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arXiv:2608.10563

Unified Exact Cylindrical Dirac Modes and Symmetry-Resolved Quantum Geometry

Zhongze Guo, Bei Xu, Qiang Gu

quant-phSubmitted Aug 11, 2026• Announced Aug 12, 2026
arXivHTMLPDF

Abstract

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.

Comments
13 pages, no figure

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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.
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.
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.
We show that a propagating electron wave in the lowest cylindrical channel carries a spin-resolved helical conserved current, and connect this current texture to its local chiral coupling. In a weakly relativistic positive-energy Dirac projection, the two sectors have the same azimuthally symmetric charge density and opposite circulation superposed on longitudinal flow, even though their nonrelativistic orbital harmonic is l=0l=0l=0. The current is radially distributed and extends into the evanescent tail. A scalar screw potential couples these modes through a derived vertex expressed by the \emph{chiral current overlap}, Jχ\mathcal J_χJχ​. Both spatial textures determine the coupling: the environmental radial profile weights the electronic circulation, while the screw harmonic selects correlated changes of spin sector and longitudinal momentum. For a uniform static harmonic, elastic energy conservation makes the first-order spin-flip amplitude vanish within the same transverse mode. Environmental energy exchange is therefore required to activate these selected channels. The same current overlap supplies the spatial kernel for heliciton-assisted inelastic scattering in a companion study and for a subsequent transport treatment of chirality-induced spin selectivity (CISS).
The hydrodynamic formulation of the Dirac equation has historically been hindered by the inability to close the system of physical variables without resorting to infinite moment hierarchies. We resolve this longstanding issue by developing a fully covariant chiral-hydrodynamic formulation of the Dirac field in curved spacetime. Working in the Weyl representation, we introduce two independent null vectors, PLμP_L^μPLμ​ and PRμP_R^μPRμ​, which decouple the left and right chiral components. This allows us to define chiral geodesic and stochastic velocities, yielding a closed system of exactly eight real equations that corresponds directly to the Dirac field degrees of freedom. Remarkably, this formulation naturally isolates the spin-orbit coupling (q/2)σμνFμν(q/2)σ^{μν}F_{μν}(q/2)σμνFμν​ while demonstrating the vanishing of the spin-gravity coupling in torsion-free general relativity. To demonstrate the analytical power of this framework, we specialize to the Schwarzschild geometry. We obtain exact radial solutions in terms of confluent Heun functions and directly compute the quasi-bound state spectrum (fermionic resonances), quasinormal mode frequencies, and greybody factors. Furthermore, by establishing an exact energy balance equation -- representing the first law of thermodynamics for Dirac fields -- we derive the Hawking radiation flux purely from chiral flux conservation at the event horizon. This work not only provides a closed hydrodynamic theory for spin-1/2 fluids but also establishes a unified framework for analyzing quantum information and fermion dynamics in strong gravitational backgrounds.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit iii is systematically replaced by the temporal Hilbert transform operator Ht\mathcal{H}_{t}Ht​, which satisfies the anti-involution property Ht2=−I\mathcal{H}_{t}^{2}=-\mathcal{I}Ht2​=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ\psiψ and its Hilbert-transformed partner ϕ=Ht[ψ]\phi=\mathcal{H}_{t}[\psi]ϕ=Ht​[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ\psi_{a}=\psi-i\phiψa​=ψ−iϕ, under which the standard U(1)U(1)U(1) gauge symmetry emerges naturally as a global SO(2)SO(2)SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2\rho=\psi^{\dagger}\psi+\phi^{\dagger}\phi=|\psi_{a}|^{2}ρ=ψ†ψ+ϕ†ϕ=∣ψa​∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
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.