Non-perturbative Quantum Dynamics on Embedded Submanifolds: From Geometric Mass to Higgs Potentials

Abstract

We establish a quantum dynamics framework for curved submanifolds embedded in higher-dimensional spaces. Through rigorous dimensional reduction, we derive the first complete Schrödinger and Klein-Gordon equations incorporating non-perturbative geometric interactions-resolving ambiguities in constrained quantization. Crucially, extrinsic curvature of the ambient manifold governs emergent low-dimensional quantum phenomena. Remarkably, this mechanism generates scalar field masses matching Kaluza-Klein spectra while eliminating periodic compactification requirements. Geometric induction concurrently produces Higgs mechanism potentials. Particle masses emerge solely from submanifold embedding geometry, with matter-field couplings encoded in curvature invariants. This enables experimental access to higher-dimensional physics at all energy scales through geometric induction. We also discuss the Higgs vacuum near small-mass black holes.