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 $\hat{D}$ on $H=H\otimes K$. The triple $(A,H,\hat{D})$ can be viewed as the quantization of the triples $(A,H,D)$.