334 THE LATER JOURNEYquantum phenomena). This was his motivation: 'It is anomalous to replace the
four-dimensional continuum by a five-dimensional one and then subsequently to
tie up artificially one of these five dimensions in order to account for the fact that
it does not manifest itself. We have succeeded in formulating a theory which for-
mally approximates Kaluza's theory without being exposed to the objection just
stated. This is accomplished by the introduction of an entirely new mathematical
concept' [E26].
The new mathematics presented by Einstein and Mayer in two papers [E24,
E27] does not involve the embedding of the Riemann manifold R 4 in a five-space.
Instead, a five-dimensional vector space M 5 is associated with each point of R 4
and the local Minkowski space (call it M 4 ) is embedded in the local M 5 , which
has (4 + l)-metric. Prescriptions are introduced for decomposing tensors in M 5
with respect to M 4. The transport of 5-tensors from one M 5 to another M 5
attached to a neighboring point in R 4 is defined. This involves a five-dimensional
connection of which (so it is arranged) some components are identified with the
Riemannian connection in R^ while, in addition, only an antisymmetrical tensor
Fu appears, which is identified with the electromagnetic field. However (as Ein-
stein noted in a letter to Pauli [E28]), one has to assume that Fkl is the curl of a
4-vector; also, the Einstein-Mayer equations are not derivable from a variational
principle. After 1932 we find no trace of this theory in Einstein's work.
In a different environment, he made one last try at a five-dimensional theory.
He was in America now. His old friend Ehrenfest was gone. The year was 1938.
This time he had in mind not to make x less real than Kaluza-Klein, but more
real. At first he worked with Peter Bergmann; later Valentin Bargmann joined
them. Altogether, their project was under active consideration for some three
years. Bergmann's textbook tells us what the motivation was:
It appeared impossible for an iron-clad four-dimensional theory ever to account
for the results of quantum theory, in particular for Heisenberg's indeterminacy
relation. Since the description of a five-dimensional world in terms of a four-
dimensional formalism would be incomplete, it was hoped that the indetermi-
nacy of 'four dimensional' laws would account for the indeterminacy relation
and that quantum phenomena would, after all, be explained by a [classical]
field theory. [Bl]
Their approach was along the lines of Klein's idea [K4] that the 5-space is closed
in the fifth direction with a fixed period. The group is again G 5 (see Eq. 17.5).
The line element (Eq. 17.1), the condition (Eq. 17.3) on yss, and the definition
(Eq. 17.6) of gik are also maintained, but Eq. 17.2 is generalized. It is still assumed
that the 7,5 (the electromagnetic potentials) depend only on x', but (and this is
new) the gik are allowed to depend periodically on x^5. The resulting formalism is
'These rules are summarized in papers by Pauli and Solomon [P2] that have been reproduced in
Pauli's collected works [P3].