Actual rods as they are moved about change their lengths compared with this absolute unit according to the route taken, and the differences correspond to the electromagnetic field. Einstein's curved space appears in a perfectly natural manner in this theory; no part of space-time is flat, even in the absence of ordinary matter, for that would mean infinite radius of curvature, and there would be no natural gauge to determine, for example, the dimensions of an electron—the electron could not know how large it ought to be, unless it had something to measure itself against.
The connection between the form of the law of gravitation and the total amount of matter in the world now appears less mysterious. The curvature of space indirectly provides the gauge which we use for measuring the amount of matter in the world.
Since the curvature is not independent of the gauge, Weyl does not identify it with the most fundamental quantity in nature. There is, however, a slightly more complicated invariant which is a pure number, and this is taken to be Action[1]. We can thus mark out a definite volume of space and time, and say that the action within it is 5, without troubling to define coordinates or the unit of measure! It might be expected that the action represented by the number 1 would have specially interesting properties; it might, for instance, be an atom of action and indivisible. Experiment has isolated what are believed to be units of action, which at least in many phenomena behave as indivisible atoms called quanta; but the theory, as at present developed, does not permit us to represent the quantum of action by the number 1. The quantum is a very minute fraction of the absolute unit.
When we come across a pure number having some absolute significance in the world it is natural to speculate on its possible interpretation. It might represent a number of discrete entities; but in that case it must necessarily be an integer, and it seems clear that action can have fractional values. An angle is commonly represented as a pure number, but it has not really this character; an angle can only be measured in terms of a unit of angle, just as a length is measured in terms of a unit of length.
- ↑ Appendix, Note 16.
