7.2 Dirac’s electron with spin, Einstein’s teleparallelism, and Kaluza’s fifth dimension

In the same year 1928 in which Einstein published his theory of distant parallelism, Dirac presented his relativistic, spinorial wave equation for the electron with spin. This event gave new hope to those trying to include the electron field into a unified field theory; it induced a flood of papers in 1929 such that this year became the zenith for publications on unified field theory. Although we will first look at papers which gave a general relativistic formulation of Dirac’s equation without having recourse to a geometry with distant parallelism, Tetrode’s paper seems to be the only one not influenced by Einstein’s work with this geometry (cf. Section 6.4.5). Although, as we noted in Section 6.4.1, the technique of using n-beins (tetrads) had been developed by mathematicians before Einstein applied it, it may well have been that it became known to physicists through his work. Both Kaluza’s five-dimensional space and four-dimensional projective geometry were also applied in the general relativistic formulation of Dirac’s equation.

7.2.1 Spinors

This is a very sketchy outline with a focus on the relationship to unified field theories. An interesting study into the details of the introduction of local spinor structures by Weyl and Fock and of the early history of the general relativistic Dirac equation was given recently by Scholz [291].

For some time, the new concept of spinorial wave function stayed unfamiliar to many physicists deeply entrenched in the customary tensorial formulation of their equations223. For example, J. M. Whittaker was convinced that Dirac’s theory for the electron

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

Some early nomenclature reflects this unfamiliarity with spinors. For the 4-component spinors or Dirac-spinors (cf. Section 2.1.5) the name “half-vectors” coined by Landau was in use224. Podolsky even purported to show that it was unnecessary to employ this concept of “half-vector” if general curvilinear coordinates are used [259Jump To The Next Citation Point]. Although van der Waerden had written on spinor analysis as early as 1929 [368Jump To The Next Citation Point] and Weyl’s [407Jump To The Next Citation Point408Jump To The Next Citation Point], Fock’s [133Jump To The Next Citation Point131Jump To The Next Citation Point], and Schouten’s [306Jump To The Next Citation Point] treatments in the context of the general relativistic Dirac equation were available, it seems that only with van der Waerden’s book [369], Schrödinger’s and Bargmann’s papers of 1932 [319Jump To The Next Citation Point6Jump To The Next Citation Point], and the publication of Infeld and van der Waerden one year later [167Jump To The Next Citation Point] a better knowledge of the new representations of the Lorentz group spread out. Ehrenfest, in 1932, still complained225:

“Yet still a thin booklet is missing from which one could leasurely learn spinor- and tensor-calculus combined.”226View original Quote ([68], p. 558)

In 1933, three publications of the mathematician Veblen in Princeton on spinors added to the development. He considered his first note on 2-spinors “a sort of geometric commentary on the paper of Weyl” [378Jump To The Next Citation Point]. Veblen had studied Weyl’s, Fock’s, and Schouten’s papers, and now introduced a “spinor connection of the first kind” ΛAB α, α = 1,...,4, with the usual transformation law under the linear transformation ψ¯A = TDAψD (A, D = 1,2) changing the spin frame:

( A ) ¯ΛC = ΛA tB + ∂tD- T C, (188 ) Dα B α D ∂x α A
where B tD is the inverse matrix − 1 T. A TD corresponds to A A B of Equation (75View Equation); however, the transformation need not be unimodular. Thus, Veblen took up Schouten’s concept of “spin density” [306Jump To The Next Citation Point] by considering quantities transforming like ψ¯A = tN T ADψD (A, D = 1,2), with t := dettBD. Then, the covariant derivative of a spinor of weight N, ψA is considered; the expression227
∂ψA ---α-+ ΛABαψB − N ΛBB αψA = ∇α ψA (189 ) ∂x
gives “the components of a geometric object which transforms like those of a spinor of weight N with respect to the index A and like those of a projective tensor with respect to α”. In his first paper [378], a further generalisation is introduced including “gauge-transformations” in an additional variable 0 x: ¯A kx0 A ψ = e f (A, D = 1,2), with the gauge transformations
0 0′ k′ k k l′ x = x − log ρ(x ), x = x (x ), (190 )
k is called the “index” of the spinor. In order to deal with 4-spinors Veblen considered a complex projective 3-space and defined 6 real homogeneous coordinates σ X, with σ = 0,...,5, through Hermitian forms of the 4-spinor components. The subspace 0 X = 0, 5 X = 0 of the quadratic (X0 )2 + (X1 )2 + (X2 )2 + (X3 )2 − (X4)2 − (X5 )2 = 0 is then tangent to the Minkowski light cone ([377], p. 515).

Veblen imbedded spinors into his projective geometry [380Jump To The Next Citation Point]:

“[...] The components of still other objects, the spinors, remain partially indeterminate after coordinates and gauge are fixed and become completely determinate only when the spin frame is specified. There are several ways of embodying this invariant theory in a formal calculus. The one which is here employed has its antecedents chiefly in the work of Weyl, van der Waerden, Fock, and Schouten. It differs from the calculus arrived at by Schouten chiefly in the treatment of gauge invariance, Schouten (in collaboration with van Dantzig) having preferred to rewrite the projective relativity in a formalism obtainable from the original one by a sort of coordinate transformation, whereas I think the original form fits in better with the classical notations of relativity theory. [...] The theory of spinors is more general than the projective relativity and is reduced to the latter by the specification of certain fundamental spinors. These spinors have been recognised by several students (Pauli and Solomon, Fock) of the subject but their role has probably not been fully understood since it has quite recently been thought necessary to give special proofs of invariance.” [380]

The transformation law for spinors is the same as before228:

0 ψ¯= ekxtN TABψB, A,B = 1,...,4. (191 )
In part, he also takes over van der Waerden’s notation (dotted indices.) As to Veblen’s papers on 2- and 4-spinors, my impression is that, beyond a more detailed presentation, alas with a less transparent notation, they do not really bring a pronounced advance with regard to Weyl’s, Fock’s, van der Waerden’s, and Pauli’s publications (cf. Sections 7.2.2 and 7.2.3). Veblen himself had a different opinion; for him the homogeneous coordinates used by Pauli seemed “to make things more complicated” (cf. the paragraphs on projective geometry in Section 2.1.3). Veblen’s inhomogeneous coordinates i x, (i = 1,2,3,4) and the homogeneous coordinates μ X, with μ = 0, ...,4, are connected by
X0 = exp (x0 ), xi = exp(x0xi). (192 )
According to Veblen,

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

7.2.2 General relativistic Dirac equation and unified field theory

After Tetrode and Wigner, whose contributions were mentioned in Section 6.4.5, Weyl also gave a general relativistic formulation of Dirac’s equation. He gave up his original idea of coupling electromagnetism to gravitation and transferred it to the coupling of the electromagnetic field to the matter (electron-) field: In order to keep quantum mechanical equations like Dirac’s gauge invariant, the wave function had to be multiplied by a phase factor [407Jump To The Next Citation Point408Jump To The Next Citation Point]. Actually, Weyl had expressed the change in his outlook, so important for the idea of gauge-symmetry in modern physics ([424], pp. 13–19), already in 1928 in his book on group theory and quantum mechanics ([406Jump To The Next Citation Point], pp. 87–88). We have noted before his refutation of distant parallelism (cf. Section 6.4.4). In his papers, Weyl used a 2-spinor formalism and a tetrad notation different from Einstein’s and Levi-Civita’s: He wrote ep(ˆk ) in place of hpˆk, and o(l;kj) for the Ricci rotation coefficients γjkl; this did not ease the reading of his paper. He partly agreed with what Einstein imagined:

“It is natural to expect that one of the two pairs of components of Dirac’s quantity belongs to the electron, the other to the proton.”229View original Quote

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

“For every attempt at establishing the quantum-theoretical field equations, one must not lose sight [of the fact] that they cannot be tested empirically, but that they provide, only after their quantization, the basis for statistical assertions concerning the behaviour of material particles and light quanta.”230View original Quote ([407Jump To The Next Citation Point], p. 332)

For many years, Weyl had given the statistical approach in the formulation of physical laws an important role. He therefore could adapt easily to the Born–Jordan–Heisenberg statistical interpretation of the quantum state. For Weyl and statistics, cf. Section V of Sigurdsson’s dissertation ([326Jump To The Next Citation Point], pp. 180–192).

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

“In the past two decades, endeavours have been made repeatedly to connect physical laws with geometrical concepts. In the field of gravitation and of classical mechanics, such endeavours have found their fullest accomplishment in Einstein’s general relativity. Up to now, quantum mechanics has not found its place in this geometrical picture; attempts in this direction (Klein, Fock) were unsuccessful. Only after Dirac had constructed his equations for the electron, the ground seems to have been prepared for further work in this direction.”231View original Quote ([135Jump To The Next Citation Point], p. 798)

In another paper [134Jump To The Next Citation Point], Fock and Ivanenko took a first step towards showing that Dirac’s equation can also be written in a generally covariant form. To this end, the matrix-valued linear form k ds = γkdx (summation over k = 1,...,4) was introduced and interpreted as the distance between two points “in a space with four continuous and one discontinuous dimensions”; the discrete variable took only the integer values 1, 2, 3, and 4. Then the operator-valued vectorial quantity γkuk with the vectorial operator uk and its derivative ds = γ vk dτ k immediately led to Dirac’s equation by replacing v k by 1( h--∂- + eA ) m 2πi∂xk c k, where Ak is the electromagnetic 4-potential, by also assuming the velocity of light c to be the classical average of the “4-velocity” vk, and by applying the operator to the wave function ψ. In the next step, instead of the Dirac γ-matrices with constant entries γ(l0), the coordinate-dependent bein-components γ := h lγ(0) ˆk ˆk l are defined; ds2 then gives the orthonormality relations of the 4-beins.

In a subsequent note in the Reports of the Parisian Academy, Fock and Ivanenko introduced Dirac’s 4-spinors under Landau’s name “half vector” and defined their parallel transport with the help of Ricci’s coefficients. In modern parlance, by introducing a covariant derivative for the spinors, they in fact already obtained the “gauge-covariant” derivative ∇ ψ := (-∂-− 2πieA )ψ k ∂xk h c k. Thus δψ = 2πieA dxk ψ h c k is interpreted in the sense of Weyl:

“Thus, it is in the law for the transport of a half-vector that Weyl’s differential linear form must appear.”232View original Quote ([134Jump To The Next Citation Point], p. 1469)

In order that gauge-invariance results, ψ must transform with a factor of norm 1, innocuous for observation, i.e., ψ → exp(i2πe σ) h c if A → A + ∂σ- k k ∂xk. Another note and extended presentations in both a French and a German physics journal by Fock alone followed suit [133Jump To The Next Citation Point131Jump To The Next Citation Point132Jump To The Next Citation Point]. In the first paper Fock defined an asymmetric matter tensor for the spinor field,

[ ( ) ] Tj = ch-- ¯ψγj ∂ψ--− Γ ψ − 1∇ (¯ψγjψ ) , (193 ) k 2πi ∂xk k 2 k
where ∑ Γ k = ˆleˆlαˆlhkˆlCˆl is related to the matrix-valued spin connection in the expression for the parallel transport of a half-vector ψ:
∑ δψ = eˆlCˆldsˆlψ. (194 ) ˆ l
The covariant derivative then is Dk = ∂∂xk − Γ k. Fock made clear that the covariant formulation of Dirac’s equation did not need the special geometry of Einstein’s theory of distant parallelism:

“By help of the concept of parallel transport of a half-vector, Dirac’s equations will be written in a generally invariant form. [...] The appearance of the 4-potential ϕl besides the Ricci-coefficients γikl in the expression for parallel transport, on the one hand provides a simple reason for the emergence of the term pl − eϕl c in the wave equation and, on the other, shows that the potentials ϕ l have a place of their own in the geometrical world-view, contrary to Einstein’s opinion; they need not be functions of the γikl.”233View original Quote ([131Jump To The Next Citation Point], p. 261, Abstract)

For his calculations, Fock used Eisenhart’s book [119] and “the excellent collection of the most important formulas and facts in the paper of Levi-Civita” [207]. Again, Weyl’s “principle of gauge invariance” as formulated in Weyl’s book of 1928 [406] is mentioned, and Fock stressed that he had found this principle independently and earlier234:

“The appearance of Weyl’s differential form in the law for parallel transport of a half vector connects intimately to the fact, observed by the author and also by Weyl (l.c.), that addition of a gradient to the 4-potential corresponds to multiplication of the ψ-function with a factor of modulus 1.”235View original Quote ([130], p. 266)

The divergence of the complex energy-momentum tensor i i i W k = T k + iU k satisfies

j j hc ∇jT k = eJlFlk, ∇jU k = ---JlRlk, (195 ) 4π
with the electromagnetic field tensor Fik, the Ricci tensor Rik, and the Dirac current J k. The French version of the paper preceded the German “completed presentation”; in it Fock had noted:

“The 4-potential finds its place in Riemannian geometry, and there exists no reason for generalising it (Weyl, 1918), or for introducing distant parallelism (Einstein 1928). In this point, our theory, developed independently, agrees with the new theory by H. Weyl expounded in his memoir ‘gravitation and the electron’.”236View original Quote ([132Jump To The Next Citation Point], p. 405)

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

“The main subject of this paper is ‘Dirac’s difficulty’237. Nevertheless, it seems to us that the theory suggested by Weyl for solving this problem is open to grave objections; a criticism of this theory is given in our article.”238View original Quote

Weyl’s paper is seminal for the further development of the gauge idea [407Jump To The Next Citation Point].

Although Fock had cleared up the generally covariant formulation of Dirac’s equation, and had tried to propagate his results by reporting on them at the conference in Charkow in May 1929239 [169], further papers were written. Thus, Reichenbächer, in two papers on “a wave-mechanical 2-component theory” believed that he had found a method different from Weyl’s for obtaining Dirac’s equation in a gravitational field. As was often the case with Reichenbächer’s work, after longwinded calculations a less than transparent result emerged. His mass term contained a square root, i.e., a ± two-valuedness, which, in principle, might have been instrumental for helping to explain the mass difference of proton and electron. As he remarked, the chances for this were minimal, however [277278].

In two papers, Zaycoff (of Sofia) presented a unified field theory of gravitation, electromagnetism and the Dirac field for which he left behind the framework of a theory with distant parallelism used by him in other papers. By varying his Lagrangian with respect to the 4-beins, the electromagnetic potential, the Dirac wave function and its complex-conjugate, he obtained the 20 field equations for gravitation (of second order in the 4-bein variables, assuming the role of the gravitational potentials) and the electromagnetic field (of second order in the 4-potential), and 8 equations of first order in the Dirac wave function and the electromagnetic 4-potential, corresponding to the generalised Dirac equation and its complex conjugate [426427].

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

“Recently, repeated attempts have been made to raise the number of dimensions of the world in order to explain its strange lawfulness (H. Mandel, G. Rumer, the author et al.). No doubt, there are weighty reasons for such a seemingly paradoxical view. For it is impossible to represent Poincaré’s pressure of the electron within the normal space-time scheme. However, the introduction of such metaphysical elements is in gross contradiction with space-time causality, although we may doubt in causality in the usual sense due to Heisenberg’s uncertainty relations. A multi-dimensional causality cannot be understood as long as we are unable to give the extra dimensions an intuitive meaning.”240View original Quote [433]

Rumer’s paper is [285Jump To The Next Citation Point] (cf. Section 8). In the paper, Zaycoff introduced a six-dimensional manifold with local coordinates x0,...,x5 where x0,x5 belong to the additional dimensions. His local 6-bein comprises, besides the 4-bein, four electromagnetic potentials and a further one called “eigen-potential” of the electromagnetic field. As he used a “sharpened cylinder condition, ” no further scalar field is taken into account. For 0 x to 4 x he used the subgroup of coordinate transformations given in Klein’s approach, augmented by ′ x5 = x5.

Schouten seemingly became interested in Dirac’s equation through Weyl’s publications. He wrote two papers, one concerned with the four-dimensional and a second one with the five-dimensional approach [306Jump To The Next Citation Point307Jump To The Next Citation Point]. They resulted from lectures Schouten had given at the Massachusetts Institute of Technology from October to December 1930 and at Princeton University from January to March 1931; Weyl’s paper referred to is in Zeitschrift für Physik [407Jump To The Next Citation Point]. Schouten relied on his particular representation of the Lorentz group in a complex space, which later attracted Schrödinger’s criticism. [305]. His comment on Fock’s paper [131Jump To The Next Citation Point] is241:

Fock has tried to make use of the indetermination of the displacement of spin-vectors to introduce the electromagnetic vector potential. However the displacement of contravariant tensor-densities of weight + 12 being wholly determined and only these vector-densities playing a role, the idea of Weyl of replacing the potential vector by pseudo-vectors of class +1 and − 1 seems much better.” ([306Jump To The Next Citation Point], p. 261, footnote 19)

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

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

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

“The joining of Dirac’s theory of the electron with general relativity has been undertaken repeatedly, such as by Wigner [419], Tetrode [344], Fock [131Jump To The Next Citation Point], Weyl [407Jump To The Next Citation Point408Jump To The Next Citation Point], Zaycoff [434Jump To The Next Citation Point], Podolsky [259]. Most authors introduce an orthogonal frame of axes at every event, and, relative to it, numerically specialised Dirac-matrices. This procedure makes it a little difficult to find out whether Einstein’s idea concerning teleparallelism, to which [authors] sometimes refer, really plays a role, or whether there is no dependence on it. To me, a fundamental advantage seems to be that the entire formalism can be built up by pure operator calculus, without consideration of the ψ-function.”242View original Quote ([319Jump To The Next Citation Point], p. 105)

The γ-matrices were taken by Schrödinger such that their covariant derivative vanished, i.e., γl∥m = ∂∂xγml− Γ rlm(g)γr + γlΓ m − γm Γ l = 0, where Γ l is the spin-connection introduced by ψ ∥l =-∂ψl − Γ lψ ∂x. Schrödinger took γ0, γi, with i = 1,2,3, as Hermitian matrices. He introduced tensor-operators Tik lm such that the inner product ψ∗γ T ikψ 0 lm instead of ψ ∗T ikψ lm stayed real under a “complemented point-substitution”.

In the course of his calculations, Schrödinger obtained the wave equation

1 √ -- R 1 √---∇k g gkl∇l − -- − --fklskl = μ2, (196 ) g 4 2
where μ = 2πmc ∕h, fkl is the electromagnetic field tensor, and skl := 1γ[kγl] 2 with the γ-matrices γk, i.e., the spin tensor. As to the term with the curvature scalar R, Schrödinger was startled:

“To me, the second term seems to be of considerable theoretical interest. To be sure, it is much too small by many powers of ten in order to replace, say, the term on the r.h.s. For μ is the reciprocal Compton length, about 11 − 1 10 cm. Yet it appears important that in the generalised theory a term is encountered at all which is equivalent to the enigmatic mass term.”243View original Quote ([319Jump To The Next Citation Point], p. 128)

The coefficient 1 − 4 in front of the Ricci scalar in Schrödinger’s (Klein–Gordon) wave equation differs from the 16 needed for a conformally invariant version of the scalar wave equation244 (cf. [257], p. 395).

Bargmann in his approach, unlike Schrödinger, did not couple “point-substitutions [linear coordinate transformations] and similarity transformations [in spin space]”[6Jump To The Next Citation Point]. He introduced a matrix α with † α + α = 0 such that l † l (αγ ) = (α γ ), with l = 0,...,3.

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

“Your fundamental memoir induced me to develop the calculational details for obtaining, from Dirac’s equations in a general gravitational field, the modified form of your four equations of second order and thus make certain the corresponding additional terms. These additional terms do depend in an essential way on the choice of the orthogonal tetrad in the space-time manifold: It seems that without such a tetrad one cannot obtain Dirac’s equation.”245View original Quote [208]

The last, erroneous, sentence must have made Pauli irate. In this paper, he pronounced his anathema (in a letter to Ehrenfest with the appeal “Please, copy and distribute!”):

“The heap of corpses, behind which quite a lot of bums look for cover, has got an increment. Beware of the paper by Levi-Civita: Dirac- and Schrödinger-type equations, in the Berlin Reports 1933. Everybody should be kept from reading this paper, or from even trying to understand it. Moreover, all articles referred to on p. 241 of this paper belong to the heap of corpses.”246View original Quote ([252Jump To The Next Citation Point], p. 170)

Pauli really must have been enraged: Among the publications banned by him is also Weyl’s well-known article on the electron and gravitation of 1929 [407Jump To The Next Citation Point].

Schrödinger’s paper was criticised by Infeld and van der Waerden on the ground that his calculational apparatus was unnecessarily complicated. They promised to do better and referred to a paper of Schouten’s [306Jump To The Next Citation Point]:

“In the end, Schouten arrives at almost the same formalism developed in this paper; only that he uses without need n-bein components and theorems on sedenions247, while afterwards the formalism is still burdened with auxiliary variables and pseudo-quantities. We have taken over the introduction of ‘spin densities’ by Schouten.”248View original Quote ([168], p. 4)

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

It is left to an in-depth investigation, how this discussion concerning teleparallelism and Dirac’s equation involving Tetrode, Wigner, Fock, Pauli, London, Schrödinger, Infeld and van der Waerden, Zaycoff, and many others influenced the acceptance of the most important result, i.e., Weyl’s transfer of the gauge idea from classical gravitational theory to quantum theory in 1929 [407408].

7.2.3 Parallelism at a distance and electron spin

Einstein’s papers on distant parallelism had a strong but shortlived impact on theoretical physicists, in particular in connection with the discussion of Dirac’s equation for the electron,

( ) k ∂ iγ ---k + μ ψ = 0, (197 ) ∂x
where the 4-spinor ψ and the γ-matrices are used. At the time, there existed some hope that a unified field theory for gravitation, electromagnetism, and the “electron field” was in reach. This may have been caused by a poor understanding of the new quantum theory in Schrödinger’s version: The new complex wave function obeying Schrödinger’s, and, more interestingly for relativists, Dirac’s equation or the ensuing Klein–Gordon wave equation, was interpreted in the spirit of de Broglie’s “onde pilote”, i.e., as a classical matter wave, not – as it should have been – as a probability amplitude for an ensemble of indistinguishable electrons. One of the essential features of quantum mechanics, the non-commutativity of conjugated observables like position and momentum, nowhere entered the approaches aiming at a geometrization of wave mechanics.

Einstein was one of those clinging to the picture of the wave function as a real phenomenon in space-time. Although he knew well that already for two particles the wave function no longer “lived” in space-time but in 7-dimensional configuration space, he tried to escape its statistical interpretation. On 5 May 1927, Einstein presented a paper to the Academy of Sciences in Berlin with the title “Does Schrödinger’s wave mechanics determine the motion of a system completely or only in the statistical sense?”. It should have become a 4-page publication in the Sitzungsberichte. As he wrote to Max Born:

“Last week I presented a short paper to the Academy in which I showed that one can ascribe fully determined motions to Schrödinger’s wave mechanics without any statistical interpretation. Will appear soon in Sitz.-Ber. [Reports of the Berlin Academy].”249View original Quote ([103Jump To The Next Citation Point], p. 136)

However, he quickly must have found a flaw in his argumentation: He telephoned to stop the printing after less than a page had been typeset. He also wanted that, in the Academy’s protocol, the announcement of this paper be erased. This did not happen; thus we know of his failed attempt, and we can read how his line of thought began ([183], pp. 134–135).

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

“the quantities sh λ250 of Einstein seem to have one foot in the macro-mechanical world formally described by Einstein’s gravitational potentials and characterised by the index λ, and the other foot in a Minkowskian world of micro-mechanics characterised by the index s. That the micro-mechanical world of the electron is Minkowskian is shown by the theory of Dirac, in which the electron spin appears as a consequence of the fact that the world of the electron is not Euclidean, but Minkowskian. This seems to us the most important aspect of Einstein’s recent work, and by far the most hopeful portent for a unification of the divergent theories of quanta and gravitational relativity.” [418]

The correction of this misjudgement of Wiener and Vallarta by Fock and Ivanenko began only one month later [134Jump To The Next Citation Point], and was complete in the summer of 1929 [134133131132].

In March, Tamm tried to show

“that for the new field theory of Einstein [8488] certain quantum-mechanical features are characteristic, and that we may hope that the theory will enable one to seize the quantum laws of the microcosm.”251View original Quote ([341Jump To The Next Citation Point], p. 288)

Tamm added a torsion term ∘ -----i i¯h (SiS )χ to the Dirac equation (197View Equation) and derived from it a general relativistic (Schrödinger) wave equation in an external electromagnetic field with a contribution from the spin tensor coupled to a torsion term252 α [iαk]S l ik. As Tamm assumed for the torsion vector Sk = ± ie-ϕk ¯hc, his tetrads had to be complex, with the imaginary part containing the electromagnetic 4-potential ϕk. This induced him to see another link to quantum physics; by returning to the first of Einstein’s field equations (170View Equation) and replacing 𝜖 in Equation (169View Equation) by iec¯h in the limit ¯h → 0, he obtained the laws of electricity and gravitation, separately. From this he conjectured that, for finite h, Einstein’s field equations might correctly reproduce the quantum features of “the microcosm” ([341], p. 291); cf. also [340].

What remained after all the attempts at geometrizing the matter field for the electron, was the conviction that the quantum mechanical “wave equations” could be brought into a covariant form, i.e., could be dealt with in the presence of a gravitational field, but that quantum mechanics, spin, and gravitation were independent subjects as seen from the goal of reaching unified field theory.

7.2.4 Kaluza’s theory and wave mechanics

For some, Kaluza’s introduction of a fifth, spacelike dimension seemed to provide a link to quantum theory in the form of wave mechanics. Although he did not appreciate Kaluza’s approach, Reichenbächer listed various possibilities: With the fifth dimension, Kaluza and Klein had connected electrical charge, Fock the electromagnetic potential, and London the spin of the electron [276]. Also, the idea of relating Schrödinger’s matter wave function with the new metrical component g55 was put to work. Gonseth and Juvet, in the first of four consecutive notes submitted in August 1927 [150Jump To The Next Citation Point148149Jump To The Next Citation Point147] stated:

“The objective of this note is to formulate a five-dimensional relativity whose equations will give the laws for the gravitational field, the electromagnetic field, the laws of motion of a charged material point, and the wave equation of Mr. Schrödinger. Thus, we will have a frame in which to take the gravitational and electromagnetic laws, and in which it will be possible also for quantum theory to be included.”253View original Quote ([150], p. 543)

It turned out that from the R55-component of the Einstein vacuum equations Rα β = 0, α, β = 1,...,5, with the identification g55 = ψ made, and the assumption that ψ,-∂ψi ∂x be “very small”, while ∂ψ- ψ, ∂x5 be “even smaller”, the covariant d‘Alembert equation followed, an equation that was identified by the authors with Schrödinger’s equation. Their further comment is:

“We thus can see that the fiction of a five-dimensional universe provides a deep reason for Schrödinger’s equation. Obviously, this artifice will be needed if some phenomenon would force the physicists to believe in a variability of the [electric] charge.”254View original Quote ([149], p. 450)

In the last note, with the changed identification g = ψ2 55 and slightly altered weakness assumptions, Gonseth and Juvet gained the relativistic wave equation with a non-linear mass term.

Interestingly, a couple of months later, O. Klein had the same idea about a link between the g55-component of the metric and the wave function for matter in the sense of de Broglie and Schrödinger. However, as he remarked, his hopes had been shattered [189Jump To The Next Citation Point]. Klein’s papers were of import: Remember that Kaluza had identified the fifth component of momentum with electrical charge [181], and five years later, in his papers of 1926 [185184], Klein had set out to quantise charge. One of his arguments for the unmeasurability of the fifth dimension rested on Heisenberg’s uncertainty relation for position and momentum applied to the fifth components. If the elementary charge of an electron has been measured precisely, then the fifth coordinate is as uncertain as can be. However, Klein’s argument is fallacious: He had compactified the fifth dimension. Consequently, the variance of position could not become larger than the compactification length −30 l ∼ 10, and the charge of the electron thus could not have the precise value it has. In another paper, Klein suggested the idea that the physical laws in space-time might be implied by equations in five-dimensional space when suitably averaged over the fifth variable. He tried to produce wave-mechanical interference terms from this approach [187Jump To The Next Citation Point]. A little more than one year after his first paper on Kaluza’s idea, in which he had hoped to gain some hold on quantum mechanics, Klein wrote:

“Particularly, I no longer think it to be possible to do justice to the deviations from the classical description of space and time necessitated by quantum theory through the introduction of a fifth dimension.” ([189Jump To The Next Citation Point], p. 191, footnote)

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

“Dr. H. T. Flint has drawn my attention to a recent paper by O. Klein [189] in which an extension to five dimensions similar to that given in the present paper is described. The corresponding part of the paper was written some time ago and without any knowledge of Klein’s work [...].” ([420], p. 441)

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

“The matrix theory leads to a very simple derivation of the first order wave equation, equivalent to Dirac’s but expressed in symmetrical form. It leads also to a wave equation which we can identify as relating to a system containing electrons with opposite spin. [...] It is interesting to note the way in which the existence of electrons with opposite spins locks the ‘fifth dimension,’ so that it cannot come into play and introduce the absolute into a world of relation. The domain of either electron alone might be rotated in a fifth dimension and we could not observe any difference.” ([61Jump To The Next Citation Point], pp. 524, 542)

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

J. W. Fisher of King’s College re-interpreted Kaluza–Klein theory as presented in Klein’s third paper [187]. He proceeded from the special relativistic homogeneous wave equation in five-dimensional space and, after dimensional reduction, compared it to the Klein–Gordon equation for a charged particle. By making a choice different from Klein’s for a constant he rederived the result of de Broglie and others that null geodesics in five-dimensional space generate the geodesics of massive and massless particles in space-time [127].

Mandel of Petersburg/Leningrad believed that

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

He now posed the question whether this would be the same for Dirac’s theory. Seemingly, he also believed that a tensorial formulation of Dirac’s equation was handy for answering this question and availed himself of “the tensorial form given by W. Gordon [151], and by J. Frenkel”255 [140Jump To The Next Citation Point]. Mandel used, in five-dimensional space, the complex-valued tensorial wave function Ψik = ψ γik + Ψ [ik] with a 5-scalar ψ. Here, he had taken up a suggestion J. Frenkel had developed during his attempt to describe the “rotating electron,” i.e., Frenkel’s introduction of a skew-symmetric wave function proportional to the “tensor of magneto-electric moment” mik of the electron by mik ψ = m0 ψik [141140]. Ψik may depend on x5; by taking Ψ periodic in x5, Mandel derived a wave equation “which can be understood as a generalisation of the Klein–Fock five-dimensional wave equation [...].” He also claimed that the vanishing of ψ made M5 cylindrical (in the sense of Equation (109View Equation[221]). As he had taken notice of a paper of Jordan [173] that spoke of the electromagnetic field as describing a probability amplitude for polarised photons, Mandel concluded that the amplitude of his Ψ-field might then represent polarised electrons as its quanta. However, he restricted himself to the consideration of classical one-particle wave equations because

“in some cases one can properly speak of a quasi-macroscopical one-body problem – think of a beam of monochromatic cathod-rays in an arbitrary external force-field.”256View original Quote

In a later paper, Mandel came back to his wave equation with a skew-symmetric part and gave it a different interpretation [222Jump To The Next Citation Point].

Unlike Klein, Mandel tried to interpret the wave function as a new discrete coordinate, an idea going back to Pauli [247]. He took “Dirac’s spin variable” ζ and the spatial coordinate x5 as a pair of canonically-conjugate operator-valued variables; ζ is linked to positive and negative elementary charge (of proton and electron) as its eigenvalues. In Mandel’s five-dimensional space, the fifth coordinate, as a “charge” coordinate, thus assumed only 2 discrete values ±e.

“This completely corresponds to the procedure of the Dirac theory, with the only difference that for Dirac the coordinate ζ could assume not 2 but 4 values; from our point of view this remains unintelligible.”257View original Quote ([222Jump To The Next Citation Point], p. 785)

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

“[...] all possible values of this quantity [5 x] still remain completely undetermined such that all its possible values from − inf to + inf are of equal probability.”

This made sense because, unlike Klein, Mandel had not compactified the fifth dimension. His understanding of quantum mechanics must have been limited, though: Only two pages later he claimed that the canonical commutation relations ¯h [p,q] = i1 could not be applied to his pair of variables due to the discrete spectrum of eigenvalues. He then essentially went over to the Weyl form of the operators p, q in order to “save” his argument [222].

Another one of the many versions of “Dirac’s equation” was presented, in December 1930, by Zaycoff who worked both in the framework of Einstein’s teleparallel theory and of Kaluza’s five-dimensional space. His Lagrangian is complicated258,

∂ ψ i 1 1 M = − i&tidle;ψ γρ---ρ + -SmJm + --SklmJklm + afmJm + -FkmJkm + μJ0, (198 ) ∂x 2 24 8
where summation is implied and Skml is the torsion tensor, fm the electromagnetic vector potential, and fik the electromagnetic field. Note that the Dirac current Jm := ψ&tidle;γm ψ couples to both the torsion vector and the 4-potential. The remaining variables in (198View Equation) are Jml := iψ&tidle;γm γ †γ0ψ l, with k ⁄= l, and &tidle; † Jklm := iψγkγ lγmψ, with k ⁄= l ⁄= m [434].

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

The Dirac equations for Riemannian space-time with electromagnetic field and mass can be written in the form of equations without field or mass αb ∇ ψ = 0 b in an R 5.

Here αb is the set of Dirac numbers defined by α (aαb) = gab,(αaαb)αc = αa (αbαc) with a,b,c = 0,...,4, and ∇ b the covariant spinor derivative defined by him.

As we mentioned above (cf. Section 6.3.2), another approach to the matter within projective geometry was taken by Pauli with his student J. Solomon [253]. After these two joint publications, marred by a calculational error, Pauli himself laid out his version of the projective theory in two installments with the first, as a service to the community, being a pedagogical presentation of the formalism connected with projective geometry [249]. The second paper, again, has the application to Dirac’s equation as a prime motivation:

“The following deductions are intended to show [...] that the unifying combination of the gravitational and the electromagnetic fields, by projective differential geometry with the aid of five homogeneous coordinates, is a general method whose range reaches beyond classical field-physics and into quantum theory. Perhaps, the hope is not unjustified that the method will stand the test as a general framework for the laws of physics also with regard to a future physical and conceptual improvement of the foundations of Dirac’s theory.”259View original Quote ([250], pp. 837–838)

Pauli started with the observation that the group of orthogonal transformations in five-dimensional space had an irreducible, four-dimensional matrix representation satisfying

αμα ν + α ναμ = 2gμν ⋅ 1, μ,ν = 1,...,5, (199 )
where αμ are 4 × 4 matrices given at the end of Section 2.1.5 in a different representation ( σ 0 ) ( μ ) α μ = 0 − σμ with μ = 1,2,3, ( 1 0 ) ( ) α4 = 0 1, and augmented by ( 0 1 ) ( ) α5 = 1 0. This had been known also to Eddington [61Jump To The Next Citation Point] and Schouten [303]. He then introduced projective spinors depending on five homogeneous coordinates without using bein-quantities. He followed the methods of Schrödinger and Bargmann [3196], i.e., used the existence of a matrix A such that A α μ is Hermitian. The transformation laws of 4-spinors Ψ and matrices α μ are coupled:
′ −1 ψ = S Ψ, (200 ) α′ = S −1α S, (201 ) μ μ A ′ = S †AS. (202 )
For transformations in the space of homogeneous coordinates X ′μ = aμ X ν ν such that α = aμ α′ ν ν μ, the quantity ⋆ aμ := Ψ Aα μΨ transforms, for fixed α μ, under changes of the spin frame, Equation (200View Equation), as a covariant vector260.

Pauli criticised an analogous attempt at formulating Dirac’s equation with the help of five homogeneous coordinates by Schouten and van Dantzig [317308Jump To The Next Citation Point309Jump To The Next Citation Point] as being “difficult to understand and less than transparent”261View original Quote. A projective spinor is defined via

l Ψ = ψF , (203 )
where ψ is a normal (“affine”) spinor (degree of homogeneity 0) and F a real scalar of (homogeneity) degree 1, i.e., μ ∂F- F = X ∂Xμ. There exist two (related) spin-connections Λk for projective spinors Ψ and R Λk for spinors ψ.

Pauli’s Dirac equation, derived from a Lagrangian, looked in five dimensions like

α μ(Ψ;μ + kX μΨ ) = 0, (204 )
with k = − imch-− iehc c√κ1r,  and l = + hiec√cκ1r. The covariant derivative is formed with the spin connection Λk. An involved calculation leads to Dirac’s equation in four dimensions:
( ) √ -- k ∂ ψ R ie mc r κ [kl] α --k-+ Λk ψ − --Φk ψ − i---α0ψ + -----Fkl α0 α ψ = 0, (205 ) ∂x hc h 8 c
with the numerical factor r, the electrical 4-potential Φk, and the electromagnetic field tensor Fkl; furthermore, α μν = α[μαν].

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

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

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