Fields over Unsharp Coordinates

It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete, or "unsharp" in one of two ways. In the literature, various quantum gravity models of space-time at short distances point towards one of these two types of unsharpness. Here, we investigate the properties of fields over such unsharp coordinates. We find that these fields are continuous--but possess only a finite density of degrees of freedom, similar to fields on lattices. As a special case we recover the Shannon sampling theorem of information theory.