D O C . 7 5 N O B E L L E C T U R E 8 1
the laws of the electromagnetic field. Unfortunately, in this endeavor, unlike the
case of the derivation of the theory of gravitation (equivalence of inertial and grav-
itational mass), we cannot base our efforts on empirical facts. Instead, we have to
base the on the criterion of mathematical simplicity, which is not free from arbi-
trariness. The attempt which at present appears to be the most successful is the one
built on the ideas of Levi-Civita, Weyl, and Eddington, to replace Riemannian ge-
ometry with the more general theory of the affine
The characteristic assumption of Riemannian geometry is the attribution of a
“distance” to two points in each others infinitesimal neighborhood. The square
of is a homogeneous function of second order in the coordinate differentials.
The validity of Euclidean geometry in any infinitesimally small region follows
from this (apart from certain conditions of reality). Therefore, to every line element
(or vector) at a point P, a parallel and equal line element (or vector) is assigned
through any given infinitesimally neighboring point P′ (affine connection). The
Riemannian metric determines an affine connection. Conversely, however, when an
affine connection (a law of infinitesimal parallel transport) is mathematically giv-
en, generally a Riemannian metric from which it could be derived does not exist.
The most important concept of Riemannian geometry, the “curvature of space,”
upon which the gravitational equations rest, is based exclusively on the “affine con-
nection.” If one defines an affine connection in a continuum without initially setting
out from a metric, one thus has a generalization of the Riemannian geometry in
which the most important quantities are still retained. By seeking out the simplest
differential equations that an affine connection can be subjected to, one may hope
to hit upon a generalization of the gravitational equations that contain the laws of
the electromagnetic field. This hope has, in fact, been satisfied, yet I do not know
whether the formal relation thus gained should be regarded as an enrichment of
physics as long as it does not deliver any new physical relations. Specifically, in my
opinion, a field theory can only be satisfactory when it allows the elementary elec-
tric bodies to be represented as singularity-free solutions.
Furthermore, it should not be forgotten that a theory of elementary electric bod-
ies cannot be separated from the issues of quantum theory. Relativity theory itself
has proved powerless before this most profound of physical problems of the present
day. Even if someday the form of the general equations may undergo modifica-
tions, however profound, through the solution of the quantum problem, even if the
quantities by means of which we represent the elementary processes change entire-
ly, the principle of relativity shall never be abandoned and the laws that have hith-
erto been derived from it shall at least be retained as limiting laws.
[p. 10]
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