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arXiv:2608.02659

Curvature as a control field in helicoidal two-body systems

Abdullah Guvendi, Hassan Hassanabadi

quant-phSubmitted Aug 1, 2026• Announced Aug 4, 2026• Revised Aug 6, 2026 (v2)
v1 · Aug 1, 2026v2 · Aug 6, 2026v3 · Sep 3, 2026
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Abstract

Geometry is increasingly recognized as an active physical resource capable of modifying the behavior of dynamical systems beyond conventional external control mechanisms. Here, we develop a curvature--gauge framework in which the geometry of an embedded manifold acts as a tunable control parameter for classical trajectories and semiclassical states. By deriving an exact reduced Hamiltonian for a charged two-body system confined to a helicoidal surface, we show that curvature modifies the effective kinetic structure, while the projected gauge field reshapes the conserved momentum landscape. This geometric modification leads to controllable transitions between distinct dynamical regimes, including bounded phase-space structures, zero-energy localization, symmetry-breaking bifurcations, and critical soft-mode behavior. A semiclassical analysis of the geometry-dependent effective potential reveals a spectral reconstruction in which harmonic confinement changes into quartic critical behavior at the localization threshold. These results establish engineered geometry as a route for controlling localization, dynamical stability, and spectral organization in curved classical and semiclassical systems, where deformation itself becomes a functional degree of freedom rather than a passive constraint.

Comments
10 pages, 4 figures

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We investigate the effective dynamics of a particle confined near a space curve. In the strict thin-layer reduction of the nondegenerate transverse ground state, torsion does not enter the local effective Hamiltonian, which contains only the curvature-induced scalar geometric potential. In contrast, for a thin guide with finite transverse width, a leading-order adiabatic projection onto the twofold-degenerate first-excited transverse band renders the rotation of the Frenet normal frame dynamically relevant and generates a matrix-valued Abelian gauge potential. Using a projection-based derivation in a co-rotating Frenet-frame basis, we show that this effective gauge potential is directly determined by the local torsion of the curve. The resulting effective Hamiltonian takes a gauge-covariant form and produces two transverse-mode branches whose parabolic dispersions are shifted in opposite directions in momentum space. For closed curves, the associated holonomy is controlled by the integrated torsion and leads to geometric interference. These results provide a direct realization of a Wilczek--Zee-type connection induced purely by spatial geometry in curved quantum waveguides. We further construct a classical-wave analogue using the degenerate bending modes of an isotropic elastic rod, demonstrating that the same torsion-induced gauge structure appears in continuum wave physics.
This work introduces a minimal variational framework in which localized energy basins and their fixed points form the foundation for emergent quantum dynamics. Rather than imposing quantization axiomatically, discrete spectra arise naturally from geometric regularity, topological invariants, and adjacency-driven transport within a continuous relational configuration space. Key contributions include: Emergent Schrödinger Dynamics: Oscillatory modes near basin fixed points admit a complex structure, leading to a Schrödinger-type evolution derived from a quadratic variational principle rather than canonical quantization. Topology-Induced Quantization: Integer-valued invariants such as winding and linking numbers enforce discrete excitation spectra without probabilistic postulates. Hydrogenic Consistency: The framework reproduces the hydrogen spectrum with emergent parameters, matching experimental wavelengths to within 0.1 nm. Conceptual Foundations: Spacetime is treated as an emergent descriptor of basin adjacency and deformation, not as a pre-imposed manifold. This approach provides a conservative and mathematically rigorous route to quantum-like behavior grounded in geometry and topology, while remaining compatible with classical correspondence in the coarse-grained limit. Extensions to relativistic regimes, gauge structures, and gravitational coupling are deferred to future work. Keywords: Emergent quantum dynamics, fixed-point geometry, basin framework, topology-induced quantization, Schrödinger evolution, variational principles.