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 [290].
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.” ([414], 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 [258]. 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 [318
, 6
], and the
publication of Infeld and van der Waerden one
year later [166
] 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”
,
, with the usual
transformation law under the linear transformation
(
) changing the spin frame:
Veblen imbedded spinors into his
projective geometry [380]:
“[...] 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 :
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“In a five-dimensional representation the use
of the homogeneous coordinates amounts
to representing the points of space-time by the straight lines
through the origin, whereas the use of
, and the
gauge variable amounts to using the system of straight lines
parallel to the
-axis for the same purpose. The
transformation (192
) given above carries
the system of lines into the other.” [382]
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
in place of
, and
for the Ricci rotation coefficients
; 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 D i r a c’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 ([325], 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 E i n s t e i
n’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
(summation over
) 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
with the vectorial operator
and its derivative
immediately led to Dirac’s equation by replacing
by
, where
is the electromagnetic
4-potential, by also assuming the velocity of light
to be the classical average of the “4-velocity”
, and by applying the operator to the wave function
. In the next step, instead of the Dirac
-matrices with constant entries
, the coordinate-dependent bein-components
are defined;
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 . Thus
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, must transform with a factor of norm 1, innocuous
for observation, i.e.,
if
. 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 besides the Ricci-coefficients
in the expression for parallel transport, on the one
hand provides a simple reason for the emergence of the term
in the wave equation and, on the other, shows that
the potentials
have a place of their own in the
geometrical world-view, contrary to Einstein’s opinion; they need not
be functions of the
.” 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” [206]. 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
-function with a factor of modulus 1.” 235
([130],
p. 266)
The divergence of the complex energy-momentum
tensor 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
Weyl’s paper is seminal for the
further development of the gauge idea [407].
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
[168],
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 [276, 277].
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 [284]
(cf. Section 8). In the paper, Zaycoff introduced a six-dimensional manifold with local
coordinates
where
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
to
he used the subgroup of
coordinate transformations given in Klein’s approach, augmented by
.
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 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
and
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 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 [318]:
“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 [258]. 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
-function.” 242
([318
], p. 105)
The -matrices were taken by
Schrödinger such that their covariant derivative vanished, i.e.,
where
is the
spin-connection introduced by
.
Schrödinger took
, with
, as
Hermitian matrices. He introduced tensor-operators
such that the inner product
instead of
stayed real
under a “complemented point-substitution”.
In the course of his calculations, Schrödinger obtained the wave equation
where“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 is the reciprocal Compton length, about
. Yet it appears important that in the
generalised theory a term is encountered at all which is equivalent
to the enigmatic mass term.” 243
([318
], p. 128)
The coefficient in front of the
Ricci scalar in Schrödinger’s (Klein-Gordon) wave equation differs
from the
needed for a conformally invariant
version of the scalar wave equation244
(cf. [255],
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
with
such that
, with
.
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
[207]
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
([251
], 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 -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
([167], p. 4)
Unlike Schrödinger’s, the wave equation derived
from Dirac’s equation by Infeld and Waerden contained a
term , with
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].
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,
where the 4-spinorEinstein 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 ([182], 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 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
, and the other foot in a Minkowskian world of
micro-mechanics characterised by the index
. 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.” [416]
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].
In March, Tamm tried to show
“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 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
. As Tamm assumed for the torsion vector
, his tetrads had to be complex, with
the imaginary part containing the electromagnetic 4-potential
. This induced him to see another link to quantum
physics; by returning to the first of Einstein’s field
equations (170
) and replacing
in Equation (169
) by
in the limit
, he
obtained the laws of electricity and gravitation, separately. From
this he conjectured that, for finite
, 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.
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 [275]. Also, the idea of
relating Schrödinger’s matter wave function with the new metrical
component 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)
It turned out that from the -component of the Einstein vacuum equations
,
, with the identification
made, and the assumption that
be “very small”, while
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.” 254
([149], p. 450)
In the last note, with the changed identification
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 -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 [188
]. Klein’s papers
were of import: Remember that Kaluza had identified the fifth
component of momentum with electrical
charge [180], and five
years later, in his papers of 1926 [184, 183], 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
, 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 [186
]. 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.” ([188], 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 [188] 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 [186]. 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.” ([220], 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
with a 5-scalar
. 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”
of the electron
by
[141, 140].
may depend on
; by taking
periodic in
, 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
made
cylindrical (in the sense of
Equation (109
) [220]). As he
had taken notice of a paper of Jordan [172] that spoke
of the electromagnetic field as describing a probability amplitude
for polarised photons, Mandel concluded that the
amplitude of his
-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 [221].
Unlike Klein, Mandel tried to interpret the wave
function as a new discrete coordinate,
an idea going back to Pauli [246]. He took
“Dirac’s spin variable” and the spatial coordinate
as a pair of canonically-conjugate operator-valued
variables;
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
.
“This completely corresponds to the procedure
of the Dirac theory, with the only difference that for Dirac the
coordinate could assume not 2 but 4 values; from
our point of view this remains unintelligible.” 257
([221
], p. 785)
In following Klein, Mandel concluded from the Heisenberg uncertainty relations that
“[...] all possible values of this quantity
[] still remain completely undetermined such that all
its possible values from
to
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 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
,
in order to “save” his
argument [221].
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 ,
where summation is implied andWhile 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 in an
.
Here is the set of Dirac numbers
defined by
with
, and
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 [252]. 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 [248]. 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
([249],
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
wherePauli criticised an analogous
attempt at formulating Dirac’s equation with the help of five
homogeneous coordinates by Schouten and van Dantzig [316, 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
withPauli 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.”