Quantum Kaluza-Klein theory with M2(ℂ)

Abstract

Abstract Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on C∞(M) ⊗ M2(ℂ) in terms of classical Riemannian geometry on a smooth manifold M , a finite quantum geometry on the algebra M2(ℂ) of 2 × 2 matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on M2(ℂ), we show that this cross term data amounts in the simplest case to a 1-form Aμ 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 Aμ. We show further that the quantum Ricci scalar on the product decomposes into a classical Ricci scalar on M, the Ricci scalar on M2(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 ϕ on M which imparts mass-splitting to some of the components of a scalar field on the product as in previous work.