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 $σ_z$ 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 $O(θ^2)$ for longitudinal polarization. Spectral reduction and internal coherence thus provide distinct controls of the observable imprint of quantum geometry.
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The paper asks why spin–orbit redistribution contributes differently to the magnetic moments of localized positive-energy Dirac wave packets and stationary monochromatic electron beams, despite their sharing the same local eigenspace. It attributes the difference to fixed-energy restriction replacing ambient momentum derivatives in position-weighted current observables with derivatives tangent to the energy shell.
For symmetrically ordered first-order observables, it derives an extension-independent Hermitian correction when the eliminated derivative has vanishing intraband symbol, $PB_NP=0$ in a neighborhood of the shell. The correction combines the observable’s inter-subspace vertex with the projector’s variation along the eliminated direction. Its expectation satisfies $|\langle\delta O\rangle|^2\leq\hbar^2\langle V_N\rangle\langle g_{nn}\rangle$, connecting the response to the quantum metric. For the free Dirac doublet, the correction is an exact non-Abelian Berry-curvature contraction, and the bound is pointwise saturable.
For longitudinal polarization, spectral reduction changes $M_z^{\rm amb}=\mu_E(\ell+2s-s\Delta)$ into $M_z^{(E)}=\mu_E(\ell+2s)$. The correction anticommutes with conserved transverse helicity, making each resolved helicity branch exactly dark to this correction. A radial coherent superposition instead gives $\delta M_z=\mu_E\theta/(2\gamma)+O(\theta^3)$ at fixed finite Lorentz factor, versus $O(\theta^2)$ for longitudinal polarization. These results follow from momentum-space projection and spectral-coordinate reduction; an independent Gordon-current calculation checks the longitudinal result.
The extension-independent rule requires the stated vanishing-symbol condition and a regular spectral chart. The velocity-current generalization assumes scalar dispersion within the isolated subspace; generally it involves gap-weighted geometry rather than Berry curvature alone. The Dirac expressions use the forward patch $p_z>0$.
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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 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.
The paper asks why spin–orbit redistribution contributes differently to the magnetic moments of localized positive-energy Dirac wave packets and stationary monochromatic electron beams, despite their sharing the same local eigenspace. It attributes the difference to fixed-energy restriction replacing ambient momentum derivatives in position-weighted current observables with derivatives tangent to the energy shell.
For symmetrically ordered first-order observables, it derives an extension-independent Hermitian correction when the eliminated derivative has vanishing intraband symbol, PBNP=0 in a neighborhood of the shell. The correction combines the observable’s inter-subspace vertex with the projector’s variation along the eliminated direction. Its expectation satisfies ∣⟨δO⟩∣2≤ℏ2⟨VN⟩⟨gnn⟩, connecting the response to the quantum metric. For the free Dirac doublet, the correction is an exact non-Abelian Berry-curvature contraction, and the bound is pointwise saturable.
For longitudinal polarization, spectral reduction changes Mzamb=μE(ℓ+2s−sΔ) into Mz(E)=μE(ℓ+2s). The correction anticommutes with conserved transverse helicity, making each resolved helicity branch exactly dark to this correction. A radial coherent superposition instead gives δMz=μEθ/(2γ)+O(θ3) at fixed finite Lorentz factor, versus O(θ2) for longitudinal polarization. These results follow from momentum-space projection and spectral-coordinate reduction; an independent Gordon-current calculation checks the longitudinal result.
The extension-independent rule requires the stated vanishing-symbol condition and a regular spectral chart. The velocity-current generalization assumes scalar dispersion within the isolated subspace; generally it involves gap-weighted geometry rather than Berry curvature alone. The Dirac expressions use the forward patch pz>0.
This work resolves a foundational tension in Dirac's formulation of relativistic quantum mechanics: momentum appears in two distinct but undifferentiated roles—as a dispersion-locked parameter generating unitary evolution in the relativistic wave equation, and as a free phase variable subject to symplectic flow in the constraint formalism. We demonstrate that this duality reflects an intrinsic Lorentzian structure in energy-momentum space, characterized by the interval dΣ² = dE²/c² − dp². This geometry divides naturally into three sectors: symmetric (dispersion-locked), null (massless, E/|p| = c), and antisymmetric (free-momentum) regimes, each governed by different geometric constraints. Dispersion emerges as the global integral of an exact action one-form, while the (E,p) symplectic structure arises locally as the exterior derivative of the action's inexact form θ_E = E dp. Energy quantization appears as a topological stability condition requiring closed spectral loops to satisfy ∮ E dp = nh, defining "proto-eigenstates" whose Fourier encodings yield standard Hilbert-space eigenstates. We recover both the Bohr spectrum and Dirac fine-structure energies from spectral phase closure, revealing α = e²/(ℏc) as a geometric action-normalization ratio between Maxwell U(1) curvature and Lorentz spinor action—not a coupling parameter from QED perturbation theory. The framework synthesizes unitary flow and symplectic uncertainty into a single bilinear Lorentzian structure, showing quantum states as objective in their symmetric unitary existence but path-dependent in their antisymmetric observed manifestation. This geometric reconstruction bridges Dirac's wave mechanics and constraint formalism through native (E,p) phase space, offering a foundation for quantum theory that treats energy-momentum as primary coordinates rather than conjugate variables derived from spacetime position.
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: K, which couples spin to transverse momentum; Km, 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 Km 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.
Electronic position controls how a crystal polarizes and responds to an external field. In crystals, it is usually described through local changes of electronic states in momentum space. This Berry framework has reshaped modern solid-state physics, but strong fields drive electrons across a finite momentum range, where coherence between different momenta becomes part of the response. Here we establish a first-principles theory of electronic position at finite momentum that retains this missing information. We show that unequal-momentum coherence can cancel under spatial averaging and still produce polarization, forming a coherence dipole. We obtain the matrix directly from material wave functions, without model bands or fitted transition elements. In Si, the finite-momentum geometry sets a material momentum scale. Comparing this scale with the momentum change driven by the field predicts when finite-momentum physics becomes active. Crossing the scale strongly reorganizes the fifth and higher harmonics, showing that momentum-space geometry, rather than emitted photon energy alone, controls the nonlinear response. HHG is the first demonstration, but the theory applies whenever driven electrons explore a finite momentum range. It therefore extends quantum geometry beyond the local Berry limit and provides a general basis for predicting field-driven phenomena in real materials.
This paper investigates whether relativistic motion imposes an intrinsic cost on certifying a system’s identity. Identity refers here to internal physical parameters, such as a spectroscopic splitting, composition marker, or spin orientation, rather than to Poincaré-invariant labels such as mass and total spin. The central result is that a fixed Lorentz boost does not destroy identity information. A fixed, parameter-independent boost acts unitarily on the complete physical state, so the global quantum Fisher information associated with an internal parameter is exactly preserved. Any apparent loss of identity information must therefore arise from restricted measurements, finite detector resources, discarded degrees of freedom, incomplete knowledge of the boost, or averaging over transformations that are not recorded. The paper develops this conclusion through classical spectroscopy, relativistic flyby estimation, photon frequency encoding, and massive spin-momentum systems. Classical spectral identity A Doppler transformation uniformly rescales the line center, line width, and internal splitting of a spectrum. For a spectral doublet, the dimensionless ratio between the splitting and the line center remains unchanged. The per-photon Fisher information about the rest-frame splitting is therefore invariant under an ideal Doppler dilation. Relativistic motion can still change the total information collected because photon number, detector resolution, observing duration, and spectral bandwidth depend on the measurement protocol. The paper compares three detector classes: An ideal spectrometer with unlimited bandwidth A spectrometer with fixed observer-frame resolution A resolution-limited detector with a finite observing band These detectors produce different information-versus-velocity laws. Relativity therefore does not impose a universal identity-collapse exponent independently of the detector and resource convention. Relativistic flyby self-calibration The paper considers a source undergoing a relativistic flyby with unknown velocity, impact parameter, time of closest approach, source amplitude, rest transition frequency, and spectral splitting. The changing line-of-sight velocity generates a Doppler chirp. The same photons can therefore constrain both the trajectory and the source’s internal identity. For a symmetric doublet observed over a window that samples both sides of the encounter, the variance penalty from marginalizing over the unknown trajectory has the form variance inflation = 1 + O[(Δ/ν₀)²], where Δ is the rest-frame splitting and ν₀ is the central rest frequency. The trajectory is determined primarily through the much larger frequency scale ν₀, while the identity parameter is encoded in the smaller separation Δ. For the numerical examples studied, this produces a Fisher-information hierarchy of approximately one million and a trajectory-marginalization penalty below approximately 0.1 percent. The result remains valid when the rest transition frequency is treated as unknown. Two mechanisms make self-calibration possible. First, the ratio of the early and late Doppler asymptotes constrains the source velocity without requiring an external frequency reference. Second, changing the central frequency translates both spectral lines together, while changing the splitting moves them symmetrically apart. For a symmetric doublet, these score directions are approximately orthogonal. The self-calibration result is conditional on adequate signal strength, a resolved spectral structure, a sufficiently long observing window, and a nonsingular nuisance-parameter Fisher matrix. Retarded-time flyby calculation The paper explicitly includes the retarded relation between emission time and observation time. This confirms the small trajectory-marginalization penalty but changes the interpretation of total information accumulation. The blueshifted approach phase is compressed into a short interval of observer time. As a result, an ideal detector can collect less total information at higher velocity, while a resolution-limited detector can benefit when blueshift moves the doublet above its instrumental resolution floor. Whether relativistic motion helps or hinders estimation is therefore protocol-dependent. Quantum frequency encoding For a single photon whose identity parameter is encoded in its frequency wavefunction, a collinear Lorentz boost acts as a unitary frequency dilation. The global quantum Fisher information about the encoded parameter is exactly invariant. This result applies to the complete frequency state. It does not imply that every reduced photonic measurement remains invariant, particularly when momentum and polarization are coupled or when the detector accesses only a restricted measurement algebra. Massive carriers and reduced-spin accessibility For a massive spin-1/2 carrier, a Lorentz boost generates momentum-dependent Wigner rotations. The complete spin-momentum state preserves its global quantum Fisher information. However, a detector that measures spin while ignoring momentum accesses only the reduced spin state. Tracing over momentum can therefore reduce the spin-only quantum Fisher information. For canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and the spin-angle encoding studied in the paper, the leading fractional accessibility loss is proportional to [2/(1 + cos²θ)] × (σₚ/mc)² × tanh²(ξ/2), where θ is the encoding angle, σₚ is the momentum spread, m is the carrier mass, and ξ is the boost rapidity. The accessibility tax has three important properties. It depends on the spin-encoding direction. It increases with momentum spread. It saturates at high rapidity because tanh²(ξ/2) approaches one. The reduced-spin information loss therefore does not grow without bound as the relative velocity approaches the speed of light. The missing information is recoverable in principle. It has been transferred into momentum dependence and spin-momentum correlations. A joint measurement of spin and momentum restores access to the invariant global information. The effect is therefore a subsystem-accessibility cost, not an irreversible destruction of identity. Localization and action-count scaling For a minimum-uncertainty wave packet, σₓσₚ = ℏ/2. The Compton wavelength is λ_C = ℏ/(mc). Using the localization scale R = σₓ, the rest-frame action-count is Ξ = ER/(ℏc) = mcσₓ/ℏ = σₓ/λ_C. At large rapidity, the reduced-spin accessibility tax scales as Ξ⁻², with an additional factor determined by the spin-encoding angle. A carrier localized over a distance much larger than its Compton wavelength therefore pays a negligible reduced-spin accessibility cost. For packets that do not saturate the uncertainty relation, the same argument gives a bound rather than an equality. Energy and identity information The paper asks whether the kinetic energy required to match the source frame can be interpreted as the price of recovering identity information. Three arguments reject a universal proportionality. First, frame-matching energy grows without bound with the Lorentz factor, while the reduced-spin accessibility loss saturates. Second, frame matching is not necessary. Classical Doppler self-calibration can recover a spectral identity parameter without accelerating the detector into the source frame. In the quantum case, expanding the measured algebra to include momentum can recover the lost spin-only information. Third, kinetic energy invested in frame matching can in principle be recovered by later deceleration, while the acquired measurement record remains. Energy constrains storage capacity, evolution speed, and thermodynamic reset costs, but the paper finds no universal energy price per unit of Fisher information acquired. Main conclusions The paper establishes that: Fixed, parameter-independent Lorentz boosts preserve global quantum Fisher information. Per-photon spectral identity information is invariant under ideal Doppler dilation. Total information remains detector- and protocol-dependent. Relativistic flyby chirps can self-calibrate unknown source kinematics. Trajectory marginalization produces only a small identity-estimation penalty for well-resolved symmetric doublets. Reduced-spin measurements can lose access to information through momentum-dependent Wigner rotations. The reduced-spin accessibility tax depends on encoding angle and saturates with rapidity. Joint spin-momentum measurements restore the invariant global information. For minimum-uncertainty packets, the reduced-spin tax scales as Ξ⁻². No universal proportionality exists between frame-matching energy and identity information. Scope The global invariance theorem applies to fixed Lorentz transformations that do not depend on the identity parameter. The paper does not analyze unknown boosts as joint quantum-estimation parameters or random boost distributions that are averaged without being recorded. The reduced-spin result is derived for a single massive spin-1/2 carrier, canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and a momentum-insensitive spin measurement. Broad or anisotropic packets, composite carriers, alternative relativistic spin observables, generic reduced photonic channels, and sequential non-collinear boosts remain open extensions. The classical flyby conclusions apply to high-signal, sufficiently resolved, symmetric spectral doublets observed over an adequate time window. Relation to the companion paper The companion paper examines the accelerated rather than inertial case: Action-Count, Observer Integrity, and Horizon Saturation: A Unified Framework from Ξ = ER/(ℏc)https://doi.org/10.5281/zenodo.19599288 Together, the two papers distinguish the operational effects of velocity and acceleration. Uniform velocity preserves global identity information but can redistribut
We show that a propagating Dirac electron with intrinsic spin generically carries a real--space helical conserved current, even in the absence of orbital angular momentum. Using exact Dirac eigenstates in cylindrical confinement, we demonstrate that this helical structure possesses definite handedness, persists into evanescent regions, and is characterized by a geometric helix pitch independent of the longitudinal de~Broglie wavelength. This intrinsic helical geometry enables a local geometric coupling between a propagating electron and a chiral environment, yielding chirality--dependent spin selectivity through current geometry rather than through a spin--orbit coupling term.