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
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
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:
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.