Geometry is increasingly recognized as an active physical resource capable of modifying the behavior of quantum systems beyond conventional external control mechanisms. Here, we establish a curvature--gauge framework in which the geometry of an embedded manifold functions as a tunable control field for classical trajectories and quantum states. By deriving an exact reduced Hamiltonian for a two-body system confined to a helicoidal surface, we demonstrate that curvature modifies the effective kinetic structure while a projected gauge field reconstructs the conserved momentum landscape. This geometric renormalization produces controllable transitions between distinct dynamical regimes, including bounded phase-space structures, zero-energy localization, symmetry-breaking bifurcations, and critical soft-mode behavior. Quantization of the geometry-dependent effective potential reveals a corresponding spectral reconstruction, where harmonic confinement evolves into quartic critical states at the localization threshold. These results establish engineered geometry as a general route for controlling localization and spectral organization in curved quantum, electronic, photonic, and synthetic platforms, where deformation itself becomes a functional degree of freedom rather than a passive constraint.
Geometry is increasingly recognized as an active physical resource capable of modifying the behavior of quantum systems beyond conventional external control mechanisms. Here, we establish a curvature--gauge framework in which the geometry of an embedded manifold functions as a tunable control field for classical trajectories and quantum states. By deriving an exact reduced Hamiltonian for a two-body system confined to a helicoidal surface, we demonstrate that curvature modifies the effective kinetic structure while a projected gauge field reconstructs the conserved momentum landscape. This geometric renormalization produces controllable transitions between distinct dynamical regimes, including bounded phase-space structures, zero-energy localization, symmetry-breaking bifurcations, and critical soft-mode behavior. Quantization of the geometry-dependent effective potential reveals a corresponding spectral reconstruction, where harmonic confinement evolves into quartic critical states at the localization threshold. These results establish engineered geometry as a general route for controlling localization and spectral organization in curved quantum, electronic, photonic, and synthetic platforms, where deformation itself becomes a functional degree of freedom rather than a passive constraint.
Spectral constraints can alter the observable manifestation of quantum geometry without changing the local eigenspace. Relativistic electron vortices provide a sharp example: the spin orbit interaction redistributes spin and orbital angular momentum, yet its magnetic contribution differs between localized three-dimensional wave packets and stationary monochromatic beams. We resolve this discrepancy by showing that fixing the energy replaces ambient momentum derivatives in position-weighted current moments by shell-tangent derivatives and derive a Hermitian selection rule for first-order observables in momentum-dependent internal subspaces. Its extension-independent form applies beyond Dirac systems, including a broad class of antisymmetric first moments of scalar-charge velocity currents. For the free positive-energy Dirac doublet, the magnetic correction is an exact non-Abelian Berry-curvature contraction with a pointwise saturable quantum-metric bound. Changing the relative phase between the two σz spinor components switches this correction from zero in a conserved transverse-helicity sector to a finite value in a radial coherent superposition. At fixed finite Lorentz factor, the radial response scales as O(θ/γ), compared with O(θ2) for longitudinal polarization. Spectral reduction and internal coherence thus provide distinct controls of the observable imprint of quantum geometry.
Exact cylindrical solutions of the free Dirac equation provide natural single-particle modes for a broad class of axially symmetric relativistic fermion systems, including electron vortices, twisted-particle scattering, rotating matter, and cylindrical field quantization, but are commonly formulated in different internal bases. We derive the general regular positive-energy cylindrical solution at fixed energy, transverse and longitudinal momenta, and total angular momentum, and show how the commonly used spin-polarized, separation, helicity, and transverse-helicity modes are embedded in the resulting two-dimensional solution space. A conserved transverse operator resolves the residual doublet into two symmetry-defined branches. Their parameter-dependent eigenspaces admit a gauge-invariant quantum-geometric characterization, with opposite Berry curvatures, a common quantum metric, and saturation of the two-level metric--curvature relation. We further derive symmetry constraints on branch conversion and the corresponding reduced two-state dynamics. The resulting framework connects mode classification, symmetry resolution, quantum geometry, and dynamics across distinct cylindrical Dirac settings.
Geometric confinement is known to modify single-particle dynamics through effective potentials, yet its imprint on the interacting quantum vacuum remains largely unexplored. In this work, we investigate the Maxwell--Klein--Gordon system constrained to curved surfaces and demonstrate that the geometric potential Σgeom(r) acts as a local renormalization environment. We show that extrinsic curvature modifies the scalar loop spectrum, entering the vacuum polarization as a position-dependent mass correction M2(r)→m2+Σgeom(r). This induces a finite, gauge-invariant ``geometry-induced running'' of the electromagnetic response. In the long-wavelength regime (∣Q∣R≪1), we derive a closed-form expression for the relative frequency shift Δω/ω, governed by the overlap between the electric energy density and the geometric potential. Applying this formalism to Gaussian bumps, cylindrical shells, and tori, we identify distinct spectral signatures that distinguish these quantum loop corrections from classical geometric optics. Our results suggest that spatial curvature can serve as a tunable knob for ``vacuum engineering,'' offering measurable shifts in high-Q cavities and plasmonic systems.
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.
We study a charged particle moving outside a spiral (screw) dislocation in a uniform magnetic field. Although the geometry is locally Euclidean, its global screw identification couples the conserved axial and azimuthal quantities through the combination J=ℓ−βk After eliminating the cyclic coordinates, the radial motion is governed by an inverse-square term determined by J and a harmonic confining term generated by the magnetic field. The corresponding reduced quantum problem is exactly solvable for qB0=0, yielding an equally spaced spectrum with an orientation-dependent energy shift. The radial probability density depends on J, and opposite values of the screw-modified quantum number have identical spatial profiles but different energies for a fixed magnetic-field orientation. The radial size scales as ⟨r2⟩n∝∣qB0∣−1/2, with the magnetic length setting the localization scale. At qB0=0, the confining term vanishes, and the problem reduces to a scale-free inverse-square system. Thus, the screw parameter enters the dynamics through the global coupling of the conserved axial and azimuthal quantities, whereas the magnetic field provides radial confinement and its orientation determines the spectral asymmetry.