Spectral noncommutative geometry and quantization: a simple example

The idea that the geometric structure of physical spacetime could be noncommutative exists in different versions. In some of versions, the noncommutativity of geometry is viewed as a direct effect of quantum mechanics, which disappears in the limit in which we consider processes involving actions much larger than the Planck constant [1]. In the noncommutative geometry approach of Connes et. al. (NCG) [2–5], on the other hand, noncommutativity is introduced as a feature of spacetime which exists independently from quantum mechanics. For instance, in the noncommutative version of the standard model [3], the theory is defined over a noncommutative spacetime, and is then quantized along conventional perturbative lines. More ambitiously, the spectral triple formulation [4] includes the gravitational field as well. For the gravitational field, however, conventional perturbative quantization methods fail [6]. The problem of going from the noncommutative, but non-quantummechanical, spectral triple dynamics to the full quantum dynamics is thus open. In a quantum theory that includes gravity, the geometric structure of spacetime is to be treated quantum mechanically. Therefore the noncommutative geometry of spacetime must be reinterpreted in quantum terms. Thus, in a theory of the physical world based on NCG and including quantum mechanics, the geometry of spacetime should be represented by a quantization of a noncommutative geometry. There should therefore be two distinct sources of noncommutativity in the theory: the noncommutativity of the elements of the algebra describing spacetime and, separately, the noncommutativity of the quantum mechanical variables. In this note, we address the problem of understanding what a quantization of a noncommutative geometry might be, and what is the relation between geometric noncommutativity and quantum noncommutativity. We study this problem using a simple model derived from [5] with the aim of developing structures and notions which, hopefully, could guide us in addressing the same issue in a full model including general relativity. In particular, we consider the spectral triple approach given in [4]. Within this approach, a dynamical model is given by the spectral triple (A,H,D), where H is a Hilbert space and A is a C∗ algebra represented on H , which are fixed once and for all; while D is a Dirac operator (in the sense of [2]) in H , which codes the value of the dynamical fields, and in particular of the spacetime metric, that is, the gravitational field. Thus D is the dynamical variable of the model and represents a classical configuration of the theory. The dynamics is then given by an action S[D]. To quantize the theory, we must find the Hilbert space K of its quantum states. A state in K will represent a quantum state of the noncommutative geometry: roughly, a probabilistic quantum superpositions of (noncommutative) geometries. Such a state will assign not a number, but rather a probability distribution, to the observable distance d(p, p′) between any two points. Observable quantities will be represented by operators on K. We construct the Hilbert space K and the dynamical operators for our simple model. From the quantum theory we obtain a concrete result: the physical distance between (certain) two points of the model, which in the classical theory can be an arbitrary nonnegative number d, turns out to be quantized as