7.3 Einstein, spinors, and semi-vectors

Ehrenfest, even after van der Waerden’s paper on spinor analysis [368Jump To The Next Citation Point], in 1932 pressed Einstein to think about a simple geometric interpretation of spinors. To this end Einstein responded, together with his assistant Mayer, by introducing the concept of semi-vector seemingly more natural to him than a spinor, and also more general:

“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.”262View original Quote ([110Jump To The Next Citation Point], 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 [110Jump To The Next Citation Point], 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.”263View original Quote ([99Jump To The Next Citation Point], 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 [110Jump To The Next Citation Point]. The basic idea of Einstein and Mayer is the possibility of a decomposition of any (proper) Lorentz transformation described by a real matrix D into a product BC of a pair of complex-conjugate, commuting matrices264 B and C. The transformations represented by B or C form a group isomorphic to the Lorentz group. In terms of infinitesimal Lorentz transformations given by an antisymmetric tensor ωik, this amounts to the decomposition into a self-dual and an anti-selfdual part: ωik = 12(ωik + iω∗ik) + 12(ωik − iω ∗ik), with the dual265 defined by ω ∗ := 1√g-𝜖ijkl ωkl ik 2.

Contravariant semi-vectors of the first and second kind now are defined by their transformation laws: i′ i k ρ = b kρ and i′ i k σ = c kσ, where i b k, i c k are the components of B, C. For real Lorentz transformation D, ¯bik = cik must hold. As B, C are both also Lorentz transformations,

“the metric tensor gik 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 ψ, χ look like266:

( ) ( ) Erστ ∂ψσ-− i𝜖ψ σϕr = ¯Cτρχ ρ, E ⋆rστ ∂χτ-− i𝜖χτϕr = − C ρσψρ, (206 ) ∂xr ∂xr
where ϕ r is the electromagnetic 4-potential and Erστ a numerically invariant tensor depending on 4 constants a(r):
√ -- Erst = grsa(t) + grta(s) − gsta(r) − g ηrstwa(w); a(w) = gwta(t). (207 )
Contrary to his idea of what a “real” unified field theory should look like, Einstein just added the Lagrangian for the semi-vector fields to the Lagrangians for the gravitational and electromagnetic fields.

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.”267View original Quote ([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.”268View original Quote [309Jump To The Next Citation Point]

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 [308Jump To The Next Citation Point]. 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.” ([309Jump To The Next Citation Point], 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 does270 (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 semivectors271. 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.” ([7Jump To The Next Citation Point], p. 68)

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

“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” [112113].

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.”273View original Quote ([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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