On the three short-distance structures which can be described by linear operators (Q1305708)

This paper contains an informal discussion of the idea that in a quantum theory of space-time the spatial information is encoded in symmetric operators on some (infinite dimensional) Hilbert space rather than self-adjoint operators (using the terminology of operational quantum physics one could equivalently say: the spatial information is described by positive-operator-valued (POV) measures rather than projection-operator-valued (PV) measures). It is discussed that these symmetric operators can only describe either continuous, discrete or `fuzzy' space-time structures. The `fuzzy' geometries are discussed in more detail. The author presents indications that the `fuzzy' geometries may be UV regular and compatible with external symmetries and that they also may play a role at the Planck scale or even in large scale gravity.