Dirac’s electron with spin, <a title="View short Biography" href="javascript:parent.bibpopup('desc_1.html')" onmouseover="status='View short Biography'; return true;" onmouseout="status='';" class="biographyLink">Einstein</a>’s teleparallelism, and <a title="View short Biography" href="javascript:parent.bibpopup('desc_3.html')" onmouseover="status='View short Biography'; return true;" onmouseout="status='';" class="biographyLink">Kaluza</a>’s fifth dimension

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 [259

]. Although van der Waerden had written on spinor analysis as early as 1929 [368

] and Weyl’s [407

, 408

], Fock’s [133

, 131

], and Schouten’s [306

] 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 [319

, 6

], and the publication of Infeld and van der Waerden one year later [167

] 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.”²²⁶ ([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” [378

]. 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:

“[...] 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 ]

“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 (192 ) 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 [407

, 408

]. 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 ([406

], 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.”²²⁹

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.”²³⁰ ([407 ], 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 ([326

], 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.”²³¹ ([135 ], p. 798)

In another paper [134

], 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.”²³² ([134 ], 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 [133

, 131

, 132

]. In the first paper Fock defined an asymmetric matter tensor for the spinor field,

“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 $γikl$ .”²³³ ([131 ], 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.”²³⁵ ([130 ], p. 266)

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

“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’.”²³⁶ ([132 ], 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.”²³⁸

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 [277 , 278 ].

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 [426 , 427 ].

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.”²⁴⁰ [433 ]

Rumer’s paper is [285

] (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 [306

, 307

]. 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 [407

]. 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 [131

] 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.” ([306 ], 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.” ([306 ], p. 265)

Two years later, Schrödinger as well became interested in Dirac’s equation. We reproduce a remark from his publication [319

“The joining of Dirac’s theory of the electron with general relativity has been undertaken repeatedly, such as by Wigner [419 ], Tetrode [344 ], Fock [131 ], Weyl [407 , 408 ], Zaycoff [434 ], 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.”²⁴² ([319 ], 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”.

“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.”²⁴³ ([319 ], 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]”[6

]. 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.”²⁴⁵ [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.”²⁴⁶ ([252 ], 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 [407

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 [306

“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.”²⁴⁸ ([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 [407 , 408 ].

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,

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].”²⁴⁹ ([103 ], 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 $sh λ$ ²⁵⁰ 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 [134

], and was complete in the summer of 1929 [134 , 133 , 131 , 132 ].

“that for the new field theory of Einstein [84 , 88 ] 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.”²⁵¹ ([341 ], p. 288)

Tamm added a torsion term $∘ -----i i¯h (SiS )χ$ to the Dirac equation (197

) 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 (170

) and replacing $𝜖$ in Equation (169

) 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 [150

, 148 , 149

, 147 ] 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.”²⁵³ ([150 ], p. 543)

“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.”²⁵⁴ ([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 [189

]. 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 [185 , 184 ], 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 [187

]. 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.” ([189 ], 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.” ([61 ], 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 ].

“a consideration in five dimensions has proven to be well suited for the geometrical interpretation of macroscopic electrodynamics.” ([221 ], 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 [140

]. 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 $mik ψ = m0 ψik$ [141 , 140 ]. $Ψik$ 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 (109

) [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.”²⁵⁶

In a later paper, Mandel came back to his wave equation with a skew-symmetric part and gave it a different interpretation [222

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.”²⁵⁷ ([222 ], 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 ,

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” ([306

], 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.”²⁵⁹ ([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

Pauli criticised an analogous attempt at formulating Dirac’s equation with the help of five homogeneous coordinates by Schouten and van Dantzig [317 , 308

, 309

] as being “difficult to understand and less than transparent”²⁶¹ . A projective spinor is defined via

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

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.”

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

7.2.1 Spinors

7.2.2 General relativistic Dirac equation and unified field theory

7.2.3 Parallelism at a distance and electron spin

7.2.4 Kaluza’s theory and wave mechanics