On the Only 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 X∗ i = 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. DAMTP-1998-38, hep-th/9806013 1 . NONCOMMUTING LIMITS AND INFINITE-DIMENSIONAL MATRICES Since limits do not have to be interchangeable, infinite-dimensional matrices possess subtle features without analogues among finite dimensional matrices. Let us first look at the trivial example k(n1, n2, a) = n1f(a) + n2g(a) n1 + n2 , n1, n2 ∈ IN (1.1) where f and g are regular functions. Now limn1→∞ limn2→∞ k(n1, n2, a) = g(a). It seems that when n1 and n2 are taken to infinity the information on f is lost. Of course it is only hidden and is not lost, because we can find f at infinity by approaching infinity on other paths, e.g. limn2→∞ limn1→∞ k(n1, n2, a) = f(a). In this sense, infinity is able to store away things in such a way that various information can be retrieved by checking at various corners at infinity. Now take an infinite dimensional matrix Xij , (i, j = 1, 2, ...,∞) which obeys Xij = X ji. On the set D of all vectors which have only a finite (but arbitrarily large) number of nonzero entries, X is clearly symmetric, i.e. all its expectation values are real: