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arXiv:2608.14041

Non-Abelian geometry and globally inequivalent symmetry reductions of the cylindrical Dirac doublet

Zhongze Guo, Bei Xu, Qiang Gu

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

Different exact cylindrical Dirac constructions are locally related within the same two-dimensional positive-energy sector, suggesting that they might be merely alternative choices of basis. We show that this equivalence can fail globally. Treating the cylindrical Dirac doublet as a rank-two bundle over momentum space, we identify transverse helicity, a mass-dressed transverse integral, and helicity as distinct symmetry-selected rank-one reductions, whose normalized restrictions organize the internal doublet through a Pauli algebra. The positive-energy Dirac $SU(2)$ connection Abelianizes exactly on fixed-azimuth meridians in the transverse-helicity basis, while its full three-dimensional curvature remains genuinely non-Abelian for nonzero mass. We derive the corresponding azimuthal Wilson-loop spectrum in closed form. The three reductions then display sharply different global structures: the transverse-helicity splitting terminates on the momentum axis, the mass-dressed splitting extends smoothly and is Chern trivial for $m>0$, whereas helicity defines line bundles with opposite unit Chern numbers. Thus a single Dirac doublet admits symmetry resolutions that are locally equivalent but globally inequivalent.

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18 pages

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Different exact cylindrical Dirac constructions are locally related within the same two-dimensional positive-energy sector, suggesting that they might be merely alternative choices of basis. We show that this equivalence can fail globally. Treating the cylindrical Dirac doublet as a rank-two bundle over momentum space, we identify transverse helicity, a mass-dressed transverse integral, and helicity as distinct symmetry-selected rank-one reductions, whose normalized restrictions organize the internal doublet through a Pauli algebra. The positive-energy Dirac SU(2)SU(2)SU(2) connection Abelianizes exactly on fixed-azimuth meridians in the transverse-helicity basis, while its full three-dimensional curvature remains genuinely non-Abelian for nonzero mass. We derive the corresponding azimuthal Wilson-loop spectrum in closed form. The three reductions then display sharply different global structures: the transverse-helicity splitting terminates on the momentum axis, the mass-dressed splitting extends smoothly and is Chern trivial for m>0m>0m>0, whereas helicity defines line bundles with opposite unit Chern numbers. Thus a single Dirac doublet admits symmetry resolutions that are locally equivalent but globally inequivalent.
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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.