are both gauge-invariant tensors. So, therefore, is R^ (which is not symmetric
now); R is a scalar but is not gauge invariant: R' =~^1 R, sinceg"' = ~lg^.
It is obvious what Weyl was after: F^ is to be the electromagnetic field. In
addition, he could show that his group leads automatically to the five conservation
laws for energy, momentum, and charge. His is not a unified theory if one
demands that there be a unique underlying Lagrangian L that forces the validity
of the gravitational and electromagnetic field equations, since to any L one can
add an arbitrary multiple of the gauge-invariant scalar ^F^F^yg dx. For a
detailed discussion and critique of this theory, see books by Pauli [PI] and by
Bergmann [Bl].
When Weyl finished this work, he sent a copy to Einstein and asked him to
submit it to the Prussian Academy [W3]. Einstein replied, 'Your ideas show a
wonderful cohesion. Apart from the agreement with reality, it is at any rate a
grandiose achievement of the mind' [E31]. Einstein was of course critical of the
fact that the line element was no longer invariant. The lengths of rods and the
readings of clocks would come to depend on their prehistory [E32], in conflict with
the fact that all hydrogen atoms have the same spectrum irrespective of their
provenance. He nevertheless saw to the publication of Weyl's paper, but added a
note in which he expressed his reservations [E33].* Weyl's response was not con-
vincing. Some months later, he wrote to Einstein, '[Your criticism] very much
disturbs me, of course, since experience has shown that one can rely on your intu-
ition' [W4].
This theory did not live long. But local gauge transformations survived, though
not in the original meaning of regauging lengths and times. In the late 1920s,
Weyl introduced the modern version of these transformations: local phase trans-
formations of matter wave functions. This new concept, suitably amplified, has
become one of the most powerful tools in theoretical physics.
17e. The Later Journey: a Scientific Chronology
The last period of Einstein's scientific activities was dominated throughout by
unified field theory. Nor was quantum theory ever absent from his mind. In all
those thirty years, he was as clear about his aims as he was in the dark about the
methods by which to achieve them. On his later scientific journey he was like a
traveler who is often compelled to make many changes in his mode of transpor-
tation in order to reach his port of destination. He never arrived.
The most striking characteristics of his way of working in those years are not
all that different from what they had been before: devotion to the voyage, enthu-
*In 1921, Einstein wrote a not very interesting note in which he explored, in the spirit of Weyl, a
relativity theory in which only g^dx'dx' = 0 is invariant [E34].
UNIFIED FIELD THEORY
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