Spectral Noncommutative Geometry and Quantization

We explore the relation between (spectral) noncommutative geometry and quantum mechanics. We consider a dynamical model based on a triple (A,H,D), which mimics aspects of the spectral formulation of general relativity. Its phase space is the space of on-shell Dirac operators. We construct the corresponding quantum theory using a covariant canonical quantization. The Connes distance between two states over A (“spacetime points”) turns out discrete and we compute its spectrum. The quantum states of the geometry form a Hilbert space K, and D is promoted to an operator D̂ on H = H⊗K. The triple (A,H,D̂) can be viewed as the quantization of the triples (A,H,D).