“In spite of the great importance which the spinor concept, as introduced by Pauli and
Dirac, has obtained in molecular physics, one cannot claim that the analysis of this
concept up to now satisfies all justified demands. Our efforts have lead to a derivation
corresponding, according to our opinion, to all demands for clarity and naturalness and
avoiding completely any not so transparent artifice. Thereby, [...] the introduction of novel
quantities was shown to be necessary, the ‘semi-vectors’, which include spinors but possess
a clearly more transparent transformation character than spinors.”^{262}
([110], p. 522)

In this first publication on the subject, Einstein and Mayer explicitly referred to the paper by Infeld and van der Waerden, of which they had received a copy several months before publication ([167], and [110], p. 25, footnote). Apparently, Einstein found the reconstruction of the spinor concept in his paper more “clear and natural” than Infeld and van der Waerden’s. Nevertheless, the approach and notation of Infeld and van der Waerden became the accepted one by physicists.

About three months before the first paper on semi-vectors was published, Einstein wrote to Besso:

“I work with my Dr. Mayer on the theory of spinors. We already could clear up the
mathematical relations. A grasp on the physics is far away, farther than one thinks at
present. In particular, I still am convinced that the attempt at an essentially statistical
theory will fail.”^{263}
([99], p. 291)

Besides the aspired-to clarity and simplicity, Einstein’s main hope was that, with his semi-vector system of equations replacing Dirac’s equation, it might be possible to explain the existence of elementary particles with opposite charge and unlike mass, i.e., of electron and proton. As noted before, he had not been able to solve this problem by his affine field theory of 1923 (cf. Section 4.3.2), nor by the approaches to unified field theory that followed. As it turned out, the positron was discovered at about the same time, and the problem dissolved while Einstein and Mayer began to reformulate the spinor concept. Einstein again seems to have been fully convinced that his new concept of “semi-vector” was superior to the spinor concept. On 7 May 1933, he wrote to De Haas:

“Scientifically Mayer and I have found one very natural generalisation of Dirac’s equation which makes it comprehensible, that there are two understandable elementary masses, while there is only one electric charge.” [69]

In the first paper in the reports of the Berlin Academy, the mathematical foundations
of the semi-vector formalism are developed [110]. The basic idea of Einstein and Mayer
is the possibility of a decomposition of any (proper) Lorentz transformation described
by a real matrix into a product of a pair of complex-conjugate, commuting
matrices^{264}
and . The transformations represented by or form a group isomorphic to the Lorentz group.
In terms of infinitesimal Lorentz transformations given by an antisymmetric tensor , this amounts to
the decomposition into a self-dual and an anti-selfdual part: with the
dual^{265}
defined by .

Contravariant semi-vectors of the first and second kind now are defined by their transformation laws: and , where , are the components of , . For real Lorentz transformation , must hold. As , are both also Lorentz transformations,

“the metric tensor is also a semi-vector of 1st kind (and of 2nd kind) with transformation-invariant components.”

Thus it can be used for raising and lowering indices of semi-vectors ([110], p. 535).

The system of equations intended as a replacement of the Dirac equation appears only in the second publication [111].
A Lagrange function for the semi-vector is found and the generalised Dirac equations for the semi-vectors look
like^{266}:

In his “Spencer Lecture” of 10 June 1933 in Oxford, Einstein embedded his point of view into the development of field theory:

“[...] Louis de Broglie guessed the existence of a wave field that could be used for the
interpretation of certain quantum properties of matter. With the spinors, Dirac found
novel field quantities whose simplest equations permitted the derivation of the properties
of the electron to a great extent. With my collaborator, Dr. Walther Mayer, I now found
that these spinors form a special case of a type of field, linked to four-dimensional space,
which we called ‘semi-vectors’. The simplest equations to be satisfied by such semi-vectors
provide a key for the understanding of the existence of two elementary particles with
different ponderable mass and like, but opposite, charge. These semi-vectors are, besides
the usual vectors, the simplest mathematical field-objects possible in a four-dimensional
metrical continuum; it appears that they naturally describe essential properties of the
electrical elementary particles.”^{267}
([100], p. 117)

That semi-vectors are in close connection with spinors and not always the simplest objects, had been noted by Schouten in a paper submitted to Zeitschrift für Physik on 15 April 1933, even before Einstein’s Spencer lecture:

“The space of semivectors of first and second kind is the manifold of two simple bivectors
in the local space-time-world, which lie on the null cone in two planes of the first and
second system of planes, respectively. [...] In the not-projective theory as well as in the
projective without an electromagnetic field, semi-vectors have an advantage, [...]. As soon
as an electromagnetic field is present, in the projective theory, calculation with spinvectors
is simpler than calculation with semivectors.”^{268}
[309]

In a previous paper, Schouten had geometrically explained the two approaches to spinor analysis followed by
van der Waerden [368], Laporte and Uhlenbeck [203], and by himself. In the first approach, the vectors of
the two invariant planes in spin space were identified; in the second, i.e., in Schouten’s, they were
taken as the basis for a four-dimensional vector space [308]. Schouten had been corresponding
with Pauli who “was so friendly as to allow me to quote this theorem from a not yet published
manuscript.”^{269}
In his second paper, Schouten placed the semi-vectors of Einstein and Mayer into his geometrical setting.
He showed that there exist two different complex-conjugated three-dimensional representations of the real
Lorentz group as a 6-parameter subgroup of the four-dimensional orthogonal group. With the help of these
he constructed four

“building blocks: the special spin vectors of 1st and 2nd kind of spin space, which at the same time are special semi-vectors in the two preferred invariant planes in semi-space. In both cases, always two vectors are joined which belong to different invariant planes, for semi-vectors two with same transformation law, for spin vectors two with conjugate complex transformations.” ([309], p. 106)

As Schouten used group theory, quaternions, sedenions, projective geometry – all not very
familiar to physicists – Pauli must have thought of a much simpler disentanglement of
semi-vectors and spinors. In fact, as Pauli pointed out in his letter to Einstein of 16 July 1933, a
semi-vector does not form an irreducible representation of the rotation group while a spinor
does^{270}
(see [252], p. 189). In this regard, the semivector concept is less “natural” than the spinor concept. In contrast
to what Schouten had shown, in his book on spinors Cartan flatly disclaimed any geometric definition for
semivectors^{271}.
Pauli then set his doctoral student V. Bargmann on the problem of comparing Einstein’s semivector
approach to the spinor calculus as developed by van der Waerden. Bargmann acknowledged Schouten’s
paper [309], “with which our presentation has quite a few points of contact”. He then proved
that

“to each semi-vector two 2-component spinors correspond, which both satisfy the same transformation law.” ([7], p. 68)

and that the generalised Dirac equations of Einstein and Mayer (206)

“decompose into two separate 4-component systems of Dirac’s type, which are distinguished only by the mass values.” ([7], p. 78)

This means that only one of these two systems is needed, and that it describes particles with opposite charge and the same mass. Thus, as van Dongen states curtly:

“It is evident from Bargmann’s analysis that the most general semi-vector Dirac system of Einstein and Mayer is nothing more than just a linear superposition of two independent Dirac spinor systems and thus cannot give insight into the fundamental nature of electrons and protons.” ([371], p. 88, and [372])

In his letter to Einstein, Pauli had also mentioned his papers to be published in Annalen der Physik and discussed here in Section 7.2.3. Two more papers were written by Einstein and Mayer before Einstein quietly dropped the subject. The last paper considered semi-vectors as “usual vectors with a different differentiation character” [112, 113].

It seems that Einstein at the time had not followed quantum field theory intensively enough to
be able to compete with that theory – leaving aside his rejection of “the statistical fad”. By
continuing to pair “electron and proton” while others speculated already about “electron and its
antiparticle”^{272}
or “proton and its antiparticle”, he was bound to run into a dead end. About one year after the Spencer
lecture, when Einstein was still publishing about semi-vectors, Pauli and Weisskopf quantised the scalar
relativistic wave equation with an external field using Bose–Einstein statistics:

“Without further hypothesis the existence follows of particles with opposite charge and
same rest mass, which can be produced or annihilated by absorption or emission of
electromagnetic radiation. The frequency of these processes is shown to be of the same
magnitude as the one for particles with the same charge and mass following from Dirac’s
hole theory.”^{273}
([255], Abstract)

Of course, Einstein’s problem was quickly solved; at about the same time the electron’s antiparticle was observed.

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