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

TL;DR

The paper develops a non-perturbative framework for quantum dynamics on curved submanifolds embedded in higher-dimensional spaces, deriving effective Schrödinger and Klein-Gordon equations whose geometry-induced terms encode masses and Higgs-like potentials without requiring compactified extra dimensions. By carefully analyzing tubular neighborhoods and the interplay of intrinsic and ambient geometry, it shows that extrinsic curvature and ambient Ricci curvature give rise to mass scales and symmetry breaking phenomena, including a geometric origin for spontaneous Higgs potential. It also explores stability criteria for manifold–matter couplings and predicts observable Higgs-mass corrections near horizons of low-mass black holes, suggesting that higher-dimensional geometry could be probed across energy scales through geometric induction. The framework yields explicit mass-generation mechanisms, curvature-driven symmetry breaking, and horizon-related vacuum stability, with potential extensions to gauge fields and spinors in future work.

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.