Quantum Kaluza-Klein theory with M2(C)

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on C-infinity(M) circle times M-2(C) in terms of classical Riemannian geometry on a smooth manifold M, a finite quantum geometry on the algebra M-2(C) of 2x2 matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on M-2(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(C), the Maxwell action parallel to F parallel to(2) of A and a higher order parallel to A.F parallel to(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.