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A Variational Stochastic Derivation of the Dirac Equation with Quantized Orbital Angular Momentum and Consequences for Probability Positivity and Spin-Statistics

Adrian Rohr

Published Mar 12, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18981025

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gr-qcCross-listedarXiv:2605.28887•May 28, 2026
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 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∞C^\inftyC∞ 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)→SO(3)SU(2) \to SO(3)SU(2)→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∞C^\inftyC∞ regular, the internal holonomy of the rotor is forced to take values in integer multiples of π\piπ. The results suggest that intrinsic angular momentum quantized in half-integer units of ℏ\hbarℏ (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.
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).
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This paper derives the Dirac equation as the logically forced relativistic completion of the de Broglie wave dynamics established in prior work of this series. Starting from the logical foundation that yields Lorentz/Poincaré invariance, we promote the non-relativistic framework (continuous spacetime, Fourier duality, local plane-wave structures, and probability conservation via statistical energy flux) to the relativistic regime. We show that the requirements of exact local relativistic plane-wave structures (with phase gradients satisfying the mass-shell condition pμpμ=m2c2p^\mu p_\mu = m^2 c^2pμpμ​=m2c2), first-order time evolution (needed for clean probability conservation), and full Lorentz covariance together force the introduction of a specific non-commuting algebraic structure and a multi-component wave function. The Dirac equation emerges as the unique linear differential operator of order at most one in the spacetime derivatives that consistently glues these local structures across smooth backgrounds while preserving a conserved four-current with positive-definite time component. This multi-component structure arises naturally as the minimal algebraic consequence of the constraints, without any separate postulate for intrinsic angular momentum. In exact analogy with the emergence of the finite invariant causal speed in special relativity, the Dirac dynamics is revealed not as an ad-hoc linearization of the Klein--Gordon equation, but as the mathematically inevitable description of indivisible particles in continuous relativistic spacetime. The derivation thereby completes the single-particle sector of the series, demonstrating that core structures of relativistic quantum mechanics flow deductively from a small number of deep, nearly tautological commitments.
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.
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