7 Geometrization of the Electron Field as an Additional Element of Unified Field Theory

After the advent of Schrödinger’s and Dirac’s equations describing the electron non-relativistically and relativistically, a unification of only the electromagnetic and gravitational fields was considered unconvincing by many theoretical physicists. Hence, in the period 1927–1933, quite a few attempts were made to include Schrödinger’s, Dirac’s, or the Klein–Gordon equation as a classical one-particle equation into a geometrical framework by relating the quantum mechanical wave function with some geometrical object. Such an approach then was believed to constitute a unification, up to a degree, of gravitation and/or electricity and quantum theory. In this section, we loosely collect some of these approaches.

The mathematicians Struik and Wiener found the task of an amalgamation of relativity and quantum theory (wave mechanics) attractive:

“It is the purpose of the present paper to develop a form of the theory of relativity which shall contain the theory of quanta, as embodied in Schrödingers wave mechanics, not merely as an afterthought, but as an essential and intrinsic part.” [338]

A further example for the new program is given by J. M. Whittaker at the University of Edinburgh [415Jump To The Next Citation Point] who wished to introduce the wave function via the matter terms:

“In addition to the wave equations a complete scheme must include electromagnetic and gravitational equations. These will differ from the equations of Maxwell and Einstein in having ‘wave’ terms instead of ‘particle’ terms for the current vector and material energy tensor. The object of the present paper is to find these equations [...].” ([415Jump To The Next Citation Point], p. 543)

Zaycoff, from the point of view of distant parallelism, found the following objection to unified field theory as the only valid one:

“It neglects the existence of wave-mechanical phenomena. By the work of Dirac, wave-mechanics has reached an independent status; the only attempt to bring together this new group of phenomena with the other two is J. M. Whittaker’s theory [415Jump To The Next Citation Point].”220View original Quote [428]

In fact, in a short note, Zaycoff presented his version of Whittaker’s theory with 8 coupled second-order linear field equations for two “wave vectors” that, in a suitable combination, were to represent “the Dirac’s wave equations”; they contain the Ricci tensor and both the electromagnetic 4-potential and field [432]. Thus, what is described is Dirac’s equation in external gravitational and electromagnetic fields, not a unified field theory. Whittaker had expressed himself more clearly:

“Eight wave functions are employed instead of Dirac’s four. These are grouped together to form two four-vectors and satisfy wave equations of the second order. It is shown [...] that these eight wave equations can be reduced, by addition and subtraction, to the four second order equations satisfied by Dirac’s functions taken twice over; and that, in a sense, the present theory includes Dirac’s.” ([415Jump To The Next Citation Point], p. 543)

Whittaker also had written down a variational principle by which the gravitational and electromagnetic field equations were also gained. However, as the terms for the various fields were just added up in his Lagrangian, the theory would not have qualified as a genuine unified field theory in the spirit of Einstein.

What fancy, if only shortlived, flowers sprang from the mixing of geometry and wave mechanics is shown by the example of H. Jehle’s

“[...] path leading, on the one hand, to electrical elementary particles and, on the other, to the explanation of cosmological problems by quantum theory.” [171]

His ad-hoc modification of Einstein’s equations was:

¯ G ψ2 − σ2R ∂ψ--∂ψ--= 0, G ψ ¯ψ − σ2R ∂ψ--∂ψ- = 0, (186 ) ik ∂xi ∂xk ik ∂xi∂xk
where ψ is the quantum mechanical state function. Although, two years later, Jehle withdrew his claims concerning elementary particles, he continued to apply

“wave-mechanical methods to gravitational phenomena, by which the curious structure of the spiral nebulae and spherical star systems may be readily understood.” [172]

An eminent voice was Weyl’s:

“It seems to me that it is now hopeless to seek a unification of gravitation and electricity without taking material waves into account.” ([408Jump To The Next Citation Point], p. 325)

Now, this posed a problem because for the representation of the electrons in the form of Dirac’s equation, the elements of spin space, i.e., spinors, had to be used. How to combine them with the vectors and tensors appearing in electromagnetic and gravitational theories? As the spinor representation is the simplest representation of the Lorentz group, everything may be played back to spin space. At the time, this was being done in different ways, in part by the use of number fields with which physicists were unacquainted such as quaternions and sedenions (cf. Schouten [315]). Others, such as Einstein and Mayer, liked vectors better and introduced so-called semi-vectors. Still others tried to write Dirac’s equations in a vectorial form and took into account the doubling of variables and equations [213415Jump To The Next Citation Point]. Some less experienced, as e.g., “Exhibition Research Student” G. Temple, even claimed that a tensorial theory was necessary to retain it relativistically:

“It is an admitted fact that Dirac’s wave functions are not the components of a tensor and that his wave equations are not in tensorial form. It is contended here that therefore his theory cannot be upheld without abandoning the theory of relativity.” ([343], p. 352)

While this story about geometrizing wave mechanics might not be a genuine part of unified field theory at the time, it seems interesting to follow it as a last attempt for binding together classical field theory and quantum physics. Even Einstein was lured into thinking about spinors by Dirac’s equation; this equation promised more success for his program concerning elementary particles as solutions of differential equations (cf. Section 7.3. )

 7.1 Unification of Maxwell’s and Dirac’s equations, of electrons and light
 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
 7.3 Einstein, spinors, and semi-vectors

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