9780192806727.pdf

Bargmann and Bergmann told me that Einstein thought that the higher Fourier

components might somehow be related to quantum fields. He gave up the five-

dimensional approach for good when these hopes did not materialize.

3. Addenda. Other attempts to use five- or more-dimensional manifolds for
   a description of the physical world continue to be made.
   a) Soon after the Einstein-Mayer theory, another development in five-dimen-
   sional theory began, known as projective relativity, to which many authors con-
   tributed. In this theory the space-time coordinates xl are assumed to be homo-
   geneous functions of degree zero in five coordinates AT^1 . A Riemann metric with
   invariant line element ds^2 = gia,dX"dX" is introduced in the projective 5-space
   (which has signature 4 + 1). The condition

where <pk are the electromagnetic potentials and F is an arbitrary homogeneous
function of degree one in the X". Thus the projective coordinates themselves are
directly related to the potentials up to a 5-gauge transformation.
The Dirac equation in projective space was discussed by Pauli [ P4]. Variational
methods were applied to this theory by Pais [P5] with the following results. Let

*For detailed references, see especially [S2]. The best introduction to this subject is a pair of papers
by Pauli [P4].
**The mathematical connection between this theory and the Kaluza-KIein theory is discussed in
[Bl].

UNIFIED FIELD THEORY 335

discussed in much detail in Bergmann's book (see also [B2] and [PI]). Two ver-

sions of the theory were considered. In the first one [E29], the field equations are

derived from a variational principle. Because of the new x" dependence, they are

integro-differential equations (an integration over xs remains). They also contain

several arbitrary constants because the action can contain new invariants (depend-

ing on derivatives of the gik with respect to x^5 ). In a second version [E30], the

variational principle is abandoned and Bianchi identities which constrain these

constants are postulated.

In theories of this kind, the g,k can be represented by (the period is normalized

to ZTT):

(17.15)

(17.16)

takes the place of the cylinder condition. The quantities 7* = dxk/dX" project
from the 5-space to the 4-space.** One proves that

(17.17)