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A Minimal Arc-Geometric Reconstruction of Single-Electron Double-Slit Interference: From the R = DA Constraint to de Broglie Wavelength and Projection Density

Frank F. (Arcman) Meng

Published May 20, 2026• 0 citations
Publisher

Abstract

This paper presents a minimal arc-geometric reconstruction of single-electron double-slit interference within the conceptual framework of Arc Theory. The central aim is not to reject the empirical success of standard quantum mechanics, but to show that a nonstandard geometric ontology can be expressed in a traditional mathematical-physical form, with explicit definitions, axioms, propositions, derivations, and stated limitations. Starting from the proposed arc-geometric constraint R = DA, where R denotes the curvature radius of the external zero-weight ring and DA denotes the intrinsic arc-radius of the electronic arc-helix, the paper derives the de Broglie relation lambdaA = h/p through phase closure and action quantization. The double-slit condition d sin(theta) = m lambdaA is then reconstructed as a consequence of arc-branch phase closure under boundary-induced splitting. The paper further formulates the double-slit process in terms of four structural operators: the arc-branching operator SAlpha, the inter-branch coherence kernel SigmaAlphaBeta, the arc-difference field DeltaAlphaBeta, and the measurement-localization operator LambdaD. In this formulation, the Born density is interpreted not as a primitive probabilistic substance, but as the observable projection density of an underlying high-dimensional arc trajectory onto a lower-dimensional detection interface. In compact form, the proposed interpretation may be written as: rhoArc(x) = |A(x)|^2 where A(x) denotes the projected arc amplitude and rhoArc(x) denotes the observable arc-projection density. A central part of the work is the quantitative formulation of topological friction. The proposed self-decoherence rate GammaSelf is expressed as a functional of curvature mismatch, torsion mismatch, and deviation from the R = DA constraint: GammaSelf = Integral over GammaA of [ gammaKappa * (DeltaKappaA)^2 + gammaTau * (DeltaTauA)^2 + gammaR * ((R - DA)/R)^2 ] dmuA This gives a precise geometric meaning to internal self-decoherence and separates it from ordinary external environmental decoherence. The paper also demystifies observational collapse by representing measurement as environment-induced arc localization. In the proposed formalism, the disappearance of interference is described by the suppression of off-diagonal coherence terms: rhoAlphaBeta -> muAlphaBeta * rhoAlphaBeta with muAlphaBeta -> 0 corresponding to complete which-path distinguishability or environmental trace formation. The scope of the paper is deliberately limited. It does not claim to derive the numerical values of Planck's reduced constant, electron mass, electric charge, magnetic moment, or the electron g-factor. Rather, it provides a mathematically disciplined reconstruction of the kinematics of single-electron double-slit interference and proposes a possible geometric foundation beneath the standard probabilistic formalism. The paper is intended as a preprint-level contribution to the foundations of quantum theory, mathematical physics, and alternative geometric models of microscopic phenomena.

DOI
10.5281/zenodo.20301431

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This paper develops a phase-first reconstruction of quantum interference within standard quantum mechanics and wave optics. Its central idea is that interference patterns become conceptually clearer when the complex phase is treated as the organizing structure and probability or intensity density is treated as the visible statistical expression of that structure. The guiding principle is: the phase structures the pattern; the density renders it visible. The paper shows how this principle applies to single-slit diffraction, double-slit interference, Airy patterns, Fourier optics, Fresnel propagation, probability current, the Madelung representation, which-way marking, reduced coherence, and quantum eraser experiments. In each case, the observed density pattern is interpreted as the visible result of coherent phase relations among amplitudes. The reconstruction is strictly conservative. It does not modify quantum mechanics, the Born rule, unitary dynamics, or standard wave optics. It introduces no new postulates and makes no new empirical prediction. Its purpose is conceptual and pedagogical: to clarify the architecture of quantum interference by placing relative phase at the center of the analysis. This work is Part I of the series The Geometry of Phase. It establishes the phase-geometric foundation for later parts of the series, where the same perspective is extended to de Broglie matter waves, proper-time phase, relativistic action, and possible curvature-coupled phase extensions.
The double-slit experiment is commonly interpreted as evidence that a single electron must be described by a spatially extended wavefunction whose path amplitudes interfere. Here, we present an alternative geometric formulation within the Bivector Standard Model, in which the observed far-field interference pattern is reproduced while the phase is attributed to an internal bivector clock carried by the electron. In this approach, each electron remains localized and produces a single detection event, while the familiar fringe pattern emerges statistically from the accumulation of many impacts. Interference arises from the comparison of clock phases associated with geometrically distinct paths, rather than from the superposition of spatial waves. The resulting probability distribution recovers the standard two-slit interference factor, the single-slit diffraction envelope, and the usual fringe-spacing relation in the Fraunhofer regime. The de Broglie wavelength emerges as the spatial manifestation of this transported phase. This formulation provides a geometric account of the origin of phase in single-electron interference, consistent with standard results while offering a distinct physical interpretation.
The double-slit experiment has stood for a century as the paradigmatic demonstration of quantum superposition: the standard interpretation holds that a particle passes through both slits simultaneously, existing in a superposition of two spatially separated states. This paper proposes a physically distinct alternative: the particle's matter wave — a real physical wave propagating through space in accordance with de Broglie's relation λ = h/p — passes through both slits, while the particle itself (matter) remains in an undetermined state until a constructive interference maximum is formed at the detection screen. The particle does not simultaneously occupy two locations; the wave does. This distinction eliminates the need for superposition as an ontological commitment while fully reproducing the observed interference pattern. We further show that the disappearance of the interference pattern upon which-path measurement is explained without invoking wavefunction collapse: the measurement interaction redirects the matter wave to a single-slit geometry, eliminating the two-wave interference condition. Three experimentally testable predictions distinguish this framework from the standard interpretation: (1) weak-measurement trajectory reconstruction should reveal single particle paths inconsistent with simultaneous two-slit passage; (2) diffraction fringe spacing should depend on the mass density of the slit material, because denser slit edges generate a steeper refractive-index gradient for the matter wave; (3) the fringe pattern should exhibit a density-dependent phase shift measurable with current matter-wave interferometry technology. This reinterpretation requires no new mathematical formalism — the Schrödinger equation and Born's rule are retained — but replaces the ontological claim of simultaneous multi-location existence with the physically consistent picture of matter-wave propagation through an undetermined-state particle.
The double-slit interference experiment remains one of the most puzzling phenomena in quantum mechanics. Standard quantum mechanics explains it through wave-particle duality, superposition, and wavefunction collapse—all mathematical postulates without physical mechanisms. Within the Huaxia Spacetime Total Conservation Theory (H-SET), the electron is not a point particle but a helical dislocation loop in the space medium with spatial extension determined by the space quantum coherence scale. Interference arises because the elastic field of a single electron loop simultaneously covers both slits, and the two emergent elastic waves interfere on the screen. The effective de Broglie wavelength is derived as the geometric extension of the dislocation loop in the momentum direction, yielding identical numerical values to standard quantum mechanics but with a physical origin. Observation-induced disappearance of interference is explained as the destruction of elastic boundary conditions at the slits by the macroscopic elastic field of the detector, not by mysterious wavefunction collapse. Three testable predictions are provided: precise fringe spacing for fullerene (C₆₀) interference, the quantitative relationship between path distinguishability and fringe visibility, and the mass limit for macromolecular interference.
This manuscript presents a deterministic boundary-phase model of the double-slit experiment. In this framework, particles follow continuous, definite trajectories while interference arises from structured curvature induced by slit boundary microstructure. The wavefunction is treated as a mathematical representation of a real boundary-generated field rather than a physical matter wave.The model preserves established interference results while introducing a falsifiable experimental prediction: coherent fringe displacement scaling with engineered slit roughness. For a baseline electron configuration (100 eV beam energy, 1 m screen distance, 1 µm slit separation), a measurable 2 µm fringe shift requires a relative phase offset of approximately 0.10 radians, corresponding to the coupling condition β_eff(δ/r) ≈ 0.10.A quantitative δ-sweep prediction is provided along with explicit experimental control requirements necessary to distinguish coherent boundary-phase displacement from conventional scattering or electrostatic effects. If the predicted scaling is observed under controlled conditions, the double-slit paradox can be resolved deterministically without invoking intrinsic randomness or wavefunction collapse.This work is presented as a falsifiable experimental proposal in quantum foundations and electron interferometry.