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

This paper derives quantum mechanics and the Dirac equation from ExB-T vortex geometry by demonstrating that the circulation quantization condition possesses a fourfold structure—encompassing circulation, integrality, spinor topology, and Lorentz covariance—which, when locked via a rotating-magnetic-field principle, establishes a strict equivalence between Madelung-type fluid dynamics and relativistic quantum theory.

Original authors: shiwei xu

Published 2026-09-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: shiwei xu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For nearly a century, the most successful theory in physics has described the universe not as a collection of solid objects, but as a landscape of probabilities. In this standard view, particles like electrons do not have definite paths until they are measured; instead, they are described by a wave function, a mathematical cloud that tells us where a particle is likely to be found. This framework, known as quantum mechanics, predicts experimental results with astonishing precision, yet it leaves a fundamental question unanswered: what is the physical reality behind the math? Is the wave function a real thing, or just a tool for calculation? For decades, a minority of physicists have argued that particles do have real trajectories, guided by a hidden, invisible field. This idea, called pilot-wave theory, suggests that the strange behavior of the quantum world arises from a fluid-like medium that pushes particles along specific paths. However, this approach has struggled to explain why the fluid must behave in a way that perfectly mimics the strange rules of quantum mechanics, particularly the requirement that certain quantities come in whole numbers rather than continuous values.

A new study by independent researcher Xu Shiwei proposes a radical solution to this puzzle by deriving the laws of quantum mechanics from a specific geometric shape found in nature. The paper suggests that the entire structure of the quantum world emerges from a rotating standing wave, a self-contained vortex where electric and magnetic fields lock together in a precise, rhythmic pattern. By analyzing the geometry of this vortex, the author demonstrates that the mysterious rules of quantum mechanics are not arbitrary laws imposed on nature, but inevitable consequences of how this wave rotates and moves. The study claims to show that the famous equation governing the behavior of electrons, the Dirac equation, can be built directly from the shape and motion of this vortex, without needing to assume the strange rules of quantum theory in the first place.

The core of this argument rests on a single, unifying concept: the circulation of the vortex. In fluid dynamics, circulation refers to the total amount of spin or flow around a closed loop. In the quantum world, this circulation is restricted to specific, discrete values, a rule that has historically been treated as a fundamental axiom of nature. Xu Shiwei argues that this restriction is not a random rule, but a geometric necessity. The paper identifies a "fourfold structure" hidden within the condition that governs this circulation. This structure consists of four interconnected facets that arise naturally from the vortex's geometry. First, there is the circulation itself, the total flow around the loop. Second, there is the requirement that the number of waves fitting into this loop must be a whole number, a condition that emerges from the symmetry of the wave's nodes rather than being an arbitrary assumption. Third, the geometry forces the wave to have a specific "spin" structure, where the wave must rotate twice to return to its original state, a property that explains why particles like electrons have half-integer spin. Fourth, the locking of the wave's rotation speed to its orbital motion creates a fixed relationship between time and space, which naturally leads to the rules of relativity that govern how objects move at high speeds.

The researcher demonstrates that these four geometric features are not separate phenomena but different aspects of a single phase-locking condition. When the vortex rotates, its self-rotation and its orbital revolution become locked in a precise ratio. This locking can only happen in two ways: either the wave completes one full turn for every orbit, creating a shape like a cylinder, or it completes half a turn for every orbit, creating a shape like a Möbius strip. This simple binary choice is the source of the entire fourfold structure. The cylinder shape corresponds to particles with integer spin, while the Möbius strip shape corresponds to particles with half-integer spin. Once this ratio is fixed, the geometry automatically enforces the other three facets: the circulation becomes quantized, the wave function acquires its spinor nature, and the relationship between time and space becomes fixed in a way that preserves the speed of light.

Using this geometric foundation, the paper traces two independent paths to the equations of quantum mechanics. The first path uses a mathematical bridge established by recent work to show that a fluid with this specific spinor and relativistic structure is mathematically identical to the Dirac equation. The second path starts with the simpler fluid equations and adds the spinor and relativistic constraints, showing that they inevitably promote the standard Schrödinger equation to the more complete Dirac equation. Both routes lead to the same destination, confirming that the geometric structure of the vortex is sufficient to generate the full formalism of quantum mechanics. The study also addresses the origin of the electric force, showing that the interaction between two such vortices naturally produces a force that follows the inverse-square law, exactly matching the Coulomb force observed in nature, without needing to assume the existence of electric charge as a fundamental property. Instead, charge appears as a label for the direction of the vortex's rotation.

To verify the validity of this derivation, the author applies the resulting equations to the simplest atom, hydrogen. By calculating the energy of the electron in its ground state, the study reproduces the value of -13.6 electron-volts, which matches the experimental measurement to a high degree of accuracy. The small difference between the theoretical value and the experimental value is explained by a standard correction for the motion of the proton, confirming that the geometric model is consistent with observed reality. The paper concludes that quantum mechanics should not be viewed merely as an effective description of a deeper reality, but as a direct corollary of the geometric structure of these rotating vortices. The strange rules of the quantum world, from the quantization of energy to the behavior of spin, are revealed as the natural consequences of a wave that is locked in a specific geometric dance. While the study focuses on the electromagnetic and quantum domains, it hints that the same geometric framework could eventually be extended to include gravity, suggesting a unified picture where all fundamental forces arise from the geometry of rotating standing waves.

Technical Summary: Deriving Quantum Mechanics and the Dirac Equation from ExB-T Geometry

Problem Statement
The paper addresses the ontological gap in standard quantum mechanics, specifically the lack of a physical mechanism for the wave function $\psi$ and the "Wallstrom obstacle." The Wallstrom obstacle highlights that the Madelung fluid equations (derived from the Schrödinger equation via polar decomposition) admit a solution space strictly larger than that of the Schrödinger equation. Specifically, the inverse mapping from fluid variables $(\rho, v)$ back to the wave function requires an ad hoc circulation quantization condition ( $\oint \nabla S \cdot dl = 2\pi\hbar n$ ) to ensure single-valuedness. Standard fluid mechanics does not naturally generate this integer constraint. Furthermore, the paper seeks to derive the Dirac equation and Lorentz covariance from a geometric foundation rather than postulating them, aiming to unify the description of quantum mechanics with a physical vortex fluid model.

Methodology
The author employs the ExB-T (Electric $\perp$ Magnetic with 90° phase difference) vortex fluid framework, which models particles as rotating standing waves (solitons) possessing three motions: orbital revolution, self-rotation, and radial oscillation (breathing). The methodology proceeds through the following steps:

Geometric Foundation: The theory starts with the "rotating-magnetic-field principle," which guarantees phase closure and self-similarity. This leads to a "phase-locking condition" ( $\delta = \pi/5$ ) and a complex identity involving the golden ratio ( $\phi$ ).
Derivation of the Fourfold Structure: The core operation is identified as the locking of the spin-to-orbit ratio $L$ (the ratio of self-rotation frequency to orbital revolution frequency). The geometry of the closed standing wave restricts $L$ to only two discrete values: $L=1$ (cylinder, integer spin) or $L=1/2$ (Möbius band, half-integer spin). This discrete two-valuedness is shown to generate a "fourfold structure" within the circulation quantization condition.
Topological and Covariant Analysis:
- The Zero-Mode Reduction Theorem is proven for a compactified space $R^3 \times S^1$ , demonstrating that the effective interaction of point sources in this geometry yields an exact $1/r$ Coulomb potential without Yukawa cutoffs.
- The Adiabatic Separation is established between the internal breathing time scale ( $\tau_{int} \sim 10^{-21}$ s) and the orbital time scale ( $\tau_{orb} \sim 10^{-16}$ s), justifying the treatment of the vortex center of mass as a particle in an effective potential.
Dual Derivation Routes: Two independent logical routes are constructed to derive the Dirac equation by utilizing the fourfold structure as a premise for established theorems:
- Route A (via Fabbri): The paper demonstrates that the spinor and covariant facets of the fourfold structure satisfy the premises of Fabbri's bijection theorem (which the author does not re-prove). By satisfying these premises within the ExB-T geometry, the strict equivalence between the Dirac equation and Madelung-type fluid equations is invoked.
- Route B (via Bohm): The paper uses the circulation and integrality facets to recover the scalar Madelung fluid, which Bohm's prior work established as equivalent to the Schrödinger equation. The paper then argues that adding the spinor and Lorentz covariance facets (derived from the fourfold structure) to Bohm's equivalence relation promotes the Schrödinger equation to the Dirac equation.

Key Contributions

The Fourfold Structure of Circulation Quantization: The paper identifies that the circulation quantization condition is not a single axiom but carries four distinct facets derived from the locking of the spin-to-orbit ratio $L$ $L$ :
- (i) Circulation: The integral value $\oint P \cdot dl = 2LQ \cdot \Delta$ .
- (ii) Integrality: The quantum number $n = 2LQ$ is an integer, derived as a geometric result of the "even-number theorem" and phase-locking, rather than an imposed axiom.
- (iii) Spinor Structure: The two-valuedness of $L$ ($1$ or $1/2$ ) corresponds to the double cover $SO(3) \to SU(2)$ , geometrically realized as cylinder vs. Möbius band topologies.
- (iv) Lorentz Covariance: The locking of the time-space mapping ratio (via the rotating-magnetic-field principle) preserves the light cone, generating the full Lorentz group $O(3,1)$ .
Revised Bohm Equivalence: The paper proposes a revision to the de Broglie–Bohm equivalence relation. It argues that the circulation quantization condition must include the spinor and covariant facets to be complete. The revised relation is: Scalar Madelung fluid + Circulation quantization condition (fourfold) = Dirac equation.
Zero-Mode Reduction Theorem: A mathematical proof showing that in the compactified space $R^3 \times S^1$ (where the compactification radius equals the particle vortex modal radius), the zero-mode Green's function is exactly $1/r$ . This provides a geometric origin for the Coulomb potential's functional form.
Geometric Origin of Quantization: The paper demonstrates that the discreteness of quantum numbers arises from the discrete zeros of spherical Bessel functions (standing wave modes in a spherical cavity), replacing the quantization postulate with geometric constraints.

Results

Derivation of the Dirac Equation: Both Route A and Route B successfully converge on the Dirac equation. Route A achieves this by showing the ExB-T geometry satisfies the premises of Fabbri's theorem, while Route B achieves this by extending Bohm's equivalence relation with the spinor and covariant facets derived from the fourfold structure.
Hydrogen Ground-State Energy: The theory reproduces the hydrogen ground-state energy of $-13.6$ eV. The calculation yields $-13.6057$ eV, which matches the experimental value ($13.5984$ eV) within the precision of the reduced-mass correction.
Multipole Corrections: The analysis shows that for a spherically symmetric proton vortex, dipole and quadrupole corrections to the effective potential vanish identically. The leading non-zero correction (octupole) is estimated to be $\sim 10^{-16}$ eV, five orders of magnitude below current experimental precision, consistent with the observed pure Coulomb spectrum.
Fine-Structure Constant: The fine-structure constant $\alpha$ is identified as the time-averaged nutation angle (geometric oblateness) of the breathing closed string, derived from the spherical Bessel solution rather than vacuum polarization.

Significance and Claims
The paper claims that quantum mechanics is not merely an effective theory of the ExB-T vortex fluid but a direct corollary of its geometric structure. By deriving the circulation quantization condition, the spinor structure, and Lorentz covariance from the geometry of a rotating standing wave, the work aims to:

Resolve the Wallstrom obstacle by showing that the quantization condition is a topological consequence of the fluid's geometry, not an external rule.
Provide an ontological basis for the wave function as a real physical field (the vortex fluid) rather than a probability amplitude.
Unify the description of quantum mechanics and the Dirac equation under a single geometric framework by demonstrating that the premises of Fabbri's theorem and the conditions for Bohm's equivalence are naturally satisfied by the ExB-T geometry.

The author emphasizes that this derivation does not invalidate standard quantum mechanics; rather, it incorporates it as the exact limit of the ExB-T framework under conditions of electromagnetic limit ( $r \gg R_p$ ) and adiabatic separation. The paper explicitly states that the derivation of the gravitational interaction (via the "breathing phase" and local scale factor) is reserved for subsequent work, positioning the current results as the completion of the quantum mechanical sector of a unified geometric framework.

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