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A Geometric Approach to the Relativistic Wallström Objection: Spin-1/2 from Covariant Fluid Regularity

Adrian Rohr

Published Mar 8, 2026• 0 citations
Publisher

Abstract

This paper explores the extension of the stochastic coherence framework (Onsager-Machlup formalism) to the relativistic domain. In previous work, it was shown that phase quantization in the non-relativistic Schrödinger regime can be understood as a consequence of requiring the probability current to be a $C^\infty$ smooth vector field across nodal surfaces (resolving the Wallström objection). Here, we examine the analogous topological constraints in Minkowski spacetime using Spacetime Algebra (STA). By replacing the scalar phase with a Lorentz rotor field, the physical observables become the probability four-current and the spin tensor, both bilinear in the rotor via the $SU(2) \to SO(3)$ double covering. Through a variational derivation of the relativistic stochastic dynamics, we show that the generalized Hamilton-Jacobi equation constrains the local density behavior near nodal tubes. When combined with the requirement that the bilinear observables remain strictly $C^\infty$ regular, the internal holonomy of the rotor is forced to take values in integer multiples of $\pi$. The results suggest that intrinsic angular momentum quantized in half-integer units of $\hbar$ (spin-1/2) can emerge naturally as a geometric and topological necessity for covariant fluid stability, recovering the Dirac hydrodynamic system without a priori postulating spinor structure.

DOI
10.5281/zenodo.18912547

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We extend the stochastic coherence framework based on the Onsager-Machlup action to Minkowski spacetime. The configuration space R^n is replaced by M4, the scalar phase by a Lorentz rotor R in Spin+(1,3), and the observables by the four-current j^mu and spin tensor S^mu nu, both bilinear in R. We show that the combination of the relativistic stochastic Hamilton-Jacobi equation and the C^infty regularity of the bilinear observables over nodal surfaces forces the orbital winding number around nodal tubes to satisfy alpha = 2k with k a natural number, yielding orbital angular momentum quantized in integer multiples of h-bar: L = k in the non-negative integers. This establishes a strict topological separation between orbital angular momentum (determined by nodal regularity on the base manifold) and intrinsic spin (determined by the fiber group Spin+(1,3)). The variational derivation from the Onsager-Machlup action produces hydrodynamic equations that match the Takabayasi-Hestenes decomposition of the Dirac equation term by term. The derivation is non-circular: the dynamical constraint and the regularity requirement are logically independent. Additionally, we establish that the positivity condition rho >= 0 (inherent to any probability density) requires the internal orientation to be described by rotors in Spin+(1,3) rather than by SO+(1,3), as only the spinorial formulation guarantees a future-directed probability current. Finally, applying the Laidlaw-DeWitt decomposition to the many-body configuration space, we show that the double-cover structure of Spin+(1,3) fixes the exchange character to chi = -1 for spinorial representations, recovering Fermi-Dirac statistics without an independent symmetrization postulate. The framework remains within the domain of the free theory; extensions to curved spacetimes and many-body interactions are left for future work.
This paper presents a complete derivation of the Schrodinger equation from the Onsager-Machlup stochastic variational principle, rigorously resolving the longstanding Wallstrom objection. Wallstrom proved that hydrodynamic equations derived via the Madelung transformation do not uniquely determine quantum mechanics unless phase circulation is quantized. We demonstrate that the quantization condition (the integral of the gradient of S along a closed loop equals 2pin*hbar) emerges as a necessary consequence of requiring finite action and topological stability of coherent structures. Configurations with non-integer winding numbers are shown to lead to divergent action costs or inadmissible multi-sheeted spatial representations. For nodal configurations where the probability density vanishes, we show that the smoothness of the probability current independently forces integer winding numbers, excluding non-quantized solutions recently constructed in the literature. This establishes the Schrodinger equation as the unique and inevitable description of probability fluid dynamics under the calibration D = hbar/(2m). Finally, the framework provides testable experimental predictions regarding wave packet spreading and interference visibility in the presence of environmental noise.
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.
This paper extends the stochastic electrodynamic framework of Papers 1, 2, and 3 [1–3] to relativistic quantum mechanics, deriving the Dirac equation from the relativistic Coulomb field. The non-relativistic Brownian motion of Paper 1 is generalised to a four-dimensional stochastic process in Minkowski spacetime, parameterised by proper time τ . The relativistic diffusion coefficient isD4D = ℏ/m—exactly twice the non-relativistic value ℏ/2m—arising from the additional contribution of the temporal component of the four-dimensional Wiener process. The covariant zero-point field spectral tensor Sμν(ω) = (ℏω^3/6π^2ϵ0c^3)gμν is derived from the relativistic Coulomb field through the Lorentz-covariant Einstein–Hopf detailed balance and the ergodic theorem, whose applicability to the SED electron-vacuum system is proved in Paper 3 [3]. Relativistic stochastic optimal control gives the Hamilton–Jacobi–Bellman (HJB) equation in four-dimensional configuration space, which yields the Klein–Gordon equation for spin-0 particles through the relativistic Itô correction (ℏ/2m)□. Linearisation of the relativistic Lagrangian −mc√uμuμ via Dirac matrices γμ satisfying {γμ, γν} = 2gμν converts the second-order HJB equation into a first-order equation—the Dirac equation: (iℏγμ∂μ−eγμAμ/c−mc)ψ = 0. The central new result is that spin- 1/2 is not introduced by the linearisation: it is already present in the geometry of the four-dimensional stochastic path. The mass-shell constraint uμuμ = −c^2 forces the relativistic Brownian path to be helical, with radius ℏ/mc and angular frequency 2mc^2/ℏ (Zitterbewegung). The angular momentum of this helix is ℏ/2 (spin- 1/2 ) and the magnetic moment gives g = 2—both following from the Coulomb field without additional assumptions. The linearisation then finds the unique first-order wave equation consistent with this pre-existing helical spin structure. The four-component Dirac spinor encodes four degrees of freedom already determined by the stochastic path: two spin states (helix handedness) and two time directions (particle/antiparticle).
We present a strictly local and deterministic geometric foundation for relativistic quantum mechanics by mapping the phenomenological theory of martensitic crystallography onto a four-dimensional pseudo-Riemannian spacetime manifold. By generalizing the classical 3D Hadamard compatibility condition to a 4D covariant tensor field framework, we demonstrate that the fundamental equations of quantum mechanics emerge naturally from the geometric constraints of a continuous lattice undergoing structural phase transformations. Defining a complex field structure from the local volume fraction density of twinned variants and the kinematic phase front normal, the principle of stationary action yields the exact, unapproximated Klein-Gordon and Dirac equations. By doing so, we provide a concrete, solid-state mechanical realization to the historical hydrodynamic formulations of Madelung and the double-solution models of de Broglie. Under this paradigm, abstract probability amplitudes are reinterpreted as real, structural domain distributions, and the Bohm quantum potential is identified as the localized interfacial surface energy density of sub-microscopic twin boundaries. Finally, we resolve the historical problem of wave-function collapse without invoking anthropocentric or non-local interventions; utilizing a framework of macroscopic structural decoherence, we prove that the perception of a single, definitive measurement outcome is an unavoidable consequence of the thermodynamic orthogonality and strict kinematic constraints governing macroscopically bound solid-state detectors embedded within a globally twinned 4D block universe.
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.