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Deriving Quantum Mechanics and the Dirac Equation from ExB-T Geometry --The Fourfold Structure of the Circulation Quantization Condition

shiwei Xu

Published Sep 18, 2026• 0 citations
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Abstract

Abstract We prove that the circulation quantization condition carries a fourfold structure, and on this basis we derive quantum mechanics and the Dirac equation from the geometry of the ExB-T (E perpendicular to B, with a 90-degree phase difference) vortex. First, we review the Madelung transformation and the Fabbri bijection. The de Broglie-Bohm pilot-wave theory recasts the Schrodinger equation, via the Madelung transformation [1], into the quantum Hamilton-Jacobi equation and the continuity equation, yielding the quantum potential [2, 3]. Wallstrom [4] pointed out that the solution space of the Madelung fluid equations is strictly larger than that of the Schrodinger equation, and that the inverse mapping requires an additional circulation quantization condition. Fabbri (2022 [5], 2025 [6]) proved, at the level of Dirac spinors, a strict two-way equivalence between the Dirac equation and Madelung-type fluid equations. Second, the central result of this paper is that the core operation underlying the circulation quantization condition (the quantization of the closed-path integral of the canonical momentum) [7, Sec. 4] is the locking of the spin-to-orbit ratio L, the ratio of self-rotation to orbital revolution, which is locked by the rotating-magnetic-field principle. L can take only the values L = 1 (cylinder, integer spin) or L = 1/2 (Mobius band, half-integer spin). The discrete two-valuedness of L yields the first three facets of the fourfold structure: (i) the circulation itself, \oint \mathbf{P} \cdot d\mathbf{l} = 2LQ \cdot \Delta; (ii) the integrality of the quantum number, n = 2LQ \in \mathbb{Z}; (iii) the spinor structure (the double cover SO(3) -> SU(2)). Once L is fixed, the ratio is locked, and this locking yields the fourth facet: (iv) Lorentz covariance (the time-space mapping ratio is fixed, preserving the light cone). The fourfold structure thus consists of four facets of one and the same phase-locking condition. Third, two independent routes lead from the fourfold structure to the Dirac equation. Route A (via Fabbri [6]): the spinor structure and the covariant structure in the fourfold structure satisfy the premises of the Fabbri bijection, and Fabbri's theorem yields the strict equivalence Dirac equation <-> Madelung fluid. Route B (via Bohm [2, 3]): the circulation and integrality facets of the fourfold structure yield a scalar Madelung fluid; Bohm already proved that the Madelung fluid is equivalent to quantum mechanics (the Schrodinger equation); adding the spinor structure and the covariant structure then promotes the Schrodinger equation to the Dirac equation. The two routes reach the same endpoint. Fourth, we propose a revision of Bohm's equivalence relation: in Bohm's era, the circulation quantization condition was regarded merely as a quantization premise (only the first two facets were recognized). We point out that the circulation quantization condition simultaneously carries the spinor and covariant facets (the latter two). Once these latter two are supplied, Bohm's equivalence relation should be revised to read: scalar Madelung fluid + circulation quantization condition (fourfold) = Dirac equation, which reduces to the Schrodinger equation in the non-relativistic limit. As a consistency check of this derivation, we reproduce the hydrogen ground-state energy -13.6 eV [8].

DOI
10.21203/rs.3.rs-11023398/v1

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Abstract We prove that the circulation quantization condition carries a fourfold structure, and on this basis we derive quantum mechanics and the Dirac equation from the geometry of the ExB-T (E perpendicular to B, with a 90-degree phase difference) vortex. First, we review the Madelung transformation and the Fabbri bijection. The de Broglie-Bohm pilot-wave theory recasts the Schrodinger equation, via the Madelung transformation [1], into the quantum Hamilton-Jacobi equation and the continuity equation, yielding the quantum potential [2, 3]. Wallstrom [4] pointed out that the solution space of the Madelung fluid equations is strictly larger than that of the Schrodinger equation, and that the inverse mapping requires an additional circulation quantization condition. Fabbri (2022 [5], 2025 [6]) proved, at the level of Dirac spinors, a strict two-way equivalence between the Dirac equation and Madelung-type fluid equations. Second, the central result of this paper is that the core operation underlying the circulation quantization condition (the quantization of the closed-path integral of the canonical momentum) [7, Sec. 4] is the locking of the spin-to-orbit ratio L, the ratio of self-rotation to orbital revolution, which is locked by the rotating-magnetic-field principle. L can take only the values L = 1 (cylinder, integer spin) or L = 1/2 (Mobius band, half-integer spin). The discrete two-valuedness of L yields the first three facets of the fourfold structure: (i) the circulation itself, \oint \mathbf{P} \cdot d\mathbf{l} = 2LQ \cdot \Delta; (ii) the integrality of the quantum number, n = 2LQ \in \mathbb{Z}; (iii) the spinor structure (the double cover SO(3) -> SU(2)). Once L is fixed, the ratio is locked, and this locking yields the fourth facet: (iv) Lorentz covariance (the time-space mapping ratio is fixed, preserving the light cone). The fourfold structure thus consists of four facets of one and the same phase-locking condition. Third, two independent routes lead from the fourfold structure to the Dirac equation. Route A (via Fabbri [6]): the spinor structure and the covariant structure in the fourfold structure satisfy the premises of the Fabbri bijection, and Fabbri's theorem yields the strict equivalence Dirac equation <-> Madelung fluid. Route B (via Bohm [2, 3]): the circulation and integrality facets of the fourfold structure yield a scalar Madelung fluid; Bohm already proved that the Madelung fluid is equivalent to quantum mechanics (the Schrodinger equation); adding the spinor structure and the covariant structure then promotes the Schrodinger equation to the Dirac equation. The two routes reach the same endpoint. Fourth, we propose a revision of Bohm's equivalence relation: in Bohm's era, the circulation quantization condition was regarded merely as a quantization premise (only the first two facets were recognized). We point out that the circulation quantization condition simultaneously carries the spinor and covariant facets (the latter two). Once these latter two are supplied, Bohm's equivalence relation should be revised to read: scalar Madelung fluid + circulation quantization condition (fourfold) = Dirac equation, which reduces to the Schrodinger equation in the non-relativistic limit. As a consistency check of this derivation, we reproduce the hydrogen ground-state energy -13.6 eV [8].
Abstract Where does the integrality of quantization conditions come from? Standard quantum mechanics treats it as a fundamental postulate, or attributes it to the single-valuedness of the wave function, without asking whether a deeper geometric reason exists. This paper argues, starting from the winding structure of closed standing waves, that integrality is a consequence of geometry. The fundamental object of the paper is the rotating standing wave, which possesses three simultaneous degrees of freedom: breathing (radial oscillation), spin, and orbital motion. First layer: the core proof of circulation quantization. The total length of the closed path is 2π2\pi2π. The even-number theorem shows that the antisymmetry condition forces zeros to appear in pairs, so an antisymmetric structure necessarily exists. The phase-locking condition states that one full rotation advances one full wave; the closed path accommodates QQQ full waves, so the total effective number of half-wave steps is 2LQ2LQ2LQ. The step length is \Delta\phi = \frac{\pi}{LQ}. The path integral gives \oint \mathbf{P} \cdot d\mathbf{l} = 2LQ \cdot \Delta, and the quantum number is n = 2LQ, where L∈{1,1/2}L \in \{1, 1/2\}L∈{1,1/2} and Q∈Z+Q \in \mathbb{Z}^+Q∈Z+, so that nnn is always an integer. Comparison with the Bohr-Sommerfeld condition \oint \mathbf{p} \cdot d\mathbf{l} = 2\pi n \hbar = n h yields the correspondence n=2LQn = 2LQn=2LQ, Δ=h=2πℏ\Delta = h = 2\pi\hbarΔ=h=2πℏ. Within the present framework, the integral part of the quantization condition is a theorem rather than a postulate; the proportionality constant Δ=h\Delta = hΔ=h is calibrated by experiment. Second layer: the geometric origin of the spin dichotomy. As the rotating standing wave completes one orbital revolution, it simultaneously spins. A spin-to-orbit ratio of 1:11{:}11:1 sweeps out a cylindrical band whose surface normal is restored after one circuit; a ratio of 1:21{:}21:2 sweeps out a Möbius band whose surface normal is restored only after two circuits. Other ratios fail to satisfy the planar gluing condition and do not close. Hence only two topologies of closed band exist, and spin admits only integer and half-integer values. Third layer: how the standing wave realizes 1:21{:}21:2. In a standing wave, EEE and BBB are spatially orthogonal, and a phase difference of exactly 90∘90^\circ90∘ corresponds to the dead point of a synchronous motor--an unstable equilibrium. Any infinitesimal perturbation introduces a phase offset, and the system, once displaced from the dead point, rotates at the synchronous speed. Each half-wave contains one pair of magnetic poles: a pole-pair number p=1p = 1p=1 corresponds to the ratio 1:11{:}11:1, and p=2p = 2p=2 to the ratio 1:21{:}21:2. Energy minimization and frequency locking jointly determine the critical state. Fourth layer: the geometric essence of parity non-conservation. The Möbius band is non-orientable and possesses no global left-right chirality. Topological inertia is unidirectional: two Möbius bands combine into a Klein bottle automatically and at low energy, whereas a cylinder is cut, flipped, and reglued into a Möbius band only at high energy and unidirectionally. The weak interaction is precisely this unidirectional flip operation. Parity non-conservation is therefore not a dynamical symmetry breaking but a manifestation of topological inertia. Fifth layer: the geometric origin of magnetism. The pole-pair number p∈Z+p \in \mathbb{Z}^+p∈Z+, and magnetic poles must appear in pairs. Within the closed-standing-wave framework, ∇⋅B=0\nabla \cdot B = 0∇⋅B=0 is demoted from a postulate to a corollary. Verification: at the particle scale, 328 massive particle modes yield a hit rate of 99.4%99.4\%99.4% (326/328326/328326/328) in the L=1/2L=1/2L=1/2 spectrum, with p=1.18×10−71p = 1.18 \times 10^{-71}p=1.18×10−71. For PDB biological macromolecules, 40 independent configurations yield a hit rate of 95.0%95.0\%95.0% (38/4038/4038/40) under the prior-LLL strategy, with p=1.21×10−6p = 1.21 \times 10^{-6}p=1.21×10−6; the counterexamples A-RNA and A-DNA share the same deviation value of 32.7∘32.7^\circ32.7∘, rooted in the disruption of phase locking by the C3'-endo sugar-ring conformation. At the macroscopic scale, 55 Galactic cool stars (Bowler 2023) yield a hit rate of 90.9%90.9\%90.9% (50/5550/5550/55) under the prior-LLL strategy, with p=8.47×10−3p = 8.47 \times 10^{-3}p=8.47×10−3. For orbital resonances, 152 pairs show a small-integer-ratio (≤10\le 10≤10) fraction of 92.8%92.8\%92.8% (141/152141/152141/152), with a direct-resonance hit rate of 100%100\%100%. The structural distribution of both hits and counterexamples jointly constitutes the verification: the theory predicts not only the phase-locking spectrum but also the mechanism by which phase locking is disrupted. This paper introduces no wave function, no Hilbert space, and no formal postulates of quantum mechanics. Its core claims are that the circulation quantization condition arises from geometry; the integrality of quantization arises from geometry; the spin dichotomy arises from the spin-to-orbit ratio; parity non-conservation arises from topological inertia; magnetism arises from the standing-wave transposition sequence; and the proportionality constant is calibrated by experiment. The paper further argues that fermions and bosons are two configurations of the same topological entity (the Möbius band and the cylinder), that the direct topological transformation between them is geometrically permitted, that quantum entanglement corresponds to topological winding, and that self-replication corresponds to the topological splitting of a twisted surface. Statement of epistemic status: this paper makes no numerical predictions and does not replace the computational power of standard quantum mechanics. It addresses the question of "why"--why the quantization condition is integral, why spin admits only two types, why parity is not conserved. The Standard Model addresses "what"--it predicts all observed values precisely. The two are complementary and do not compete. The mode of verification in this paper is structural: if the theory is correct, the phase-locking spectrum should appear in three-scale data, and counterexamples should cluster around the theoretically predicted disruption mechanisms. This paper derives the geometric origin of the integer nnn in the quantization condition and does not claim to derive the numerical value of Planck's constant, which is calibrated by experiment.
Abstract We consider the Dirac equations in polar form proving that they can equivalently be re-configured into a system of equations consisting of derivatives of the velocity density plus the Hamilton–Jacobi equation, giving the momentum in terms of relativistic quantum potentials (i.e. displaying first-order derivatives of the two degrees of freedom of the spinor field): this system is said to have Madelung structure. Conservation laws, second-order equations and multi-valuedness are also discussed.
Madelung showed that the Schrodinger equation can be rewritten, away from the zeros of the wave function, as hydrodynamic equations for a density and a velocity potential. Wallstrom [Phys. Rev. A 49, 1613 (1994)] showed that the rewriting is not an equivalence: the hydrodynamic system admits solutions whose circulation around nodes is an arbitrary real multiple of h/m, and equivalence requires imposing circulation quantization loop by loop, by hand. We examine a single kinematic postulate, the compact-phase postulate: the phase is fundamentally circle-valued, and the circle has action-circumference h. We prove three statements. First, a continuous circle-valued phase field winds an integer number of times around every loop in the node complement, so the fractional-circulation solutions are excluded kinematically. Second, compactness alone quantizes circulation in an arbitrary unit; the postulated circumference fixes the unit to h/m, and the circumference h is displayed independently by measured ladders in superconducting rings (h/2e) and in superfluid helium (h/m), and by the Aharonov-Bohm period (h/e), all in coherent-condensate systems. Third, under an integer-free regularity condition at the nodes, the postulate together with the Madelung equations is equivalent to exact scalar quantum mechanics at strong-solution (H1_loc) grade. We state the caveat explicitly: over the node complement, given continuity, the circle-valued phase is logically equivalent to the quantization condition it subsumes. The gain is at the level of axioms: one universal kinematic postulate with a calibrated, independently measured unit in place of a per-solution family of integral constraints.
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).
We present a three-level organization of a bilocal relational theory inwhich structures usually associated with quantum mechanics arise atlogically distinct stages. The first level is purely geometric. Starting from the bilocalconfiguration space M×MM\times MM×M and its canonical cotangent bundle,the relative sector admits a radial--angular decomposition. For N=3N=3N=3,the angular phase space is T∗S2T^*S^2T∗S2. Its non-singular sector isnaturally organized asR+×SO(3)\mathbb R_+\times SO(3)R+​×SO(3) and therefore possesses the canonical doublecoverR+×SU(2)≃C2∖{0}.\mathbb R_+\times SU(2)\simeq\mathbb C^2\setminus\{0\}.R+​×SU(2)≃C2∖{0}. A classical two-component SU(2)SU(2)SU(2)-spinorial carrier thus appearsbefore any Lorentzian dynamics or quantum postulate; its identificationwith a Lorentz-covariant Weyl spinor is not assumed at this stage.Reduction at fixed internalCasimir yields the Hopf fibration with ∣c1∣=1|c_1|=1∣c1​∣=1. For N=3N=3N=3, definingpμ=(pt,P)p_\mu=(p_t,\mathbf P)pμ​=(pt​,P) andμint=κ1+iκ2\mu_{\rm int}=\kappa_1+i\kappa_2μint​=κ1​+iκ2​rewrites the full constraint aspμpμ+∣μint∣2=0.p_\mu p^\mu+|\mu_{\rm int}|^2=0.pμ​pμ+∣μint​∣2=0.Its Clifford factorization then admits an exact chiral first-order formwith a two-dimensional complex kernel. At this stage the constructionremains classical: the momenta are phase-space variables and the kernelspinor is not yet a quantum wave function.The third level promotes this geometric and dynamical structure toquantum spinor dynamics. Geometry supplies an integer topological classbut not a universal scale of action; the identificationI=nℏ/2I=n\hbar/2I=nℏ/2 therefore requires the quantum scale ℏ\hbarℏ.Together with the operator rulepμ↦−iℏDμp_\mu\mapsto-i\hbar D_\mupμ​↦−iℏDμ​, the classical chiral factorization becomesa quantum Dirac-type equation. If the internal complex contributionadmits a sector or limit in which it reduces to a real constantmmm, the standard massive Dirac equation(iℏγμDμ−m)ΨD=0(i\hbar\gamma^\mu D_\mu-m)\Psi_D=0(iℏγμDμ​−m)ΨD​=0is recovered. The construction therefore separates three ingredients that arenormally introduced together: spinorial and topological statestructure, Lorentzian dynamics, and quantum operator dynamics. We donot derive quantum mechanics from classical geometry. Rather, we showthat the spinorial carrier and the algebraic architecture of the Diracequation can arise before quantization, while the quantum scale andoperator realization remain independent inputs. This identifiesprecisely which elements of quantum spinor dynamics are geometric,which depend on the Lorentzian dynamical postulate, and which remaingenuinely quantum assumptions.