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

connection.[8]

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.

ds

ds

[p. 10]