Space, Time, Matter by Hermann
Weyl
God Is The Geometer (Maybe)
A classic of physics ... the first systematic presentation of Einstein’s theory
of relativity—British Journal for Philosophy and Science. Long one of the
standard texts in the field, this excellent introduction probes deeply into
Euclidean space, Riemann’s space, Einstein’s general relativity,
gravitational waves and energy, laws of conservation.
Personal Review: Space, Time, Matter by Hermann Weyl
Weyl's book is most famous for introducing gauge theory, which was later
reborn in the form of phase transformations in quantum theory. Weyl did
not live guite long enough to hear of the latter being applied by Yang and
Mills, though he socially interacted with Yang in his last year at Princeton.
Einstein and Pauli both criticized Weyl's original unified theory based on
general relativity using a length gauge, both as implying false empirical

consequences (Einstein), since it implied tiny changes of length dependent
on path and as untestable (Pauli). (Obviously it could not both be
empirically false and non-empirical.) Yet Eddington and later Einstein
himself revived similar theories. Eddington claimed that the length
differences were to tiny as to be undetectable, but also that his own gauge
theory could be thought of not as literal space/time structure but as a
geometrization of an abstract background theory for specific space/time
structures.
Thomas Ryckman's The Reign of Relativity: Philosophy in Physics
1915-1925 (Oxford Studies in the Philosophy of Science)has an excellent
eighty page discussion of these ideas of Weyl in relativity, as well as
chapters on those of Eddington in the 1920s.
Another novelty is Weyl's suggestion that General Relativity could be tied
to observation via the conformal structure as representing light cones and
the projective structure as particles in free fall. This alternative to the rods
and clocks approach, on the basis of which Weyl was criticized, has been
developed by Ehlers (who edited the new German edition of this work)
Pirani and Schild.
Weyl also introduces what he calls "tensor densities" which Shouten called
"Weyl tensors" and Synge and Schild call oriented tensors, often called
twisted tensors. These are analogous to and include "axial vectors."
Weyl's introduction of the "affine connection" after criticism of Levi-Civita's
notion of parallelism led the way to further notions of connections and
generalization of the notion of connection as such by Elie Cartan and
others.
These are but a few of the intellectual gems in this work.
The philosophical parts are, unfortunately, almost uniformly mistranslated.
The phenomenological introduction is re-translated in Kockelmans and
Kisiel, eds. Phenomenology and the Natural Sciences (SPEP). (Courant
suggested Weyl as successor to Husserl in the philosophy chair at
Goettingen!) This together with the misprints in formulas, makes it
desirable that the whole book be retranslated.
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