Now, with the restrictions we have imposed, it may appear that there is nothing left of Weyl’s theory, since we have deprived the Weylian space-time of all its characteristics by invoking the tendency of adjustment in place of that of persistence. This view would, however, be too extreme. It is true that Weyl’s identification of the electromagnetic potentials, the
’s, with the
’s or particularities of structure of the continuum, now appears more in the light of a graphical representation; but as such it remains useful. Indeed, we must remember that the original urge which rendered a theory such as Weyl’s desirable was the necessity of reaching a more unified understanding of nature by including at least the representation of electrical manifestations in the structure of the fundamental continuum of space-time, instead of having to regard them as foreign occurrences. From this standpoint, Weyl’s theory appears secure.
But it is when we come to consider the problem of the Action of the universe, that Weyl’s theory opens up the most interesting possibilities. We remember that the action would constitute, so to speak, the nucleus whence all the laws of nature would issue. In fact, were the action known, all we should have to do would be to express its stationary condition, and the laws of matter and electricity would be revealed. But the trouble is, we do not know what the action may be; we can only guess at it. In this respect Weyl’s theory proves of help. For whereas Einstein’s theory restricts the action to being a space-time invariant, Weyl’s theory allows us to narrow down the choice by imposing the more stringent conditions of in-invariance. As a result, the Gaussian curvature
or
, which Einstein had assumed to constitute the function of action of the gravitational field, must be discarded and replaced by