I can only think of one interpretation of a fractional number which can have an absolute significance, though doubtless there are others. The number may represent the probability of something, or some function of a probability. The precise function is easily found. We combine probabilities by multiplying, but we combine the actions in two regions by adding; hence the logarithm of a probability is indicated. Further, since the logarithm of a probability is necessarily negative, we may identify action provisionally with minus the logarithm of the statistical probability of the state of the world which exists.
The suggestion is particularly attractive because the Principle of Least Action now becomes the Principle of Greatest Probability. The law of nature is that the actual state of the world is that which is statistically most probable.
Weyl's theory also shows that the mass of a portion of matter is necessarily positive; on the original theory no adequate reason is given why negative matter should not exist. It is further claimed that the theory shows to some extent why the world is four-dimensional. To the mathematician it seems so easy to generalise geometry to dimensions, that we naturally expect a world of four dimensions to have an analogue in five dimensions. Apparently this is not the case, and there are some essential properties, without which it could scarcely be a world, which exist only for four dimensions. Perhaps this may be compared with the well-known difficulty of generalising the idea of a knot; a knot can exist in space of any odd number of dimensions, but not in space of an even number.
Finally the theory suggests a mode of attacking the problem of how the electric charge of an electron is held together; at least it gives an explanation of why the gravitational force is so extremely weak compared with the electric force. It will be remembered that associated with the mass of the sun is a certain length, called the gravitational mass, which is equal to 1.5 kilometres. In the same way the gravitational mass or radius of an electron is 7.10-56 cms. Its electrical properties are similarly associated with a length 2.10-13 cms., which is called the electrical radius. The latter is generally supposed to correspond to the electron's actual dimensions. The theory suggests that the ratio
