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