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:

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

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

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:

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

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

An eminent voice was Weyl’s:

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 [213, 415]. Some less experienced, as e.g., “Exhibition Research Student” G. Temple, even claimed that a tensorial theory was necessary to retain it relativistically:

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. )