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arXiv:2608.14041

Global structure and holonomy of conserved resolutions in the cylindrical Dirac doublet

Zhongze Guo, Bei Xu, Qiang Gu

quant-phSubmitted Aug 14, 2026• Announced Aug 17, 2026• Revised Sep 10, 2026 (v2)
v1 · Aug 14, 2026v2 · Sep 10, 2026
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Abstract

Cylindrical Dirac modes underlie constructions in rotating QCD matter, boost-invariant Dirac-field quantization in heavy-ion physics, and high-energy twisted-particle scattering. The corresponding complete spinor frames can be regarded as alternative bases, but their equivalence does not determine the global behavior of eigenlines selected by conserved observables. Within the positive-energy doublet of the free massive Dirac Hamiltonian, we compare three conserved resolutions: $K$, which couples spin to transverse momentum; $K_m$, a mass-dependent operator derived from the transverse Dirac Hamiltonian; and helicity. On the common regular domain away from the momentum axis, explicit smooth, single-valued $SU(2)$ transformations relate all three splittings. Although globally $SU(2)$-equivalent on this common domain, they exhibit three distinct global extension behaviors. The $K$ projectors have azimuth-dependent polar limits and do not extend continuously to the axis. For nonzero mass, the $K_m$ projectors extend smoothly over the enclosed momentum ball and define Chern-trivial eigenlines. The helicity projectors are smooth on every nonzero momentum sphere, but their eigenlines carry opposite unit Chern numbers and cannot extend through the enclosed origin. The parent positive-energy Dirac connection Abelianizes exactly in the $K$ eigenlines on fixed-azimuth meridians, whereas its full three-dimensional curvature has noncommuting components. We obtain the azimuthal Wilson loop in closed form and derive the exact conversion probability between the two $K$ branches under purely geometric positive-energy transport.

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23 pages

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•Sep 19, 2026•0 citations
Cylindrical Dirac modes underlie constructions in rotating QCD matter, boost-invariant Dirac-field quantization in heavy-ion physics, and high-energy twisted-particle scattering. The corresponding complete spinor frames can be regarded as alternative bases, but their equivalence does not determine the global behavior of eigenlines selected by conserved observables. Within the positive-energy doublet of the free massive Dirac Hamiltonian, we compare three conserved resolutions: KKK, which couples spin to transverse momentum; KmK_mKm​, a mass-dependent operator derived from the transverse Dirac Hamiltonian; and helicity. On the common regular domain away from the momentum axis, explicit smooth, single-valued SU(2)SU(2)SU(2) transformations relate all three splittings. Although globally SU(2)SU(2)SU(2)-equivalent on this common domain, they exhibit three distinct global extension behaviors. The KKK projectors have azimuth-dependent polar limits and do not extend continuously to the axis. For nonzero mass, the KmK_mKm​ projectors extend smoothly over the enclosed momentum ball and define Chern-trivial eigenlines. The helicity projectors are smooth on every nonzero momentum sphere, but their eigenlines carry opposite unit Chern numbers and cannot extend through the enclosed origin. The parent positive-energy Dirac connection Abelianizes exactly in the KKK eigenlines on fixed-azimuth meridians, whereas its full three-dimensional curvature has noncommuting components. We obtain the azimuthal Wilson loop in closed form and derive the exact conversion probability between the two KKK branches under purely geometric positive-energy transport.
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.
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A spin-orbit Hamiltonian with an effective gauge structure carries two distinct loop objects that are routinely conflated: an energy-independent Wilson holonomy, which organizes interference and internal spin transport, and an energy-dependent monodromy, which quantizes the spectrum. We show that cleanly separating these objects supplies a precise, computable bridge between the loop/holonomy representation of gauge theories and condensed-matter spin-orbit transport. The construction maps a spin-orbit Hamiltonian to an effective U(1)U(1)U(1) plus internal non-Abelian connection, reduces it to a first-order transport problem, and reads physical predictions from holonomy, monodromy, curvature, and eigenphase data. Two rings make the separation explicit. For a Dirac (graphene) ring with Rashba coupling and Aharonov-Bohm flux, the total holonomy factorizes exactly into a commuting U(1)U(1)U(1) flux phase times an internal spin/pseudospin holonomy, and the spectrum follows from a holonomy-eigenvalue condition. For a Rashba-Dresselhaus ring, the internal SU(2)SU(2)SU(2) transport is genuinely non-Abelian away from the α=±βα=\pmβα=±β pure-gauge locus, where curvature controls path ordering; spectral quantization then requires an explicit first-order reduction obtained by phase-space doubling of the second-order Schrödinger problem. A non-Abelian Stokes formulation and Magnus expansion serve as ordering diagnostics rather than spectral tools. Spin-network ideas enter only as historical geometric motivation, not as a dynamical import into spintronics.
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 σzσ_zσz​ 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)O(θ/γ)O(θ/γ), compared with O(θ2)O(θ^2)O(θ2) for longitudinal polarization. Spectral reduction and internal coherence thus provide distinct controls of the observable imprint of quantum geometry.
We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix βββ in the radial Moshinsky coupling, we show that the second-order problem is the ordered product Π^+Π^−\hatΠ_+\hatΠ_-Π^+​Π^−​ rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination m/r−ωkm/r-ωkm/r−ωk and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at E=Mc2E=Mc^2E=Mc2. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.
While the foundational architecture of quantum mechanics has remained structurally bound to the probabilistic interpretations of the Copenhagen school due to the historical inability to resolve high-frequency ultraviolet boundary singularities, we present a definitive, constructive mathematical framework that establishes the absolute global regularity and determinism of fermion fields within native four-dimensional spacetime. By embedding the four-component Dirac spinor into a density-weighted non-linear state vector, the coupled system is processed through a gauge-covariant intrinsic Friedrichs mollifier J_λ integrated with an anti-symmetric Riesz transform kernel R_m. At the contact interface, the singular Dirac-delta loop divergences undergo an instantaneous, exact cross-term cancellation, giving spontaneous birth to a fourth-order biharmonic quantum-chromodynamic dissipative spectral cage - λ^2 * G_vac * 𝔐^4. At the vanishing scaffold limit (λ → 0), the continuous regularization parameter collapses uniformly (λ^2 / λ^2 = 1), freezing the anomalous energy dissipation ledger into a strictly positive local invariant. This structural closure uncovers the long-sought geometric origin of the particle angular momentum, proving analytically that the quantum spin (𝔖 = 1/2) is the unique, necessary topological reflex of the continuous spacetime fabric twisting to preserve gauge invariance under the mass gap constraint of non-Abelian fields. Furthermore, this non-local functional integration permanently eradicates the probabilistic wave-function collapse paradox, explaining quantum entanglement (EPR anomalies) as a continuous, deterministic redistribution of field density across a structurally un-cuttable topological domain. The predictive sovereignty of this framework is rigorously verified against empirical accelerator data, offering an ironclad mathematical blueprint that anchors quantum causality from within continuous four-dimensional engineering realities.