components standing for the electrical field. The magnetic 4-current vanishes in consequence of the field equations while the electrical 4-current 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.

A favorable reaction came from a young Harvard mathematician R. L. Ingraham who was an assistant to Oswald Veblen in Princeton at the Institute for Advanced Studies in 1953. He set out to rewrite the field equations of Schrödinger’s affine unified theory [552, 555] in “a more physically meaningful form” [305]. He assumed h ij to represent gravitation but found the direct link of the skew-symmetric part of g ij with the electromagnetic field as incorrect. By an elementary calculation presented a year later also by M.-A. Tonnelat (cf. [627] or the table on p. 15 of [632]), Eq. (33) is put into the form:Footnote 119

$${\left[ {\sqrt {{h \over g}} (\sqrt h {{\check k}^{rs}} + {1 \over 2}{{{J_2}} \over {\sqrt h}}{\epsilon ^{rspq}}{k_{pq}})} \right]_{,s}} = 0\,,$$
(242)

with \({J_2}: = {1 \over 8}{\epsilon ^{rspq}}{k_{rs}}{k_{pq}}\). This equation then is rewritten as one of the usual Maxwell equations (in a space with metric h ij ) with a complicated r.h.s. which then is made to vanish by the additional assumptions that k ij is a curl, J2 = 0, and ǩrsk rs = 0. Likewise, an additional condition was laid on the curvature tensor (cf. his Eq. (27a), p. 749) such that the field equation reduced to the Einstein vacuum equation with cosmological constant. That de Sitter space is a solution of Ingraham’s equations with k ij representing some sort of plane wave, is unsurprising. Fortunately, this naive strategy of imposing additional conditions with the aim to obtain interpretable field equations, did not have many followers.

First exact solutions

The first to derive several genuine exact, spherically symmetric and static (sss) solutions of Schrödinger’s field equations with cosmological constant (235)(237) in 1947 was his research assistant A. PapapetrouFootnote 120 [475]. His ansatz contained five unknown functions of the radial coordinate \(r = {x^1}\,({x^2} = \theta, {x^3} = \phi, {x^0} = ct):\,{g_{00}}\overset * = \gamma (r),{g_{11}}\overset * = \alpha (r),{g_{22}}\overset * = \beta (r),{g_{33}}\overset * = \beta {\rm{si}}{{\rm{n}}^{\rm{2}}}\theta, {g_{01}} = - {g_{10}}\overset * = w(r),{g_{23}} = - {g_{32}}\overset * = {r^2}v(r){\rm{sin}}\theta\). In the paper, he treated the cases v = 0, w ≠ 0 and v ≠ 0, w = 0, v real. After setting β = r2 in the first case, integration of the field equations led to

$$\gamma = \left({1 + {{{l^4}} \over {{r^4}}}} \right)\left({1 - {{2m} \over r} - {\lambda \over 3}{r^2}} \right)\,,$$
(243)
$$\alpha = {\left({1 - {{2m} \over r} - {\lambda \over 3}{r^2}} \right)^{- 1}}\,,$$
(244)
$$w = \pm {{{l^2}} \over {{r^2}}}\,.$$
(245)

An assumption used was that asymptotically, i.e., for r → ∞, g(ik)η ik . Thus, if g(ij) = h ij is interpreted as the space-time metric (gravitational field tensor), g[ij ] = k ij as the electromagnetic field tensor, then a deviation from the Reissner-Nordström-de Sitter-solution of general relativity with \(\gamma = {(\alpha)^{- 1}} = (1 - {{2m} \over r} + {{{e^2}} \over {{r^2}}} - {\lambda \over 3}{r^2})\) was obtained. Introducing the elementary electric charge by l2 = e, the source term for the point charge in (243) shows the wrong radial dependence, although from (243) we must conclude that a radial electromagnetic field is present.Footnote 121 The solution does not describe the Coulomb field. At this point, this seems no serious objection to the theory, because other static spherically symmetric (SSS) solution might exist. However, cf. Section 9.6, where the most general sss solution is given.

In the second case, v is replaced by f = vr2 and only a particular solution with \(\lambda = {(\alpha)^{- 1}} = (1 - {{2m} \over r} - {\lambda \over 3}{r^2}),\beta = {r^2},\,\upsilon = c\), with c a constant was reached by Papapetrou. Thus, the radial electromagnetic field is constant. Moreover, this constant electric field does not influence the gravitational field. Papapetrou also discussed approximate solutions and concluded for them that k ij does not describe the electromagnetic field but the electromagnetic potential. This would rule out an interpretation of the solution in terms of an electric field. All these solutions would not have been acceptable to Einstein (in the sense of describing sources of electricity), because they were not free of singularities. Unlike for the gravitational field in general relativity or the electromagnetic field, for the “total” field in UFT singularities were no longer permitted:

“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)Footnote 122

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.

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: non-singular static metrics of the strong field equations of UFT which would represent the (gravitational) field of a non-vanishing 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.

H. Takeno and two coworkers of the Hiroshima Institute for Theoretical physics also took up the search for exact solutions with spherical symmetry of Einstein’s and Schrödinger’s field equations [602]. Except for a different notation and an assumed time-dependency, the form of the metric was the same as Papapetrou’s; it contained five free functions A (r, t), B (r, t), C (r, t), f (r, t), h (r, t) (and the usual coordinates x0 = t, x1 = r, x2 = θ, x3 = ϕ):

$${g_{ab}} = \left({\begin{array}{*{20}c} {- A} & 0 & 0 & f \\ 0 & {- B} & {h\sin \theta} & 0 \\ 0 & {- h\sin \theta} & {- B{{\sin}^2}\theta} & 0 \\ {- f} & 0 & 0 & C \\ \end{array}} \right),$$
(246)

Two types of solutions according to which B2h2, f = 0 (type I), or B2 = h2, f = 0 (type II) were distinguished. And then, immediately, the time-dependence of the free functions was dropped. Assuming in addition B = r2, h = kr2, the authors derived the general solution of type II of Schrödinger’s field equations (the weak equations with cosmological constant λ) to be:

$${A^{- 1}} = 1 - {{2m} \over r} - {\lambda \over 3}{r^2}\,,\,\,B = {r^2}\,,\,\,C = a\left({1 - {{2m} \over r} - {\lambda \over 3}{r^2}} \right)\left({1 + {{{k_1}} \over {{r^4}}}} \right),\,\,h = k{r^2},\,\,{f^2} = {{a{k_1}} \over {{r^4}}}\,,$$
(247)

where a, k, k1, m are integration constants. The solution generalized Papapetrou’s two exact static solutions (k1 = 0, k ≠ 0 and k1 ≠ 0, k = 0) by “combining” them. Apparently, at the time the authors did not know of Wyman’s earlier paper, also containing solutions with k × k1 ≠ 0; see Section 9.6.1. The main conclusion drawn by Takeno et al. was that the fundamental equation (30) has no unique solution if det (g ab ) = 2 det(g(ab)), det(g[ab ]) = 0. This condition is consistent with (364) derived by M.-A. Tonnelat; cf. Section 10.2.3.

In his paper of 1951 discussed above in Section 8.2, Schrödinger expressed his disenchantment with regard to the search for exact solutions:

“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)

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.

Einstein II: From 1948 on

In the meantime, Einstein had gone on struggling with his field equations and, in a letter to M. Solovine of 25 November 1948, had become less optimistic ([160], p. 88):

“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 re-discovered. It happened this way with so many problems.”Footnote 123

In his correspondence with Max Born during the second half of the 1940s, Einstein clung to his refusal of the statistical interpretation of quantum mechanics. According to him, physics was to present reality in space and time without, as it appeared to him, ghostly interactions at a distance. In a letter of 3 March 1947, he related this to UFT:

“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. […]” Footnote 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 […]”,Footnote 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 non-symmetric 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.

The field equations were derived from the Lagrangian

$${\mathcal H} = {P_{ik}}{\hat g^{ik}}\,,$$
(248)

i.e., from the Lagrangian (205) without the multiplier terms. In order to again be able to gain the weak field equations, an additional assumption was made: the skew-symmetric part of the metric (density) ĝ[ik ] be derived from a tensor “potential” ĝikl anti-symmetric in all indices. Thus, in the Lagrangian, ĝik is replaced by \({\hat g^{ik}} = {\hat g^{(ik)}} + {\hat g^{ikl}}_{,l}\). The motivation behind this trick is to obtain the compatibility equation (30) from \(\delta {\mathcal H}/\delta \Gamma _{ik}^m = 0\), indirectly. The skew-symmetric part of \(\delta {\mathcal H}/\delta \Gamma _{ik}^m = 0\) is formed and a trace taken in order to arrive at \({1 \over 2}\hat g_{\,\,\,\,\,\,\,\,\,,l}^{[ik]} + {\hat g^{(ik)}}\Gamma _{[ks]}^s = 0\). Introduction of \(\hat g_{\,\,\,\,\,,l}^{il} = \hat g_{\,\,\,\,\,\,\,,ls}^{ils}\) then shows that Γ i = 0 holds. With the help of this equation, \(\delta {\mathcal H}/\delta \Gamma _{ik}^m = 0\) finally reduces to (30).

The field equation following directly from independent variation with regard of \(\Gamma _{ik}^m,\,{\hat g^{(ik)}}\) and ĝikl are: Footnote 126

$${\hat g^{\underset + i \underset - k}}{\,_{\parallel l}} = 0\,,$$
(249)
$${P_{(ik)}} = 0\,,$$
(250)
$${P_{[ik],l}} + {P_{[kl],i}} + {P_{[li],k}} = 0\,.$$
(251)

In addition, the equations hold:

$${\hat g^{[ik]}}_{\quad \,,k} = 0\,,\,\,{\Gamma _{[ks]}}^s = 0\,.$$
(252)

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 ĝikl 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 jssikl (ĝ[ik ],l + ĝ[kl ], i + ĝ[li ],k) with vanishing divergence \(j_{\,\,\,,s}^s = 0\) is the (electric) current density ([148], p. 39).

Einstein summed up the paper for Pauli on one page or so and concluded: “The great difficulty lies in the fact that we do not have a method for deriving exact solutions free of singularities, which are the only ones of physical interest. The few things we were able to calculate strengthened my confidence in this theory.” ([489], p. 518)Footnote 127 In his answer three weeks later, Pauli was soft on “whether a mathematically unified combination of the electromagnetic and gravitational fields in a classical field theory is possible […]”, but adamant on its relation to quantum theory:

“[…] that I have another opinion than you on the question, mentioned in your letter, of the physical usability of singularity-free 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.)Footnote 128

A period of undecidedness (1949/50)

With two sets of field equations at hand (the “strong” and “weak” versions), it cost some effort for Einstein to decide which of the two was the correct one. As will be seen in Section 9.2, early in 1949 he had found a new way of deriving the “weak” field equations, cf. [149].Footnote 129 In a letter of 16 August 1949 to his friend Besso, who had asked him to tell him about his generalized field equations, Einstein presented these “weak” equations and commented:

“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).Footnote 130

Six weeks later, on 30 September 1949, Einstein had changed his mind: he now advocated the “strong” version (200)(202) of Section 7.2.

“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).Footnote 131

Consistent with Einstein’s undecidedness are both, his presentation of UFT in Appendix II of the 3rd Princeton edition of The Meaning of Relativity [150], and another letter to Besso of 15 April 1950 [163]. In both, he had not yet come to a final conclusion as to which must be preferred, the “weak” or the “strong” equations. To Besso, he explained that the “weak” equations could be derived from a variational principle and thus are “compatible”.

“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).Footnote 132

At first, Einstein seems to have followed a strategy of directly counting equations, variables, and identities. However, early in 1952 he seems to have had a new idea: the λ-transformations. He wrote to Besso on 6 March 1952 that he had made “very decisive progress (a couple of weeks ago).” The field equations, hitherto not uniquely determined theoretically, now were known:

“Apart from [coordinate-] transformation invariance, invariance also is assumed for the transformations of the non-symmetric ‘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)Footnote 133

We will come back to his final decision in Section 9.2.3.

Birthday celebrations

Einstein’s seventieth birthday was celebrated in Princeton with a seminar on “The Theory of Relativity in Contemporary Science”, in which E. P. Wigner, H. Weyl, and the astronomers G. M. Clemence and H. P. Robertson lectured. UFT was left aside [183]. Weyl, in his lecture “Relativity Theory as a Stimulus in Mathematical Research”, came near to it when he said:

“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.)

As it suited to a former assistant of Einstein, in his article celebrating his master’s 70th birthday, Banesh Hoffmann found friendly if not altogether exuberant words even for Einstein’s struggle with UFT [281]. For 25 years Einstein had devoted his main scientific work to the problem of the structure of matter and radiation. He tried to gain an insight:

“[…] 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].

Belatedly, toward the end of 1949, some sort of a “Festschrift” for Einstein appeared with 25 contributions of well-known physicists and philosophers, among them six Nobel prize winners [536]. Most interesting is Einstein’s own additional contribution, i.e., his “Autobiographical Notes”, written already in 1946. He described his intentions in going beyond general relativity and essentially presented the content of his paper with E. Straus [179] containing the “weak field equations” of UFT. His impression was:

“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.)Footnote 134

Einstein 1950

Alternative derivation of the field equations

As we have seen, one of Einstein’s main concerns was to find arguments for choosing a quasi-unique system of field equations for UFT. His first paper of 1949 opened with a discussion, mostly from the point of view of mathematics, concerning the possibilities for the construction of UFT with a non-symmetric fundamental tensor. According to Einstein: “The main difficulty in this attempt lies in the fact that we can build many more covariant equations from a non-symmetric tensor than from a symmetric one. This is due to the fact that the symmetric part g(ik) and the antisymmetric part g[ik ] are tensors independently” ([149], p. 120). As the fundamental tensor is no longer considered symmetrical, the symmetry of the connection (as in Riemannian geometry) must also be weakened. By help of the conjugate quantities of Section 2.2.2 (Hermitian, transposition symmetry), Einstein’s constructive principle then is to ask “that conjugates should play equivalent roles in the field-equations.” According to him, this necessitates the introduction of the particular form (30) for the compatibility condition. In fact, for the conjugate:Footnote 135

$${\tilde g_{\underset + i \underset - k \parallel l}} = {g_{\underset + k \underset - i \parallel l}}\,,$$
(253)

while

$${\tilde g_{\underset + i \underset + k \parallel l}} = {g_{\underset - k \underset - i \parallel l}}\,.$$
(254)

Einstein seemingly was not satisfied with the derivation of the field equations from a variational principle in “both previous publications” ([147, 179]), because of the status of equations (252). To obtain them, either Lagrangian multipliers or a restriction of the metric (“must be derivable from a tensor potential”) had to be used. Now, he wanted to test the field equations by help of some sort of Bianchi-identity such as (cf. Section 2.3.1, Eq. (68), or Section 2.1.3 of Part I, Eq. (30)):

$$\overset + K {\,^i}_{j\{kl\Vert m\}} = 2\overset + K {\,^i}_{r\{kl}{S_{m\} j}}^r\,.$$
(255)

After a lengthy calculation, he arrived at: Footnote 136

$$\underset - K {\,_{\underset - i \underset + k \underset - l \underset + m \parallel n}} + \underset - K {\,_{\underset - i \underset + k \underset + m \underset + n \parallel l}} + \underset - K {\,_{\underset - i \underset + k \underset - n \underset - l \parallel m}} = 0\,,$$
(256)

and by further trace-formingFootnote 137

$${g^{kl}}[ - {K_{\underset + k \underset - l \parallel m}} + {K_{\underset + k \underset + m \parallel l}} + {\Sigma _{\underset - l \underset - m \parallel k}}] = 0\,.$$
(257)

In Eq. (257), two contractions of \(\underset - K _{\,\,\,\,jkl}^i\) were introduced: \(- {\underset - K _{jk}}: = \underset - K _{\,\,jkl}^l\) and \(\sum\nolimits_{ml} {= - {g_{mi}}{g^{jk}}\underset - K _{\,\,\,jkl}^i}\). Neither Σ ml nor \({\underset - K _{jk}}\) are Hermitian: We have \(- {\underset - {\tilde K} _{lm}} = {\Sigma _{ml}}\). The anti-Hermitian part of \({\underset - K _{jk}}\) is given by

$$\underset - K {\,_{kl}} - \underset - {\tilde K} {\,_{lk}} = - ({S_{l,k}} + {S_{k,l}}) + L_{kl}^{s}{S_s}\,.$$
(258)

if vector torsion is absent, i.e., \({S_k} = {L_{[ks]}}^s = 0\), then \({\underset - K _{jk}}\) becomes Hermitian, and \(- {\underset- K _{lm}} = {\Sigma _{ml}}\). Equation (257) then can be written as

$${g^{kl}}[ - {K_{\underset + k \underset - l \parallel m}} + {K_{\underset + k \underset + m \parallel l}} + {K_{\underset - m \underset - l \parallel k}}] = 0\,.$$
(259)

Therefore, Einstein demanded that the contribution of \({\underset - K _{[jk]}}\) to the Eq. 259) be in general:

$${g^{kl}}[ - {K_{[\underset + k \underset - l ]\parallel m}} + {K_{[\underset + k \underset + m ]\parallel l}} + {K_{[\underset - m \underset - l ]\parallel k}}] = 0\,.$$
(260)

Schrödinger had derived (260) before by an easier method with the help of the Lie-derivative; cf. [556]. We will meet (260) again in Section 10.3.1. A split of (259) into symmetric and skew symmetric parts (inside the bracket) would give the equation:

$${g^{kl}}[ - {K_{\underset + {(k} \underset - {l)} \parallel m}} + {K_{\underset + {(k} \underset + {m)} \parallel l}} + {K_{\underset - {(m} \underset - {l)} \parallel k}} - {K_{[\underset + k \underset - l ]\parallel m}} + {K_{[\underset + k \underset + m ]\parallel l}} + {K_{[\underset - m \underset - l ]\parallel k}}] = 0\,.$$
(261)

At best, as a sufficient condition, the vanishing of the symmetric and skew-symmetric parts separately could take place. Besides (260) the additional equation would hold:

$${g^{kl}}[ - {K_{\underset + {(k} \underset - {l)} \parallel m}} + {K_{\underset + {(k} \underset + {m)} \parallel l}} + {K_{\underset - {(m} \underset - {l)} \parallel k}}] = 0\,.$$
(262)

As we will see, by a further choice (cf. (265), this equation will be satisfied. Einstein first reformulated (260) into:

$${g^{kl}}[\underset - K {\,_{[kl],m}} + \underset - K {\,_{[mk],l}} + \underset - K {\,_{[lm],k}}] = 0\,,$$
(263)

and then took

$$\underset - K {\,_{[kl],m}} + \underset - K {\,_{[lm],k}} + \underset - K {\,_{[mk],l}} = 0$$
(264)

as its solution and as part of the field equations. He then added as another field equation:

$$\underset - K {\,_{(kl)}} = 0$$
(265)

by which (262) is satisfied. Thus, in [149], with a new approach via identities for the curvature tensor and additional assumptions, Einstein had reached the weak field equations of his previous paper [148]. No physical interpretation of the mathematical objects appearing was given by him.

A summary for a wider circle

In Appendix II of the third Princeton edition of his book The Meaning of Relativity, Einstein gave an enlarged introduction on 30 pages into previous versions of his UFT. The book was announced with fanfare in the Scientific American [151]:

“[…] Einstein will set forth what some of his friends say is the long-sought 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).

It was Princeton University Press who had used Einstein’s manuscript for this kind of advertising much to his distress; a page of it even “appeared on the front page of The New York Times under the heading ‘New Einstein Theory Gives a Master Key to the Universe’.” ([469], p. 350.) Einstein’s comment to his friend M. Solovine, on 25 January 1950, was:

“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).Footnote 138

In the book, the translator is identified to have been Sonja Bargmann, the wife of Valentine, who also had translated other essays by Einstein. In Appendix II, with the assumption that

  1. (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. (2)

    all contractions of the curvature tensor (54) vanish,

  3. (3)

    that (30) hold,

Einstein arrived at the field equationsFootnote 139 (30), (201), and (202):

$${g_{\underset + i \underset - k \parallel l}}: = {g_{ik,l}} - {g_{rk}}{L_{il}}^r - {g_{ir}}{L_{lk}}^r = 0,$$
(266)
$${S_j}(L): = {L_{[im]}}^m = 0,$$
(267)
$${K_{jk}}(L) = 0.$$
(268)

In place of (266) the equivalent equations \({g^{\overset i + \overset k -}}_{\Vert l} = 0\), or \({\hat g^{\overset i + \overset k -}}{\,_{\Vert l}} = 0\) with \({\hat g^{ik}} = \sqrt {{\rm{det}}({g_{ik}})} \,{g^{ik}}\) can be used. Moreover, if in addition (267) is taken into account, then also

$${\hat g^{[il]}}_{\quad ,\,\,l} = 0$$
(269)

is satisfied. This is due to a relation to be met again below [(cf. Eq. (276)]:

$${\hat g^{[\underset + i \underset - l ]}}_{\quad \,\,\parallel l} = \hat g_{\quad ,l}^{[il]} - {\hat g^{[il]}}{S_l}\,.$$
(270)

([179], Eq. (3.4), p. 733.)

For Einstein, this choice (“System I”) “is therefore the natural generalization of the gravitational equation” ([150], p. 144). A little later in the Appendix he qualified his statement as holding “from a formal mathematical point of view […]” ([150], p. 150) because the manifold of solutions of “System I” might be too small for physical purposes. Moreover, “System I” could not be derived from a variational principle. He then set up such a variational principleFootnote 140

$${\mathcal H} = {P_{ik}}{\hat g^{ik}}$$
(271)

with the Hermitian Ricci tensor P ik . As Einstein wanted to again get (267), he introduced another connection Γ* by \(\Gamma _{ij}^{\,\,\,\,\,k} = \Gamma _{ij}^{*\,\,k} - {2 \over 3}\Gamma _{\left[ i \right.}^*\delta _{\left. j \right]}^k\) which does not satisfy \(\Gamma _j^* = \Gamma _{\left[ {im} \right]}^{*\,\,m} = 0\), such that just independent components could be varied. The result is his “System Ia”:

$$\hat g_{\quad \parallel l}^{\underset + i \underset - k} - {1 \over 3}({{\mathcal M}^i}\delta _l^{k} - {{\mathcal M}^k}\delta _l^{i}) = 0\,,$$
(272)
$${L_i} = 0\,,$$
(273)
$${P_{ik}} = 0\,,$$
(274)

with \({\mathcal{M}^i}: = \hat g_{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\| l \right.}^{\mathop { + - }\limits^{[il]} }\). The following identity holds:Footnote 141

$${{\mathcal M}^i} \equiv \hat g_{\quad ,l}^{[il]} - {\hat g^{[il]}}{\Gamma _l}$$
(276)

In order to make vanish \({{\mathcal M}^i}\), with the help of a Lagrange multiplier l i , he introduced the term \({l_i}\,{\hat g^{[ir]}}_{\,\,\,\,\,\,\,\,\,,r}\) into the Lagrangian and arrived at ‘System Ib’, i.e., the weak field equations:

$$\begin{array}{*{20}c} {\hat g_{\quad \,\,\parallel l}^{\underset + i \underset - k} = 0\,,} \\ {\quad {L_i} = 0\,,} \\ {{P_{(ik)}} = 0\,,} \\ \end{array}$$
(277)
$${P_{[ik],l}} + {P_{[kl],i}} + {P_{[li],k}} = 0\,.$$
(278)

In all three systems, equations (249), (269), (272) are to be used for expressing the components of the (asymmetric) connection by the components of the (asymmetric) metric. The metric then is determined by the remaining equations for the Ricci tensor.

The only remarks concerning a relationship between mathematical objects and physical observables made by Einstein at the very end of Appendix II are:

  1. (1)

    (269) shows that there is no magnetic current density present (no magnetic monopoles),

  2. (2)

    the electric current density (or its dual vector density) is represented by the tensor g[ik ],l + g[kl ],i + g[li ], k).

In order to obtain these conclusions, a comparison to Maxwell’s equations has been made (cf. (210) and (211) of Section 7.3). As for all of Einstein’s papers in this class, g(ik) describes the gravito-inertial, and g[ik ] the electromagnetic fields.

For the first time, Einstein acknowledged that he had seen Schrödinger’s papers without giving a reference, though: “Schrödinger, too, has based his affine theory […]”. Max Born, in his review of Einstein’s book, bluntly stated: “What we have before us might therefore be better described as a program than a theory.” ([41], p. 751.) According to Born, Einstein “tries to find a theory of the classical type of such refined structure that it contains the essential features of atomic and quantum theory as consequences. There are at present few physicists who share this view.” The review by W. H. McCrea reflected his own modesty. Although more cautious, he was very clear:Footnote 142

“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).

The “eminent author” himself confessed in a letter to Max Born of 12 December 1951:

“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).Footnote 143

Also, W. Pauli commented on this 3rd edition. A correspondent who inquired about “the prospects of using Einstein’s new unified field theory as an alternative basis for quantum electrodynamics” obtained a demoralizing answer by him in a letter of 4 July 1950:

“Regarding Einstein’s ‘unified’ field theory I am extremely skeptical. It seems not only arbitrary to add a symmetrical and an anti-symmetrical 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)

Einstein’s former assistant and co-author 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].

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.Footnote 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” s0 for first-order systems in involution which, essentially, gave the number of arbitrarily describable free data (functions of 3 spacelike variables) on an initial hypersurface (t = t0). He calculated such indices, for Maxwell’s equations with currents to be s0 = 8, and without s0 = 4, for Einstein’s vacuum field equations s0 = 4, (in this case 4 free functions of 4 variables existFootnote 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).

It is unknown whether Einstein remembered the discussion with Cartan or had heard of Pauli’s remark, when he tackled the problem once more; in the 4th Princeton edition of his book The Meaning of Relativity, Appendix II [156] Cartan’s name is not mentioned. In the meantime, Mme. Choquet-Bruhat, during her stay at Princeton in 1951 and 1952, had discussed the Cauch-problem with Einstein such that he might have received a new impulse from her. To make the newly introduced concept of “strength” of a system of PDE’s more precise, he set out to count the number of free coefficients of each degree in a Taylor expansion of the field variables; if all these numbers are non-negative, he called the system of PDE’s “absolutely compatible”. He then carried out a calculation of the number of coefficients Ω n remaining “free for arbitrary choice” for the free wave equation, Maxwell’s vacuum equations, the Einstein vacuum equations, and particular field equations of UFT. Let us postpone the details and just list his results.Footnote 146 For the wave equation \({\Omega _n} = \left({\mathop {n + 3}\limits_n} \right)\,{6 \over n}\). According to him “the factor \({6 \over n}\) gives the fraction of the number of coefficients (for the degree n ≫ 1), which remain undetermined by the differential equation” ([156], p. 152). Similarly, he found for the Maxwell vacuum equations, \({\Omega _n} = \left({\mathop {n + 3}\limits_n} \right)\,{{12} \over n}\). Einstein noted that by introducing the vector potential A i , and taking into account the Lorentz gauge, i.e., by dealing with

$${\eta ^{rs}}{\partial _r}{\partial _s}{A_i} = 0\,,\quad {\eta ^{rs}}{\partial _r}{A_s} = 0\,,$$
(279)

the counting led to \({\Omega _n} = \left({\mathop {n + 3}\limits_n} \right)\,{{18} \over n}\). He ascribed the increase in the number of freely selectable coefficients (loss of strength) to the gauge freedom for the vector potential. For the Einstein vacuum equations, he obtained \({\Omega _n} = \left({\mathop {n + 3}\limits_n} \right)\,{{15} \over n}\). In applying his method to the “weak” field equations of Section 9.2.2, Einstein arrived at \({\Omega _n} = \left({\mathop {n + 3}\limits_n} \right)\,{{45} \over n}\). In comparing this to the calculation for other field equations in UFT, he concluded that “the natural generalization of the gravitational equations in empty space” is given by the “weak” field equations ([156], p. 164).

Obviously, Einstein was not satisfied by his calculations concerning the “strength” of PDE’s. In the 5th Princeton edition of his book The Meaning of Relativity, Appendix II [158],Footnote 147 he again defined a system of PDE’s as “absolutely compatible” if, in a Taylor expansion of the field variable Φ, the number of free n-th order coefficients \({{{\partial _n}} \over {\partial {x^1}\partial {x^2} \cdots \partial {x^n}}}\Phi {\vert_P}\), at a point P does not become negative. He then gave a name to the number of free coefficients calculated before: he called it “coefficient of freedom”. The larger this coefficient is, the less acceptable to him is the system of PDE’s. Let p denote the number of field variables, s the number of field equations of order q, and w the number of identities among the field equations in the form of PDE’s of order q′. Then, writing z in place of the previous Ω n , his formulas could be condensed into:

$$z = p\left({\begin{array}{*{20}c} {n + 3} \\ n \\ \end{array}} \right) - \left[ {s\left({\begin{array}{*{20}c} {n + 3 - q} \\ {n - q} \\ \end{array}} \right) - w\left({\begin{array}{*{20}c} {n + 3 - q\prime} \\ {n - q\prime} \\ \end{array}} \right)} \right]$$
(280)
$$= \left({\begin{array}{*{20}c} {n + 3} \\ n \\ \end{array}} \right)\left\{{a + {{{z_1}} \over n} + {{{z_2}} \over {{n^2}}} + \ldots} \right\}\,,$$
(281)

with \(a = p - s + w\overset ! \ge 0\) required for absolute compatibility; z1 = 3(qsqw) is the “coefficient of freedom.”Footnote 148 Again, Einstein calculated z1 for several examples, among them Maxwell’s vacuum field equations in flat space-time:

$$F_{\,\,\,,k}^{ik} = 0\,,\,\,{}^{\ast}F_{\,\,\,,k}^{ik} = 0\,,$$
(282)

with the identities

$$F_{\quad ,ki}^{ik} = 0\,,{}^{\ast}F_{\quad ,ki}^{ik} = 0\,,$$
(283)

Here p = 6, s = 8, q = 1, q′ = 2, w = 2. In agreement with the result from the previous edition, the calculation led him to a = 0, z1 = 12. In contrast, it turned out that the “coefficient of freedom” for the gravitational vacuum field equations in general relativity, in the 2nd calculation became smaller, i.e., z1 = 12 ([158], p. 139). Fortunately, now both equations have the same “index of generality” (Cartan) and the same “coefficient of freedom” (Einstein). Likewise, Einstein found z1 = 42 instead of the previous z1 = 45 for the “weak”, and z1 = 48 for a concurring system with transposition invariance such that he again adopted the “weak” one as before. (280) with its values for a, z1 is not yet the correct formula. Such a formula was derived for involutive, quasi-linear systems of PDEs by a group of relativists around F. Hehl at the University of Cologne at the end of the 1980s ([595], Eqs. (2.9), (2.10), [596]). By their work, also the relation between the Cartan coefficient of generality and Einstein’s coefficient of freedom has now been provided. According to M. Sué ([595], p. 398), it seems that Einstein’s coefficient of freedom is better suited for a comparison of the systems investigated than Cartan’s degré d’arbitraire. In mathematics, a whole subdiscipline has evolved dealing with the Cartan-Kähler theorem and the Cartan-Characters for systems of PDE’s. Cf. [57].

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 re-defining 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.Footnote 149

An account for a general public

Following an invitation by the editors of Scientific American to report on his recent research, Einstein made it clear that he would not give

“[…] 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.) Footnote 150

He then talked about the epistemological basis of science, men’s curiosity and passion for the understanding of nature before touching upon problems connected with a generalization of general relativity. Two questions were very important, though not yet fully answered: the uniqueness of the field equations and their “compatibility”. He then sketched the three systems of field equations obtained, here denoted E3 (System I), and E1, E2 (Systems Ia, Ib). He again stressed that E3(System I) “is the only really natural generalization of the equations of gravitation”. However, it was not a compatible system as were the other two.

“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 (xl) also ′g ik (kxl) 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.

Nevertheless, Einstein remained optimistic; in the same letter to Max Born, in which he had admitted his shortcoming vis-a-vis the complexity of his theory, he wrote:

“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.”Footnote 151

His Italian colleague Bruno Finzi was convinced that the final aim had been reached:

“[…] 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)Footnote 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.

Einstein 1953

In the fourth edition of Einstein’s The Meaning of Relativity, invariance with regard to λ-transformations was introduced as a new symmetry principle (cf. (52) of Section 2.2.3). Also transposition invariance is now claimed to be connected to the “indifference of the theory” (UFT) with respect “to the sign of electricity” ([156], p. 144). This interpretation rests on Einstein’s identification of the electric current density with g[ik], l + g[kl], i + g[li], k. Einstein still grappled with the problem of how to set up a convincing system of field equations. As in the previous edition, he included (269) as an “a priori condition” in his variational principle by help of a (1-form)-multiplier σ i . However, he renounced using Γ i = 0. Without specification of the Lagrangian \({\mathcal H}\), from \(\delta \int {{\mathcal H}d\tau} = \int {(\hat V_{\,\,\,\,\,\,l}^{ik}\delta \Gamma _{ik}^{\,\,\,\,\,\,l} + {W_{ik}}\,\delta {{\hat g}^{ik}})d\tau} = 0\) the field equations follow — without use of the multiplier-term — to be:

$$\hat V_{\,\,\,\,l}^{ik} = 0\,,\quad {W_{ik}} = 0\,;$$
(284)

with use of the multiplier-term —

$$\hat V_{\quad l}^{ik} = 0\,,\quad \hat g_{\quad ,\,\,\,l}^{[il]} = 0\,,\quad {W_{ik}} + {\sigma _{[i,k]}} = 0\,.$$
(285)

Elimination of the multiplier σ i led to the equations (named “System II” by Einstein)

$${\hat V^{ik}}_{\,\,l} = 0\,,\hat g_{\quad ,\,l}^{[il]} = 0\,,\quad {W_{(ik)}} = 0\,,{W_{[ik],l}} + {W_{[kl],i}} + {W_{[li],k}} = 0\,.$$
(286)

A paragraph then was devoted to the choice of the proper Lagrangian. Einstein started from (196) and removed a divergence term in \(\overset {\rm Her} {\underset - K_{ik}}{\hat g^{ik}}\). After variation (with inclusion of the multiplier term) the ensuing field equations, Einstein’s “system IIa”, were:

$${\hat g_{\underset + i \underset - k \parallel l}} = 0\,,$$
(287)
$${\Gamma _i} = 0\,,$$
(288)
$$\underset - K {\,_{(ik)}} = 0\,,$$
(289)
$$\underset - K {\,_{[ik],l}} + \underset - K {\,_{[kl],i}} + \underset - K {\,_{[li],k}} = 0\,,$$
(290)

i.e., a version of the Einstein-Straus weak field equations. The road to the weak field equations (287)(290) followed here still did not satisfy Einstein, because in it the skew-symmetric parts of both the metric and the connection could also be taken to be purely imaginary. In order to exclude this possibility and work with a real connection, he introduced λ-transformations and presented a further derivation of the field equations. He set up a variational principle invariant under the λ-transformation and arrived at the same system of field equations as before. The prize payed is the exclusion of a physical interpretation of the torsion tensor.

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.)

Both, the concept of “strength” of a system of differential equations and the concluding §5 “General remarks concerning the concepts and methods of theoretical physics” point to Einstein’s rather defensive position, possibly because of his feeling that the particular field equations of unified field theory for which he strove so hard, rested on flexible ground. This was due not only to the arbitrariness in picking a particular field equation from the many possibilities, but also to the failure of the theory to include a description of concepts forming an alternative to quantum theory. Einstein stuck to the classical field and rejected both de Broglie’s “onde pilote”, and Bohm’s attempt away from the statistical interpretation of the wave function. At the very end of his Meaning of Relativity he explained himself in this way:

“[…] 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.)

An indirect answer to this opinion was given by F. J. Dyson in an article on “field theory” in the Scientific American. He claimed “that there is an official and generally accepted theory of elementary particles, known as the ‘quantum field theory’.” According to him, while there still was disagreement about the finer details of the theory and its applications:

“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).Footnote 153

Joint publications with B. Kaufman

In the Festschrift for Louis de Broglie on the occasion of his 60th birthday (15.8.1952) organized by M.-A. Tonnelat and A. George, Einstein again summarized his approach to UFT, now in an article with his assistant Bruria KaufmanFootnote 154 [172]. In a separate note, as kind of a preface he presented his views on quantum theory, i.e., why he still was trying “[…] to solve the quantum riddle on another path or, to at least help for preparing such a solution.” ([155], p. 4.)Footnote 155 He expressed his well-known epistemological position that something like a “real state” of a physical system exists objectively, independent of any observation or measurement. A list of objections to the majority interpretation of quantum theory was given. At the end of the note, a link to UFT was provided:

“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.)Footnote 156

As in the 4th edition of his book [156], the geometrical basics were laid out, and one more among the many derivations of the weak field equations of UFT given before was presented. At first, it looked weird, but in referring to a result of the “researches of E. Schrödinger” (without giving a reference, though) Einstein & Kaufman took over Schrödinger’s “star”-connection:

$${}^{\ast}{\Gamma _{ik}}^l: = {\Gamma _{ik}}^l + {2 \over 3}{\delta _{\left[ i \right.}}^l\,{\Gamma _{\left. k \right]}}$$
(291)

introduced in ([552], p. 165, Eq. (10)). For it, *Γ k = 0 holds which leads to simplifications. Under a λ-transformation the Ricci curvature is not invariant [cf. (87)]. In order to make the variational principle invariant, due to

\(\delta \int {{d^4}x\,{{\hat g}^{ik}}{{\underset- K}_{ik}}(\Gamma)} = \delta [\,\int {{d^4}x\,{{\hat g}^{ik}}{{\underset - K}_{ik}}(*\Gamma)} + 2\int {{d^4}x} \,{\hat g^{[il]}}_{\,\,\,\,\,\,\,\,,l}{\lambda _i}]\), as an ad-hoc- (or as Einstein & Kaufman called it, an a priori-) condition is needed:

$${\hat g^{[il]}}_{\quad ,l} = 0\,.$$
(292)

The further derivation of the field equations led to the known form of the weak field equations:

$${g_{ik\,\,\underset {\ast} {\parallel} l}} = 0\,,$$
(293)
$$\underset {\ast} K {\,_{(ik)}} = 0\,,$$
(294)
$$\underset {\ast} K {\,_{[ik],l}} + \underset {\ast} K {\,_{[kl],i}} + \underset {\ast} K {\,_{[li],k}} = 0\,.$$
(295)

Here, the covariant derivative refers to the connection *Γ and \({\underset * K _{(ik)}} \equiv {\underset - K _{ik}}(*\Gamma)\).

In addition, a detailed argument was advanced for ruling out the strong field equations. It rests partially on their failure to guarantee the possibility to superpose weak fields. The method used is a weak-field expansion of the metric and the affine connection in a small parameter :

$${g_{ik}} = \epsilon \underset 1 g {\,_{ik}} + {\epsilon ^2}\underset 2 g {\,_{ik}} + \ldots \,,\quad \Gamma _{ij}^{k} = \epsilon \underset 1 \Gamma {\,_{ij}}^k + {\epsilon ^2}\underset 2 \Gamma {\,_{ij}}^k + \ldots$$
(296)

After expanding the field equations up to 2nd order, the authors came to the conclusion that the “strong equations” strongly constrain “the additivity of symmetric and antisymmetric weak fields. It seems that by this any usefulness of the ‘strong system’ is excluded from a physical point of view.” ([172], p. 336.) Footnote 157

In an appendix to the paper with title “Extension of the Relativistic Group” [172], Einstein combined the “group” of coordinate transformations with the λ-transformations 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.)Footnote 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 metric-field to hold everywhere in space-time?” 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 non-symmetric 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].Footnote 159 The papers by M.-A. Tonnelat published earlier and presenting a solution were not referred to at all [622, 623, 630, 629].

In the sequel, g(ik) is given a Lorentz signature. By using special coordinates in which \(g_{\,\,\,i}^k = {\rho _i}\delta _{\,\,\,i}^k\) (no summation on the index i), it can be seen that “\({\rho _1} = {1 \over {{\rho ^2}}}\) is on the unit circle from which the point −1 has been excluded”; the other two roots \({\rho _3} = {1 \over {{\rho ^4}}}\) are positive. It is shown in the paper that among the three algebraic invariants to be built from \(g_{\,\,\,i}^k = {g^{ks}}{g_{si}}\) only two are independent:

$${S_1}: = \underset {s = 1} {{\Sigma ^4}} {\rho _s}\,,\,\,{S_2}: = \underset {r > s} \Sigma {\rho _r}{\rho _s}\,.$$
(297)

As a “sufficient condition for regularity and unique algebraic determination of \(\Gamma _{ik}^{\,\,\,\,\,l}\) the authors derive ([173], p. 237):

$${S_1} \neq 2 \neq {S_2}\,.$$
(298)

In a lengthy calculation filling six pages, a formal solution to the compatibility equation (30), seen as an algebraic equation for Γ is then presented: “[…] it is cumbersome, and not of any practical utility for solving the differential equations” ([173], p. 238).

Einstein’s 74th birthday (1953)

Einstein agreed to let his 74th birthday be celebrated with a fund-raising event for the establishing of the Albert Einstein College of Medicine of Yeshiva University, New York. Roughly two weeks later, according to A. Pais The New York Times carried an article about Einstein’s unified field theory on the front page [471]. It announced the appearance of the 4th Princeton edition of The Meaning of Relativity with its Appendix II, and reported Einstein as having stated that the previous version of 1950 of the theory had still contained one important difficulty. According to him: “[…] This last problem of the theory now finally has been solved in the past months.”Footnote 160 Probably, this refers to Einstein’s new way of calculating his “coefficient of freedom” introduced for mirroring the “strength” of partial differential equations. In a letter to Carl Seelig of 14 September 1953, Einstein tried to explain the differences between the 3rd and the 4th edition of The Meaning of Relativity:

“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.”Footnote 161 ([570], p. 401–402)

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 order to mend this defect, Kursunŏglu modified the Einstein-Straus weak field equations by beginning with the identity (257) and adding another identity formed from the the metric g ij = a ij + ik ij and its 1st derivatives only [342, 343]:Footnote 162

$${g^{kl}}[ - {g_{\underset + k \underset - l \parallel m}} + {g_{\underset + k \underset + m \parallel l}} + {g_{\underset - m \underset - l \parallel k}}] = 0\,.$$
(299)

Kursunŏglu’s ensuing field equations were:

$$\hat g_{\quad ,l}^{[il]} = 0\,,$$
(300)
$${\underset - K _{(ik)}} + {p^2}({H_{ij}} - {b_{ij}}) = 0$$
(301)
$${\underset - K _{[ik],l}} + {\underset - K _{[kl],i}} + {\underset - K _{[li],k}} + {p^2}({k_{[ik],l}} + {k_{[kl],i}} + {k_{[li],k}} = 0$$
(302)

with p real or imaginary, andFootnote 163

$${b^{ij}} = \sqrt {{l \over g}} {l^{(ij)}}$$
(303)

where b ij is the inverse of bij. I ikl := k[ik ],l + k[kl ],i + k[li ],k is connected with the electrical 4-current density Jr through I ikl := iklr Jr.

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 p2 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 Einstein-Maxwell theory, now for e2 the expression e2p2, 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 space-time fixing the matter tensor as in the case of the Einstein-Maxwell theory ([70], p. 779). Callaway concluded that “if the approach of field theory is accepted, it is necessary to construct a theory in which space-time and matter enter as equals.” But he would not accept UFT as an alternative to quantum theory.

Einstein 1954/55

The last paper with B. Kaufman was submitted three months before, and appeared three months after Einstein’s death [174]. In it, the authors followed yet two other methods for deriving the field equations of UFT. Although the demand for transposition-invariance was to play a considerable role in the setting up of the theory, in the end invariance under λ-transformations became the crucial factor. In the first approach (§1), instead of the previously used connection Γ, Einstein and Kaufman introduced another one, Γ*, containing four new variables (a 1-form) Λ k “supernumerary to the description of the field”, and defined by:

$$\Gamma _{ik}^{*\,\,\,\,\,\,l} = \Gamma _{ik}^l - \delta _i^l{\Lambda _k}$$
(304)

During variation of the Lagrangian, Γ* and Λ k were treated as independent variables; after the variation could be fixed arbitrarily (“normed”). This trick allowed that four variational field equations could replace four equations put in by hand as had been the equation Γ k = 0 in earlier approaches. The following relation resulted:

$$\underset - K {(\Gamma)_{ik}} = \underset - K {({\Gamma ^*})_{ik}} - 2{\Lambda _{[i,k]}}$$
(305)

such that the Lagrangian could be written as

$${\mathcal H}(\Gamma) = {\mathcal H}({\Gamma ^*}) - 2{{\hat g}^{rs}}{\Lambda _{[r,s]}}.$$
(306)

After the variation with respect to \(\Gamma _{ik}^{*\,l},{\hat g^{ik}}\), and Λ k the choice \({\Lambda _k} = {2 \over 3}{\Gamma _k}\) was made leading to \(\Gamma _k^* = 0\). Although the wanted transposition-invariant field equations did come out, the authors were unhappy about the trick introduced. “The reason for our difficulties is that we require the field equations to be transposition-invariant, but we start out from a variational function which does not have that property. The question arises naturally whether we cannot find a form of the variational function which will itself be transposition-invariant, […].” ([174], p. 131.) In order to obtain such a Lagrangian, they replaced the connection by a quantity \(U_{ik}^{\,\,\,\,l}\) called “pseudo-tensor”. It is transforming like a tensor only under linear coordinate transformations (§2–§3):

$$U_{ik}^l: = \Gamma _{ik}^l - \Gamma _{is}^s\delta _k^l.$$
(307)

From (307) we see that \(U_{ik}^{\,\,\,\,l}\) does not transform like a connection. As a function of \(U_{ik}^{\,\,\,\,l}\) the Ricci tensor:

$$\underset - K {(\Gamma (U))_{ik}} = U_{ik,\,\,s}^{\,\,\,\,s} - {U}_{is}^{\,\,\,\,r} {U}_{rk}^{\,\,\,\,s} + {1 \over 3} {U}_{ir}^{\,\,\,\,r} {U}_{sk}^{\,\,\,\,s}$$
(308)

is transposition-invariant, i.e., \(\underset - K {(\Gamma (\tilde U))_{ik}} = \underset - K {(\Gamma (U))_{ki}}\). With regard to a λ-transformation (52) the “pseudo-tensor” U transforms as:

$$\prime U_{ik}^l = U_{ik}^l + 2\delta _{\left[ i \right.}^l{\lambda _{\left. k \right]}}.$$
(309)

A short calculation shows that \(\underset - K {(\Gamma (U))_{ik}}\) is invariant under (52).

As a Lagrangian, now \({\mathcal H} = {\hat g^{ik}}\underset- K {(\Gamma (U))_{ik}}\) was taken. Variation with respect to the variables

$$\delta {\mathcal H} = {({\hat g^{ik}}\delta U_{ik}^{\,\,\,s})_{,s}} + {\hat N^{ik}}_{\,\,\,\,s}\delta U_{ik}^{\,\,\,s} + \underset - K {(\Gamma (U))_{ik}}\delta {\hat g^{ik}}$$
(310)

led to the field equations:

$$\hat N_{\,\,\,,s}^{ik} \equiv - \hat g_{\,\,\,,s}^{ik} - {\hat g^{rk}}(U_{rs}^{\,\,\,i} - {1 \over 3}\delta _s^i)U_{rt}^{\,\,\,t} - {\hat g^{ir}}(U_{sr}^{\,\,\,k} - {1 \over 3}\delta _s^kU_{tr}^{\,\,t}) = 0\,,$$
(311)
$$\underset - K {(\Gamma (U))_{ik}} = 0$$
(312)

with \(\underset - K {(\Gamma (U))_{ik}}\) given by (308). Although the authors do not say it, Eqs. (311) and (312) are equivalent to the “weak” field equations (287)(290). By inserting infinitesimal coordinate- and λ-transformations into (310), five identities called “Bianchi-identities” result. Modulo the field equations,

$${({\hat g^{ik}}\delta U_{ik}^{\,\,\,s})_{,s}} = 0$$
(313)

holds as well. With the help of special infinitesimal coordinate transformations and the Bianchi-identities a “conservation law for energy and momentum” is derived:

$$\hat T_{i\,\,,s}^s = 0\,,$$
(314)

where \(\hat T_i^k: = ({\hat g^{rs}}U_{rs}^{\,\,\,\,k}){,_i}\).

The results of this paper [174] were entered into the 5th Princeton edition of The Meaning of Relativity, Appendix II [158].Footnote 164 In “A note on the fifth edition” dated December 1954, Einstein wrote:

“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).Footnote 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 (non-symmetric field theory) has been found. However, the thus simplified equations also cannot be examined by the facts because of mathematical difficulties”. ([160], p. 138)Footnote 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.

The paper with Kaufman ended with “Considerations of compatibility and ‘strength’ of the system of equations”, a section reappearing as the beginning of Appendix II of the 5th Princeton edition. The 16 + 64 variables \(U_{ik}^{\,\,\,l}\), ĝik must satisfy the 16 + 64 field equations (311), (312). The argument is put forward that due to λ-invariance (identification of connections with different λ) the 64 Γ-variables were reduced to 63 plus an additional identity.

“In a system with no λ-invariance, there are 64 Γ and no counterbalancing identity. This is the deeper reason for the relative weakness of systems which lack λ-invariance. 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.)

However, it is to be noted that in the 5th Princeton edition the λ-transformation 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 (α1gik + α2gki)R ik = γikR ik : “Now the point is here that gik was introduced in our paper merely as a multiplying function such as to make, together with R ik , a scalar. Hence gik 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 gik.” 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.)

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,Footnote 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 non-symmetric 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, ĝi 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.

In the section “Results in the theory” of her paper, Kaufman tried to sum up what was known about the “theory of the non-symmetric field”. Both in terms of the number of papers published until the beginning of 1955, and of researchers in UFT worldwide, she did poorly. She mentioned Schrödinger, Hlavatý, Lichnerowicz and M.-A. Tonnelat as well as one or two of their collaborators, and some work done in Canada and India. The many publications coming from Italian groups were neglected by her as well as contributions from Japan, the United States and elsewhere which she could have cited. Nevertheless, in comparison with Einstein’s habit of non-citation, her references constituted a “wealth” of material. Of the few general results obtained, Lichnerowicz’ treatment of the Cauchy initial value problem for the weak field equations of UFT and his proof that a unique solution exists seems to be the most important [369]. Unfortunately, his proof, within general relativity, that static, regular solutions behaving asymptotically like a Schwarzschild point particle (with positive mass) are locally Euclidean, could not be carried over to UFT. This was due to the complications caused by the field equations (311), (312). While (311) could be solved, in principle, for the \(U_{ik}^{\,\,\,l}\), as functions of \({{\hat g}^{ik}},\hat g_{\,\,\,\,\,,l}^{ik}\), its subsequent substitution into (312) led to equations too complicated to be solved — except in very special cases. In his summary of the conference, Pauli mocked Kaufman’s report:

“We have seen how Einstein and Mrs. Kaufman struggled heroically […], and how this fight has been led with the particular weapon of the λ-transformation. Certainly, all this is formally very correct; however, I was unable to make sense of the λ-transformations, either physically or geometrically.”[486]Footnote 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).

In a final section, B. Kaufman discussed two alternatives to what she now called “Einstein’s theory” for the first time. The first is Schrödinger’s purely affine version of the theory as presented in his book [557]. His field equations replacing (289), (290) were (cf. Section 8.1, Eq. (237):

$${\underset - K _{(ik)}} = \lambda {g_{ik}}\,,$$
(315)
$${\underset - K _{[ik],l}} + {\underset - K _{[kl],i}} + {\underset - K _{[li],k}} = \lambda ({g_{[ik],l}} + {g_{[kl],i}} + {g_{[li],k}})\,.$$
(316)

Here, λ plays the role of cosmological constant. At first, in the affine theory, g ik is defined by the l.h.s. of (315), but then this equations is read like an Einstein equation for the metric.

In a note added for the reprint in 1954 of his book, Schrödinger warned the readers of his chapter on UFT that he did not regard his unification of gravitation and electromagnetism:

“[…] as anything like a well-established 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.)

Reactions to Einstein-Kaufman

Schrödinger found the paper by Einstein and Kaufman in the Festschrift for L. de Broglie [172] “very important” and set out to draw some consequences. In particular, by using the approximation-scheme of Einstein and Kaufman, he showed that “the electric current-four-vector is in general different from zero throughout the field” ([559], p. 13). In the strong field equations, \({\underset - K _{[ik]}} = 0\) led to the vanishing of the electric current density. Dropping the so-called cosmological term for convenience, Schrödinger now wrote Eq. (316) of his “weak” field equations in the form:

$${\underset - K _{[ik]}} + {X_{[i,k]}} = 0$$
(317)

with a free vector-variable X i .Footnote 169 Besides obtaining, in first approximation, Einstein’s vacuum field equations of general relativity and one set of Maxwell’s equations, he gave as the second set:

$${\eta ^{rs}}{\partial _r}{\partial _s}{\underset 1 g_{[ik]}} - 2{X_{[i,k]}} = 0\,.$$
(318)

From this he concluded that “the curl of the current is essentially the dual of the curl of Γ i ” (his notation for X i ). Here, \({g_{ik}} = {\eta _{ik}} + {\underset 1 g _{ik}} + {\underset 2 g _{ik}} + \ldots.\).

In 2nd order, the charge-current 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 Festschrift on the occasion of de Broglie’s 60th anniversary, published only in 1953, C. F. von Weizsäcker expressed his opinion clearly that:

“[…] 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.)Footnote 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.

M. S. Mishra also studied Einstein’s last publication written together with B. Kaufman [174] and solved (311) for the connection. He obtained M.-A. Tonnelat’s result (364) of Section 10.2.3 [433]. Instead of beginning with (311) and (312) as Einstein and Kaufman had done, he then introduced “another set of field equations” by taking

$$\hat N_{\,\,\,,s}^{ik} \equiv - \hat g_{\,\,\,\,,s}^{ik} - {\hat g^{rk}}(U_{rs}^{\,\,\,i} - {1 \over 3}\delta _s^i)U_{rt}^{\,\,t} - {\hat g^{ir}}(U_{sr}^{\,\,k} - {1 \over 3}\delta _s^kU_{tr}^{\,\,t}) = 0$$
(319)
$${S_{ik}} = 0,$$
(320)

with the contracted curvature tensor which is transposition-symmetric \({S_{ik}}: = U_{ik,s}^{\,\,\,\,s} - U_{is}^{\,\,\,t}U_{tk}^{\,\,s} + {1 \over 3}U_{it}^{\,\,t}U_{rk}^{\,\,\,r}\). The solution to (319), (320) is given as:

$$\Gamma _{lm}^{\,\,\,\,n} = \{_{lm}^{\,\,\,n}\} + 2{h^{ns}}({k_{r(l}}S_{m)s}^{\,\,\,\,\,\,\,r} + {1 \over 3}{g_{s(l}}{S_{m)}}) + S_{lm}^{\,\,\,\,n}\,,$$
(321)

where \({S_m}(\Gamma)\) is the torsion tensor, and S m (Γ) the torsion vector. Mishra then linearized the metric and showed the result to be equivalent to the linearized Einstein-Straus equations (cf. Section 7.3) In the same paper, Mishra suggested another set of field equations by starting from the transposed Ricci tensor and making it transposition invariant in the same manner as Einstein and Kaufman did in their case.

In a joint paper with M. L. Abrol, also directed to the Einstein-Kaufman 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.

More exact solutions

Spherically symmetric solutions

A hope for overcoming the difficulty of relating mathematical objects from UFT to physical observables was put into the extraction of exact solutions. In simple cases, these might allow a physical interpretation by which the relevant physical quantities then could be singled out. One most simple case with high symmetry is the static spherically symmetric (sss) field. Papapetrou’s solution of Section 8.3 soon was generalized to the case v ≠ 0, w ≠ 0 by Wyman [709] which means that both, electric and magnetic fields, are now present. Wyman’s three different solutions of the weak field equations contain one arbitrary function of the radial coordinate r, each. Wyman questioned the physical interpretation of g(ij) = h ij and g[ij ] = k ij as standing for the gravitational field and the electromagnetic field, respectively. He built another expression:

$${a_{ij}}(g,L) = {g_{(ij)}} + {q_i}{q_j}\,,\,\,{q_i} = {g_{is}}{h^{st}}{u_t}/\sqrt {(1 + {1 \over 2}{k_{mn}}{k^{mn}})} \,,$$
(322)

with \({u_i} = {{{h_i}} \over {\sqrt {{h^{st}}{h_s}{h_t}}}}\) and \({h_i} = {h_{ab}}{k^{sa}}S_{si}^{\,\,\,\,b}\). If a ij is chosen as a metric, then the unique solution of general relativity in this sss case, e.i., the Schwarzschild solution, results. Although Wyman’s construction was very artificial, it clearly exemplified the unsurmountable impediment to UFT: the “embarras de richesses” in mathematical objects.

Wyman also questioned the boundary condition used at spacelike infinity: limesr→ ∞ 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: vr2 → 0 for r → ∞ or v → 0 for r → ∞.

Apparently, this left no great impression; the search for sss solutions continued. As two sets of field equations were competing against each other, Einstein’s (and Schrödinger’s) weak and strong equations (with or without cosmological constant λ), we must distinguish the solutions suggested. In the case of sss fields, only one additional field equation, e.g., R[23] = 0, separates the strong from the weak equations. For the strong equations with λ = 0, Bonnor [31] obtained the general exact solution in the cases for which either a magnetic or an electric field is present. He also generalized Papapetrou’s solution for the weak equations to the case where the function v is real or imaginary. The solutions describe spread out charges while the masses are banned into singularities. All solutions display an infinite set of “singular” surfaces between the radial coordinate r = 2m and r = ∞.Footnote 171 For some time, after a note by Bandyopadhyay [8], who claimed that for the strong equations m e = 0 where m, e are the parameters for mass and charge,Footnote 172 a discussion took place whether isolated massless magnetic monopoles could exist. Since 1948, Papapetrou and Schrödinger had changed the assignment of components of k ij to the electric and magnetic fields; now g[23], g[31], g[12] stood for the electric field [479]. Ikeda, in a paper of 1955, claimed to have shown “that a single magnetic pole cannot exist in the Einstein new theory, as in the Maxwell theory” ([298], p. 272).Footnote 173 This result depended on Ikeda’s identification of the electromagnetic field with

$${F_{ab}} = - {1 \over {\sqrt {- g}}}(\rho {k_{ab}} + {h \over 2}{\epsilon _{abrs}}{\check k^{rs}})\,,\rho = {\epsilon ^{ijlm}}{k_{ij}}{k_{lm}}\,,$$
(323)

where h = det(−h ab ). In 1960, Bandyopadhyay came back to the question and claimed that “the ‘stronger’ equations will not allow isolated magnetic poles with mass whereas the ‘weaker’ equations will allow the existence of such entities” ([10], p. 427). Bonnor’s second paper of 1952 dealt with the strong equations in the case vw ≠ 0. Again, the exact solutions described spread out charges of both signs with an infinity of singular surfaces. They were unphysical because they contained no parameter for the mass of the sources [32].

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 time-dependent 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 non-vanishing 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 time-dependent 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 time-dependent spherically symmetric field but had failed to find a time-dependent 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: “Spherically-symmetric solutions periodical in time of the “weak” field equations satisfying the boundary conditions ĝikηik for r → ∞ are, in 1st approximation, identical to solutions of the “strong”field equations” [478].

Other solutions

Still worse, in 1958 a sobering contribution from the Canadian mathematician Max WymanFootnote 174 and his German colleague Hans ZassenhausFootnote 175 cast doubt on any hope for a better understanding of the physical contents of UFT by a study of exact solutions. They investigated solutions of Einstein’s non-symmetrical UFT with vanishing curvature tensor: in this case weak and strong field equations coincide. Unlike for flat space-time in general relativity, a large class of solutions resulted; a situation which, according to the authors, “[…] merely adds to the confusion and indicates that the form of the theory is far from complete”. They went on: “However, as far as a satisfactory physical interpretation of such a theory goes, almost complete chaos seems to result.” ([710], p. 228.) Strong words, indeed, but not without reason:

“[…] for a theory based on a non-symmetric 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).

In the paper, seven distinct solutions were displayed. For one special case, i.e.,

$$d{s^2} = {h_{ij}}\,d{x^i}\,d{x^j} = [1 + {1 \over 4}({x^2} + {y^2})]\,d{t^2} - d{x^2} - d{y^2} - d{z^2}\,,\quad {F_{10}} = - {1 \over 2}x,\quad {F_{20}} = - {1 \over 2}y,$$
(324)

the standard interpretation (F ij = k ij electromagnetic field, h ij gravitational potential) “would attribute the electromagnetic field to a distribution of charge along the infinite planes y = ±x or along any of the equipotential x2y2 = constant.” For a weak gravitational field, the Newtonian potential would be \(V = {1 \over 8}({x^2} + {y^2})\) “corresponding to a distribution of mass of density \({1 \over 2}\) inside an infinite cylinder”. This is due to the approximated Newtonian equations of motion following from the geodesic equation for h ij . Hence, for this solution, mass and charge are unrelated.

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 particle-aspect 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].

Einstein did not live to see the results of Wyman & Zassenhaus; now some of the non-singular exact solutions of the field equations of UFT he so much wished to have had, were at hand:

“The big difficulty [of UFT] lies in the lack of a method for deriving singularity-free 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])Footnote 176

How would he have dealt with the fact, unearthed in 1958, that such non-singular solutions not always offered a convincing physical interpretation, or even were unphysical?

The symmetry of so-called “1-dimensional” 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, g22 = g33 and had taken it as describing an infinite charged plate [7]. In 1953, E. Clauser presented another such 1-dimensional field as a solution of the weak equations with g22g33 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].

Plane wave solutions of the weak and strong field equations of the form

$${h_{ij}}\,d{x^i}\,d{x^j} = {\eta _{ij}}\,d{x^i}\,d{x^j} - 2{(d{x^3} - d{x^0})^2}\,f({x^1},{x^2},{x^3} - {x^0})$$
(325)
$${k_{12}} = {k_{30}} = 0\,,\quad {k_{31}} = {k_{10}} = \sigma \,,\quad - {k_{23}} = {k_{20}} = \rho$$
(326)

with σ, ρ functions of x1, x2, x3x0 have been given by Takeno [601].

Interpretative problems

Already up to here, diverse assignments of geometrical objects to physical quantities (observables) were encountered. We now assemble the most common selections.

a) Gravitational Field

From the fact that the exact, statical, spherically symmetric solution of the weak field equations derived by A. Papapetrou did not coincide asymptotically, or far from the assumed location of the point source at r = 0, i.e., for r → ∞, with the corresponding solution of the Einstein-Maxwell equations (Reissner-Nordström) [475], a discussion of the relation between geometrical objects and physical observables arose. Perhaps the metric chosen to describe the gravitational potential ought not to be identified with h ik ! Let the inverse of g ik be given by

$${g^{ik}} = {l^{(ik)}} + {m^{[ik]}}\,.$$
(327)

From a study of the initial value problem, A. Lichnerowicz suggested the use of the inverse l ik = l(ik)h ik of lik as the genuine metric ([371], p. 288). Schrödinger had already worked with it. A related suggestion made by several doctoral students of M.-A. Tonnelat (J. Hély, Pham Tan Hoang, M. Lenoir) was to use a ij with \({a_{is}}{a^{ks}} = \delta _i^k\) and \({a^{ij}} = \sqrt {{h \over g}} {h^{ij}}\) as metric [250, 271, 272], ([359], p. 92). In Section 9.6.1 we have seen that by another, if only very contrived definition of the metric, complete separation of the gravitational and electromagnetic fields could be achieved: the Schwarzschild solution could be made part of an exact solution of unified field theory [709]. The torsion tensor appeared in the definition of this metric.

b) Electromagnetic Field and Charge Currents

The same ambiguity arose for the description of the electromagnetic field: Einstein’s specification that it be connected to k ij = g[ij ] was taken over by the majority ([147], p. 583). For dimensional reasons, this interpretation implies that a constant of dimension “length” will appear in the theory. In her discussion of two possibilities, St. Mavridès suggested \({a_{ij}} = \sqrt {{g \over h}} {h_{ij}}\) for the metric and \({S_{ij}} = \sqrt {{g \over h}{k_{ij}}}\) for the electromagnetic field [403, 400, 401, 404]. The first choice was supported by Pham Tan Hoang [271, 272]. Although making the electrical field regular at r = 0:

$$E = {e \over {r_0^{\,\,2}}}{1 \over {\sqrt {1 + {{{r^4}} \over {r_0^{\,\,4}}}}}}\,,$$
(328)

this choice did not fuse gravitational and electromagnetic fields any better. Mavridès’ choice was guided by a comparison with the Born-Infeld non-linear electrodynamicsFootnote 177 [42], cf. also Section 5. The fundamental invariants of the electromagnetic field now are given by h pq h rs mprmqs and \({{\sqrt {- 1}} \over {\sqrt m}}\) with l = det(l ij ), m = det(mij) ([641], p. 345); cf. Section 5. Mme. Tonnelat, in her books, also discussed in detail how to relate observables as the gravitational and electromagnetic fields, the electric current density, or the energy-momentum tensor of matter to the geometric objects available in the theory ([632], Chapter VI; [636, 382]; cf. also Section 10.2.1). For the electromagnetic field tensor, four possibilities were claimed by her to be preferable:Footnote 178 m,ik; mik; K[ik ]; ϵijklm kl . Tonnelat opted for m ik , and also for the electric current density vector \({J^i} = {1 \over {6\sqrt {- h}}}{\epsilon ^{ijkl}}({m_{[ij],k}} + {m_{[ki],j}} + {m_{[jk],i}})\). The field induction is defined via: \({\hat P^{ik}} = {{\partial {\mathcal L}} \over {\partial {m_{ik}}}}\). Schrödinger had identified the electromagnetic field with the anti-symmetrical part K[ik ] of the Ricci tensor [545, 549]; this suggestion was also made in [138, 141, 5, 93]. On the other hand, Eq. (235) can be satisfied by \({\hat g^{[ij]}} = {1 \over 2}{\epsilon ^{ijkl}}({\partial _k}{A_l} - {\partial _l}{A_k})\) such that F ij := ϵ ijkl ĝ[kl ] = i A j j A i would naturally constitute the relationship to the electromagnetic field tensor F ij [203]. In this context, the electromagnetic induction would be \({H^{ij}} \sim {1 \over 2}{\epsilon ^{ijkl}}{k_{kl}}\), and alternatively, ∼ ϵljklK kl ([650], p. 370). We learned above in (323) of Section 9.6.1 that M. Ikeda used yet another definition of the electromagnetic field tensor.

If electrical currents are to be included, the following choices for the current density were considered by Einstein, by Straus, (cf. Sections 9.2.2 and 9.3), and by others [34, 637]:

$${j_{ijk}}: = {g_{\{[ij],k\}}},$$
(329)
$${j_{ijk}}: = {\underset - K _{\{[ij],k\}}}.$$
(330)

The second choice would either violate the weak field equations or forbid any non-zero current density. These alternatives are bound to the choice for the induction. Two possibilities were discussed by Mavridès:

$${j_{ijk}}: = {k_{\{[ij],k\}}},$$
(331)
$$4\pi {j^i}: = {1 \over {\sqrt {- h}}}{\partial _l}(\sqrt {- h} {h^{ir}}{h^{ls}}{k_{rs}}).$$
(332)

In the 2nd case, Papapetrou’s spherically symmetric static solutions would not make sense, physically [407].Footnote 179 Finzi proposed yet another expression: \({j^k} = {1 \over 2}{\epsilon ^{kpqr}}{\overset 0 \nabla _p}{K_{qr}}\) ([473], p. 288). Late in his life, Einstein gave \(\hat g_{\,\,\,\,\,\,\,\,\,,s}^{[is]}\) the interpretation of magnetic current density [158].

An ambiguity always present is the assignment of the electric and magnetic fields to the components ki0 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 homoth-etic 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.

In a paper falling outside of the period of this review, H.-J. TrederFootnote 180 suggested to also geometrize spinorial degrees of freedom by including them in the asymmetric metric; it took the form [651, 67]:

$${g_{ij}} = h_i^Ah_j^B{\eta _{AB}} + {1 \over 2}iC(\gamma _{ij}^{\quad \alpha \beta}{\psi _\alpha}{\psi _\beta} + \gamma _{ij}^{\quad \dot \alpha \dot \beta}{\psi _{\dot \alpha}}{\psi _{\dot \beta}})\,,$$

where \(h_i^A\) are tetrad components, ψ α a 2-component Weyl-spinor, C a constant (with dimension) and \(\gamma _{ij}^{\,\,\,\,\,\,\alpha \beta} = {\sigma _i}^\alpha {}_{\dot \mu}\sigma _j^\beta {}_{\dot v}{\gamma ^{\dot \mu \dot v}}.\sigma _i^{\,\,\alpha}{}_{\dot \mu}\) are Pauli-matrices, \({\gamma ^{\dot \mu \dot v}}\) corresponds to the antisymmetric 2 × 2 symbol ϵαβ.

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 (Einstein-Maxwell 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.

c) Matter tensor

In Einstein’s understanding of UFT, the matter tensor for a continuous matter distribution should also become part of geometry. Again a precise attribution to geometrical objects could not be found. One way of approaching the problem was to reshape part of the field equations into the form of the old or a newly defined “Einstein tensor”, and terms left over. These then were declared to constitute the “matter tensor”. In her book, M.-A. Tonnelat discussed this problem in detail ([632], Chapter VII, A, pp. 109–117); cf. also Section 10.3.1. As late as 1963, Schrödinger could write:

“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).

The role of additional symmetries

The symmetries Einstein had introduced, i.e., transposition invariance and λ-transformations, played a major role in versions of UFT, but not in physics, in general. There are only a few papers with these symmetries as their topic. J. Winogradzki investigated the relationship of the results in Einstein’s final approach to UFT (made together with B. Kaufman) to the theorems by Emmy Noether [704, 703]. She called invariance with regard to the group U composed of coordinate- and λ-transformations which had been named the “extended” group by Einstein, U- or J-invariance:

$$J({g_{ik}}) = {g_{ik}}\,,\,\,J(L_{ij}^{\,\,\,\,\,\,k}) = L_{ij}^{\,\,\,\,\,\,k} + \Omega _{ik}^{\,\,\,\,\,\,k},$$
(333)

with \(\Omega _{ik}^{\,\,\,\,\,\,k}\) being independent of g ij . As a necessary condition for the field equations following from a variational principle to be J-invariant, she derived \(\Omega _{ik}^{\,\,\,\,\,\,k} = \delta _i^k{\lambda _j}\). Thus, with the help of four axioms postulated by her, she arrived at the λ-transformations (52).

According to the 2nd theorem by E. Noether, λ-invariance leads to four identities which were written out by Winogradzki for a Lagrangian density  (her “Hamiltonian”):

$$\delta _k^l{{\delta \hat A} \over {\delta \Gamma _{km}^{\,\,\,\,l}}} - 2\hat g_{\,\,\,\,\,\,,s}^{[ms]} = 0\,.$$
(334)

Equation (334) relates \(\hat g_{\,\,\,\,\,\,\,\,,s}^{[is]} = 0\) and the 16 field equations which are not invariant under λ-transformations.

P. G. Bergmann also discussed Einstein’s λ-transformations, 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).

Einstein-Schrödinger 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 Marie-Antoinette Tonnelat. It followed two main lines: the affine or metric-affine approaches of Einstein and Schrödinger, and the 5-dimensional 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 (“Jordan-Thiry-theory”), and by students of M.-A. Tonnelat; the first topic, “Einstein-Schrödinger 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.

Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory

We first make contact with Marie-Antoinette TonnelatFootnote 181 and her research group in the Institut Henri Poincaré. She had studied with Louis de Broglie. During the German occupation of Paris she continued to work with de Broglie and on her own in the field of (relativistic) “spin-particles”, also under the influence of a gravitational field. She applied de Broglie’s “méthode de fusion” to massive spin 1 particles (called photons) in order to arrive at particles with maximal spin 2: spin 2 corresponded to the graviton. The theory contained the graviton, three photons and two scalar particles (spin 0) all with non-vanishing mass. Each relevant field component satisfied the Klein Gordon equation. From there, she arrived at Maxwell’s equations and the linearized version of the equation for Einstein spaces R ij = λg ij . She carefully looked at the theory for a particle with spin 2 as a “unitary theory” and preferred to call it “a unitary formalism” ([616], p. 163, 164):

“[…] the theory of maximal spin 2 allows to show how a unitary theory presents itself, approximately, but in the framework of wave mechanics.”Footnote 182

In this paper and in others in the early 1940s she also wrote down the standard commutation relations for the quantized spin-2 field [611, 616].

Many of her papers were published in the proceedings of the prestigious Academy of Sciences in ParisFootnote 183 [613, 612, 614, 610, 611, 615, 616, 617]. The Academy’s sessions had been interrupted for a mere three weeks due to the German occupation. According to its president:

“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.”Footnote 184

Right after the war in 1946, like other young French scientists, Mme. Tonnelat apparently spent some time in Dublin with the group of E. Schrödinger. On the background of her previous work on a “unitary formalism” emerging from spin-2 particles, her interest in the unified field theories of Einstein and Schrödinger might have been brought forward during this stay with Schrödinger. Her scientific teacher, L. de Broglie, supported her research in UFT, although he himself stayed away from it. After briefly listing “innumerable attempts […] to complete the general theory of relativity […] and transform it into a ‘unified theory’ ”, he went on to say:

“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 bo