Canonical quantization and the spectral action; a nice example

We study the canonical quantization of the theory given by Chamseddine–Connes spectral action on a particular finite spectral triple with algebra _M2(C)⊕C$M_2\left(\mathbb{C}\right)\oplus \mathbb{C}$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge invariant. The results are similar in the two cases, but the gauge invariance of the spectral action manifests itself by the presence of a non-trivial degeneracy structure for our distance operator.