  Following steps analogous to classical Kaluza-Klein theory, we solve for the
quantum Riemannian geometry on $C^\infty(M)\otimes M_2(\mathbb{C})$ 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 $||F||^2$ of $A$ and a higher order $||A.F||^2$ 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.
