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

We point out that if spatial information is encoded through linear operators Xi, or ‘infinite-dimensional matrices’ with an involution Xi∗ = Xi then these Xi can only describe either continuous, discrete or certain “fuzzy” space-time structures. We argue that the fuzzy space structure may be relevant at the Planck scale. The possibility of this fuzzy space-time structure is related to subtle features of infinite-dimensional matrices which do not have an analogue in finite dimensions. For example, there is a slightly weaker version of self-adjointness: symmetry, and there is a slightly weaker version of unitarity: isometry. Related to this, we also speculate that the presence of horizons may lead to merely isometric rather than unitary time evolution.