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

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.