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 where are coordinates of space-time, is an additional parameter (a gauge variable) and a constant named “index”.40 transforms as . The auxiliary 5-dimensional space appearing has no physical significance. A projective symmetric metric of index was given by where is an arbitrary projective scalar of index . In addition, a third symmetric tensor , the gravitational metric, appeared. Here, is a projective vector. Likewise, the Levi-Civita connections with with and were used. For arbitrary index , the field equations were derived from the curvature scalar calculated from the connection . One equation could be written in the form of a wave equation:, abstract). How Planck’s constant might be brought in, is left in the dark.
During the 1940s, meson physics became fashionable. Of course, the overwhelming amount of this research happened in connection with nuclear and elementary particle theory, outside of UFT, but sometimes also in classical field theory. Cf. the papers by F. J. Belinfante on the meson field, in which he used the undor-formalism41 [16, 15]. In his doctoral thesis of 1941, “Projective theory of meson fields and electromagnetic properties of atomic nuclei” suggested by L. Rosenfeld, Abraham Pais in Utrecht kept away from UFT and calculated the projective energy momentum tensor of an arbitrary field. Although citing the paper of Veblen and Hoffmann, in projective theory he followed the formalism of Pauli; in his application to the Dirac spinor-field, he used Belinfantes undors . After this paper, he examined which of Kemmer’s five types of meson fields were “in accordance with the requirements of projective relativity” (, 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 was defined by Hoffmann via: with and given above. Its space-time components form an affine vector from which the vector meson field tensor 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, 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 .42 It turned out that a 6-dimensional auxiliary space was needed. We shall denote the coordinates in this by . The components of a similarity tensor are , where are the number of covariant and contravariant coordinate indices while again is named the index of the tensor. In place of the transformations in projective geometry, now, p. 1045).
It was Pascual Jordan who in physics re-applied projective geometry (cf. Section 220.127.116.11 of Part I) by showing that the transformation group of the 4-potential in electrodynamics, composed of the gauge transformations[316*]: 43 Equivalently, the new coordinates are homogeneous functions of degree 1 of the old and transform like a vector: Jordan defined projector-components to transform under (106*) like tensor-components which are homogeneous functions of degree in the . Thus, itself is a projector just as the Minkowski (Euclidean) metric of with the invariant:
Pauli had browsed in Ludwig’s book and now distanced himself from his own papers on projective relativity of 1933 discussed briefly in Section 7.2.4 of Part I.44 He felt deceived:
“The deception consists in the belief that by the projective form, i.e., the homogeneous coordinates, the shortcomings of Kaluza’s formulation have been repaired, and that one has achieved something beyond Kaluza. At the time, in 1933, I did not know explicitly the transition from Kaluza to the projective form (as in [20*]); it is too simple and banal to the effect that the factual contents of both equivalent formulations could be somehow different.” (letter of W. P. to P. Jordan, [490*], p. 735):45
Toward the end of the second world war, Kaluza’s five- dimensional theory and projective relativity emerged once again as vehicles for a new physical theory which, much later, came to be known as “scalar-tensor theory of gravitation.”46 Cosmological considerations related to the origin of stars seem to have played the major role for the building of a theory by P. Jordan in which the gravitational constant is thought to be varying in (cosmological) time and thus replaced by a scalar function 47. The theory nicely fit with Dirac’s “large number hypothesis” [122, 123]. The fifteenth field variable in Kaluza’s theory was identified by Jordan with this function, or in projective relativity, with the scalar: by setting [321*]. In space-time, the field equations for the gravitational field , the electromagnetic 4-potential , and the -variable were derived by Jordan and Müller48 to be:Jordan & Müller denoted the Ricci-tensor in space-time by . 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:
It may be that at the time, they just did not have an idea for a physical interpretation like the one suggested by P. Jordan. Although there were reasons for studying the theory further, Bergmann pointed out that there is an “embarras de richesses” in the theory: too many constructive possibilities for a Lagrangian. Nonetheless, in his subsequent paper on “five-dimensional cosmology”, P. Jordan first stuck to the simplest Lagrangian, i.e. to the Ricci scalar in five dimensions [318*]. In this paper, Jordan also made a general comment on attempts within unitary field theory of the Einstein–Schrödinger-type to embed corpuscular matter into classical field theory (cf. chapter 6 with Section 6.1.1 below):
“The problem of the structure of matter can only be attacked as a problem in quantum mechanics; nevertheless, investigations of the singularities of solutions of the field equations retain considerable importance in this framework. […] the wave functions of matter must be taken into account. Whether this program can be carried through, and to which extent, in the sense of an extension of geometry (to which Schrödinger’s ideas related to the meson field seem to provide an important beginning) is such a widespread question […]”.49 (, 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*)]:[319*], p. 139): [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 equally well with arbitrary function could serve as a metric.50 This conformal invariance of the theory is preserved in the case that an electromagnetic field forms the matter tensor. A problem for the interpretation of mathematical objects as physical variables results: by a suitable choice of the conformal factor , 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 from looking at Jordan’s field equations. His Lagrangian corresponded to with . On the other hand, in the MKSA system of physical units, the fine-structure constant is . Thus, the fine-structure constant would depend on .
Neither Pauli nor Fierz seem to have known that the mathematician Willy Scherrer at the university of Bern had suggested scalar-tensor theory already in 1941 before P. Jordan, and without alluding to Kaluza or Pauli’s projective formulation.53 Also, in 1949, Scherrer had suggested a more general Lagrangian :. The ensuing field equations correspond to those following from Jordan’s Lagrangian if his parameter ([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 () and Heckmann, Jordan & Fricke () .
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 . Pauli was loath to get involved and asked the editor of this journal, the very same M. Fierz, what the most appropriate answer to Scherrer could be. He also commented:
“Because according to my opinion all “unified field theories” are based on dubious ideas – in particular it is a typically suspect idea of the great masters Einstein and Schrödinger to add up the symmetric and antisymmetric parts of a tensor – I have to pose the question […].” (W. Pauli to M. Fierz, 15 Dec. 1953) ([491*], p. 390–391)54
Scherrer’s paper eventually was published in Zeitschrift für Physik . 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 . For studies of Kaluza’s theory in Paris (Jordan–Thiry theory) cf. Section 11.1.
As described in Section 6.3 of Part I, since 1927 Einstein and again in 1931 Einstein and Mayer55, within a calculus using 5-component tensorial objects in space-time, had studied Kaluza’s approach to a unification of gravitation and electromagnetism in a formal 5-dimensional space with Lorentz-signature. A decade later, Einstein returned to this topic in collaboration with his assistant Peter Bergmann [167*, 166]. The last two chapters of Bergmann’s book on relativity theory are devoted to Kaluza’s theory and its generalization ([20*]. Einstein wrote a foreword, in which he did not comment on “Kaluza’s unified field theory” as the theory is listed in the book’s index. He admitted that general relativity “[…] has contributed little to atomic physics and our understanding of quantum phenomena.” He hoped, however, that some of its features as were “general covariance of the laws of nature and their nonlinearity” could contribute to “overcome the difficulties encountered at present in the theory of atomic and nuclear processes” ([20*] p. V). Two years before Bergmann’s book appeared, Einstein already had made up his mind against the five-dimensional approach:
“The striving for most possible simplicity of the foundations of the theory has prompted several attempts at joining the gravitational field and the electromagnetic field from a unitary, formal point of view. Here, in particular, the five-dimensional theory of Kaluza and Klein must be mentioned. Yet, after careful consideration of this possibility, I think it more proper to accept the mentioned lack of inner unity, because it seems to me that the embodiment of the hypotheses underlying the five-dimensional theory contains no less arbitrariness than the original theory.”56
Nonetheless, in their new approach, Einstein and Bergmann claimed to ascribe “physical reality to the fifth dimension whereas in Kaluza’s theory this fifth dimension was introduced only in order to obtain new components of the metric tensor representing the electromagnetic field” (, 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 .57 With the fifth dimension being compact, this lessened the need for a physical interpretation of its empirical meaning. Now, the authors partially removed Kaluza’s ‘cylinder condition’ (cf. Section 4.2 of Part I, Eq. (109)): they set , but assumed and : the electrodynamic 4-potential remains independent of . 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 , Einstein & Bergmann list three further quadratic invariants: where . The ensuing field equations for the fourteen variables and 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 , i.e., the
hypersurface cannot be generated by the set of tangent spaces in each point. Besides the metric of
space-time , a non-integrable differential form
defining the hypersurface was introduced together with the additional assumption . The path of
a particle with charge , mass and 5-vector was chosen to be a
geodesic tangent to the non-holonomic hypersurface. Thus , and Vranceanu then took
. The electromagnetic field was defined as . 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
be the sum of two tensors one of which is due to the electromagnetic field […]”. ([681*],
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 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).” (,
More than a decade later, K. Yano and M. Ohgane generalized the non-holonomic UFT to arbitray
dimension: -dimensional space is a non-holonomic hypersurface of -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” (, 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
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 . For a detailed discussion of Klein’s contribution cf. .
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:
At first he was very much fascinated by the renewed approach to unified field theory by way of Kaluza’s idea. We learn this from the letter of 8 August 1938 to his friend Besso:
“After twenty years of vain searching, this year now I have found a promising field theory which is a quite natural sequel to the relativistic gravitational theory. It is in line with Kaluza’s idea about the essence of the electromagnetic field.” ([163*], p. 321)61
In 1943, Einstein had come to the conclusion that the failure of “finding a unified theory of the physical field by some generalization of the relativistic theory of gravitation” seemed to require “a decisive modification of the fundamental concepts” ([165*], p. 1). He wanted to keep the four-dimensional space-time continuum and the diffeomorphism group as the covariance group, but wished to replace the Riemannian metric by a generalized concept. Together with the assistant at the Institute for Advanced Studies, Valentine Bargmann , he set out to develop a new scheme involving “bi-vector fields”. Unlike the concept of a bi-vector used by Schouten in 1924 (, 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 is defined viafirst order derivatives like the tensorial quantity: . In order to cut down on this wealth, a new operation called “rimming” was introduced which correlated a new “bi-vector” with by multiplying it from left and right by tensors of full rank where each is taken from one of the two manifolds (now Greek indices refer to the two different points)62: 118*), (119*) remain invariant under rimming, the relations , 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”:122*) leads to a scalar equation: ).
In another paragraph, the authors returned to the “single space”-version. Here, a symmetry condition is demanded: . Now, the tensorial p-point equations admit the special case that the two points (coordinates ) coincide. By a suitable rimming operation was reached where is the matrix diag . The rimming operations were performed with representations of the Lorentz group. Also, mixed bi-vectors were introduced and a tensorial three-point equation had to be satisfied:
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 is interpreted as the (non-infinitesimal) affine connection because the relation63 Two conditions are to be fulfilled: the displacement of a vector from to 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 125*) and (126*) are invariant with respect to rimming. Again a tensorial three-point equation was written down which, in matrix form, reads as: 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: Bargmann have meanwhile succeeded in proving that (129*), too, admits only “flat space” solutions”(, p. 23). He slightly changed the equation into:
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” (, p. 143).
He rewrote Einstein’s mixed bi-vector with the help of the tetrads used in teleparallel geometry:127*) to the curvature tensor: the three-point matrices of a distant affinity perform a transfer from the point via back to the starting point . This is analogous to parallel transfer around a triangle formed by the points ; 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 (, 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 . 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 .
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 -space has an ’inverse image’ in the -space, in the first instance the laws of relativity” (, p. 327). In this note in Nature, Born added a Lorentzian metric 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 alone, but may depend on 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” (, 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. ) 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 ) depended on “two sets of real variables” (in the same space) [719, 720, 721]. According to Yukawa: “ coincide with the ordinary coordinates in the limit of local field, so that the dependence on describes the asymptotic behavior and the dependence on characterizes the internal structure or inertial motion” (, 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.” (, 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.” (, 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 […]. All amounts to replace your fundamental invariant by the symmetrical quantity where . is an operator, the integer eigenvalues of which are the distances […].” ([168*], p. 242.)64
This story shows that a very loose kinship existed between the Einstein–Bargmann “bi-vector” method and Yukawa’s non-local field theory with Born’s reciprocity theory in the middle. Although some of those involved were in direct personal contact, no concrete evidence for a conscious transfer of ideas could be established.