Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on $C^{\mathrm{\infty }}(M)\otimes M_2(\mathbb{C})$ in terms of classical Riemannian geometry on a smooth manifold $M$, a finite quantum geometry on the algebra $M_2(\mathbb{C})$ of $2\times 2$ matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on $M_2(\mathbb{C})$, we show that this cross term data amounts in the simplest case to a 1-form $A_{\mu }$ on $M$, which we regard as like a gauge-fixed background field. We show in this case that a real scalar field on the product algebra with its noncommutative Laplacian decomposes on $M$ into two real neutral fields and one complex charged field minimally coupled to $A_{\mu }$. We show further that the quantum Ricci scalar on the product decomposes into a classical Ricci scalar on $M$, the Ricci scalar on $M_2(\mathbb{C})$, the Maxwell action $\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }{\mathrm{\mid }}^2$ of $A$ and a higher order $\mathrm{\mid }\mathrm{\mid }A\mathrm{.}F\mathrm{\mid }{\mathrm{\mid }}^2$ term. Another solution of the QRG on the product has $A=0$ and a dynamical real scalar field $\phi$ on $M$ which imparts mass-splitting to some of the components of a scalar field on the product as in previous work.