The first is that we are led to a theory with differential wave propagation. The field functions are continuous functions of continuous parameters x and t, and the changes in the fields at a point x are determined by properties of the fields infinitesimally close to the point x. For most wave fields (for example, sound waves and the vibrations of strings and membranes) such a description is an idealization which is valid for distances larger than the characteristic length which measures the granularity of the medium. For smaller distances these theories are modified in a profound way. The electromagnetic field is a notable exception. Indeed, until the special theory of relativity obviated the necessity of a mechanistic interpretation, physicists made great efforts to discover evidence for such a mechanical description of the radiation field. After the requirement of an “ether” which propagates light waves had been abandoned, there was considerably less difficulty in accepting this same idea when the observed wave properties of the electron suggested the introduction of a new field. Indeed there is no evidence of an ether which underlies the electron wave. However, it is a gross and profound extrapolation of present experimental knowledge to assume that a wave description successful at “large” distances (that is, atomic lengths ≈ 10 cm) may be extended to distances an indefinite number of orders of magnitude smaller (for example, to less than nuclear lengths ≈ 10 cm). In the relativistic theory, we have seen that the assumption that the field description is correct in arbitrarily small space-time intervals has led—in perturbation theory—to divergent expressions for the electron self-energy and the bare charge. Renormalization theory has sidestepped these divergence difficulties, which may be indicative of the failure of the perturbation expansion. However, it is widely felt that the divergences are symptomatic of a chronic disorder in the small-distance behaviour of the theory. We might then ask why local field theories, that is, theories of fields which can be described by differential laws of wave propagation, have been so extensively used and accepted. There are several reasons, including the important one that with their aid a significant region of agreement with observations has been found. But the foremost reason is brutally simple: there exists no convincing form of a theory which avoids differential field equations.