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

TL;DR

This paper develops a non-perturbative quantum dynamics framework for curved embedded submanifolds, revealing how geometry induces particle masses and Higgs potentials, linking higher-dimensional physics to observable low-dimensional phenomena.

Contribution

It provides the first complete derivation of Schrödinger and Klein-Gordon equations with geometric interactions, showing mass generation and Higgs potentials from embedding geometry.

Findings

Mass spectra match Kaluza-Klein predictions

Higgs potentials arise from extrinsic curvature

Particle masses depend on submanifold geometry

Abstract

We establish a quantum dynamics framework for curved submanifolds embedded in higher-dimensional spaces. Through rigorous dimensional reduction, we derive the first complete Schr\"{o}dinger 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.