Appendix I

Hermann Weyl

Hermann Weyl revised the geometry of Reimann with its  “…residual element of finite geometry–without any substantial reason, as far as I can see.” This element is the preservation of lengths. If direction is not preserved, then why the assumption that lengths should be preserved?

Such an assumption being recognized as false, a geometry comes into being, which, when applied to the world, explains in a surprising manner not only the phenomena of gravitation, but also those of the electromagnetic field. According to the theory which now takes shape, both classes of phenomena spring from the same source, and in fact we cannot in general make any arbitrary separation of electricity from gravitation. In this theory all physical quantities have a meaning in world geometry. In particular the quantities denoting physical effects appear at once as pure numbers. The theory leads to a world-law which in its essentials is defined without ambiguity. It even permits us in a certain sense to comprehend why the world has four dimensions. (H. Weyl, from "Gravitation and Electricity", in The Principle Of Relativity -- see Bibliography)

Recognizing the implications, he goes on: “I shall now first of all give a sketch of the amended geometry of Riemann without any thought of its physical interpretation. Its application to physics will then follow of its own accord.” (bold emphasis is mine) 


In 1918 the outstanding German Mathematician Hermann Weyl—then a professor at the ‘Zurich Poly—proposed an extension of the general theory of relativity that was so natural and ingenious that it deserved a better fate than the one that befell it.

Weyl proposed that general relativity needed a variable for length variation!

and he introduced this sort of change of size as a possibility in curved space-time thus making a fundamental alteration in its geometric structure... He showed that with this new geometrical structure of space-time he could, in a natural way, link Einsteinian gravitation with Maxwellian electrodynamics. This at once excites our interest. For when Einstein treated gravitation as curvature he was unable to give electromagnetism a correspondingly fundamental geometrical role. But Weyl, with his changes of lengths, had made electromagnetism too an aspect of geometry--a geometrical partner of gravitational curvature. He had thus constructed what we call a unified field theory. (from Albert Einstein, Creator and Rebel, pp. 223)

This length variable, “…this sort of change of size ”, was too much for even Einstein to accept. “…I do not believe that his theory will hold its ground in relation to reality” (Einstein, Sidelights on Relativity, pp 23). However, "Apart form the agreement with reality, it is at any rate a grandiose achievement of the mind". (Pais's, Subtle is the Lord, pp 341)

The Encyclopaedia Britannica says: “He (Weyl) produced the first unified field theory for which the Maxwell electromagnetic field and the gravitational field appear as geometrical properties of space-time” (Under Weyl, H.).  Also, “A ‘gauge invariant geometry’ extended the class of transformations to which the geometrical quantities characterizing space-time must be invariant; he could thus introduce four quantities identifiable with the potentials in classical electromagnetism… This theory is unsuitable because it leads to field equations of the fourth differential order, whereas, there is reason to believe that the correct equations are of the second order.” (from Unified field theory, pp. 259, vol. X).

"There is reason to believe" does not constitute any kind of proof whatsoever. While this objection may be significant in a three dimensional world with a three dimensional "way of thought", it turns to straw in four dimensionality and its way of thought.

Einstein's initial argument against the theory was that "...the lengths of objects would depend on their pasts", in apparent disagreement with spectral analysis.

With apparent difficulty, Hoffmann tries to translate (from the original German) Einstein's response to Weyl. Alluding to the quote above, he begins with a note of suspicion: "Such was Einstein's official argument against Weyl's theory...But it leaves something hidden. Here is an excerpt from a letter to Weyl in 1918 that shows a deeper Einstein objection":

Could one really accuse the Lord God of being inconsistent if he passed up the opportunity discovered by you to harmonize the  physical world? I think not. If he had made the world according to your plan [I would have said] to him reproachfully: "Dear God, if it did not lie within Thy power to give an objective meaning to the [equality of sizes of separated rigid bodies] why has Thou, Oh Incomprehensible One, not disdained ... to [preserve their shapes]?" Sic (Banesh Hoffman, Albert Einstein, Creator and Rebel, pp. 224; all brackets in original)

The theory was simply dropped for reasons of incredulity. It was not, and never has been, disproved. It was “revised” for quantum theory (gauge theory) however, and used to justify the “mathematical fiction” of renormalization!  Renormalization is where you subtract an artificial infinity from a calculated infinity to get a usable, finite answer.

Nobody, it seems, is willing to change what Heisenberg calls "our way of thought". I.E., three dimensionality into four-dimensionality. Even Einstein couldn't take his fourth dimension serious as an existential aspect of existence. And we cannot forget the fact that if Weyl was right, then Einstein's very reason to be, to find the unified theory himself, would end. At age 39.