I know that electromagnetism is force via curvature in a U(1)-bundle. I am now trying to literally visualize this, and write down equations that make this manifest.
KK (Kaluza-Klein) theory is the only promising candidate, because it describes (classical) motion of charged particles as geodesics in some U(1)-bundle. If we project this phase space down to the base space (spacetime), and then apply the Feynman quantization procedure, do we recover the quantum mechanics of a charged particle?
Is there some non-relativistic formulation of KK theory, that say, describes only magnetism, and involves only 3D space-manifolds, which we can Feynman-quantize (i.e. write a path integral over all trajectories), and yields quantum mechanics of a charged particle in a magnetic field? This would suffice as a proof of the statement
EM Force is curvature: i.e., quantum mechanical partition function is a path-integral with action $$S[\gamma]=\int h(\dot{\gamma},\dot{\gamma})\,dt$$ where h is the KK metric on the U(1)-bundle, I.e., we have replaced the force of electromagnetism with just the curvature of a higher-dimensional space.
