D O C . 1 7 0 O N O V E R D E T E R M I N A T I O N 1 6 5
Can one do justice to this insight about the natural
processes,[7]
to which we
surely must assign general importance, with a theory based on partial differential
equations? Quite certainly; we only have to “overdetermine” the field variables
with
equations.[8]
This means that the number of differential equations must be
greater than the number of field variables described by them. (In the case of the
general theory of relativity, the number of independent equations must only be
greater than the number of field variables reduced by 4, because according to this
theory, owing to the free choice of coordinates, the field variables are determined
by the equations only up to 4 of them.) Riemannian geometry shows us a fine ex-
ample of overdeterminacy that also seems to be substantially related to our prob-
lem. If one requires that all components of Riemann’s curvature tensor van-
ish, the manifold is Euclidean, hence completely defined, and tolerates no “initial
conditions” at all. The 4-dimensional continuum involves 20 algebraically mutual-
ly independent equations that are satisfied by the 10 -coefficients of the qua-
dratic metric form.
Analogously, let us try to overdetermine with equations the events in the electro-
magnetic and gravitational field, whereby the options are constrained by the fol-
lowing conditions:
1. The equations must be generally covariant; and only the -components of
the metric field and the ’s of the electric field should appear in them.
2. The sought-after equation system must in any case contain the one satisfying
gravitational theory and Maxwell’s theory, namely, the system of equations
where signifies the curvature tensor of second order.[10]
3. The sought-after system of equations, which overdetermines the field, must
however allow the static spherically symmetric solution that, according to the
above equations, describes the positive, or as the case may be, the negative
electron.[11]
If we succeed in overdetermining the whole field sufficiently with differential
equations while fulfilling these three conditions, we may hope that with these equa-
tions the mechanical behavior of singularities (electrons) will also be
determined[12] in such a way that the initial states of the field and the singularities
are also subject to the limiting conditions.
If it is at all possible to solve the quantum problem by differential equations, we
may hope to reach our goal along this route. In what follows I would like to present
what I have tried up to now in this direction, without being able to claim that the
[p. 361]
Rik,
lm
gμν
gμν
φμν
Ril κTil –=
Til φiαφlα
1
4
--gilφαβαβ - + =
Ril
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