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Uniquely Distinguishing an Electron's Spin from Two Quantum States via Riemann Surface Guidance

Satoshi Hanamura

Published Mar 31, 2019• 0 citations
Publisher

Abstract

[3 Supplement] This 2019 study represents an early and ambitious attempt to extend the two-kernel "oscillatory black-body electron" model (Zenodo:10.5281/zenodo.16759284), first introduced in *A Model of an Electron Including Two Perfect Black Bodies*, into the framework of complex analysis and Riemann surfaces. In the earlier model, the sum of the spinors' thermal potential energy (TPE) and the photon's kinetic energy remained constant and followed the algebraic form:E0 = E0 ( cos⁴(θ/2) + sin⁴(θ/2) + 1/2 sin²(θ) )This paper explores how the two spinor bases could be placed on separate sheets of a Riemann surface, leading to four possible composite spin states within a single electron: |↑↑⟩, |↑↓⟩, |↓↑⟩, |↓↓⟩. A magnetic field gradient or a measurement process is proposed to "disconnect" the analytic continuation between the sheets, collapsing the superposition into a restricted two-state system (|↑↑⟩ or |↓↓⟩). After this work, our research accepted a redefinition of spin via the reinterpretation of Thomas precession. As of 2025, the examination of the four possible composite spin states proposed here is no longer in active use.Within the author's series, this work marks a transitional phase: it preserves the black-body kernel mechanics from the first paper but begins to treat spin as part of a structure embedded in a two-sheeted analytic space. Although later studies refined the interpretation, this paper stands as the first to test the Riemann surface as a mathematical stage for the electron's internal state space. Its domain-disconnection mechanism remains an early foundation for later explorations of measurement and symmetry breaking in the 0-Sphere model. This paper serves as the logical foundation for a subsequently submitted work, Redefining Electron Spin and Anomalous Magnetic Moment Through Harmonic Oscillation and Lorentz Contraction (DOI: 10.5281/zenodo.16861903), which provides predictions that electron Zitterbewegung oscillates at approximately 4% of the speed of light. While some mathematical refinements remain, the work opens a new path for interpreting fundamental equations through internal electron dynamics rather than particle–antiparticle dualism. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers

DOI
10.5281/zenodo.16788576

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[3 Supplement] This 2019 study represents an early and ambitious attempt to extend the two-kernel "oscillatory black-body electron" model (Zenodo:10.5281/zenodo.16759284), first introduced in A Model of an Electron Including Two Perfect Black Bodies, into the framework of complex analysis and Riemann surfaces. In the earlier model, the sum of the spinors' thermal potential energy (TPE) and the photon's kinetic energy remained constant and followed the algebraic form:E0 = E0 ( cos⁴(θ/2) + sin⁴(θ/2) + 1/2 sin²(θ) )This paper explores how the two spinor bases could be placed on separate sheets of a Riemann surface, leading to four possible composite spin states within a single electron: |↑↑⟩, |↑↓⟩, |↓↑⟩, |↓↓⟩. A magnetic field gradient or a measurement process is proposed to "disconnect" the analytic continuation between the sheets, collapsing the superposition into a restricted two-state system (|↑↑⟩ or |↓↓⟩). After this work, our research accepted a redefinition of spin via the reinterpretation of Thomas precession. As of 2025, the examination of the four possible composite spin states proposed here is no longer in active use.Within the author's series, this work marks a transitional phase: it preserves the black-body kernel mechanics from the first paper but begins to treat spin as part of a structure embedded in a two-sheeted analytic space. Although later studies refined the interpretation, this paper stands as the first to test the Riemann surface as a mathematical stage for the electron's internal state space. Its domain-disconnection mechanism remains an early foundation for later explorations of measurement and symmetry breaking in the 0-Sphere model. This paper serves as the logical foundation for a subsequently submitted work, Redefining Electron Spin and Anomalous Magnetic Moment Through Harmonic Oscillation and Lorentz Contraction (DOI: 10.5281/zenodo.16861903), which provides predictions that electron Zitterbewegung oscillates at approximately 4% of the speed of light. While some mathematical refinements remain, the work opens a new path for interpreting fundamental equations through internal electron dynamics rather than particle–antiparticle dualism. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers
In 1925 the concept of electron spin was introduced to explain two experimental observations: the splitting of silver atom beams in an inhomogeneous magnetic field (Stern–Gerlach, 1922), and the fine structure of atomic spectra. The physical picture proposed — an electron spinning on its own axis — was immediately shown by Hendrik Lorentz to require surface velocities exceeding the speed of light. The paper was published anyway. Dirac subsequently rescued the two-valuedness mathematically, whereupon the physical question was declared irrelevant. For one hundred years the field has been unable to state what spin physically is. This paper traces the sequence of observations, inferences, and decisions that produced this situation and identifies two distinct wrong turns. The first wrong turn was publishing a model known to be physically impossible. Lorentz's calculation was correct — at the classical electron radius (2.818 fm), the required surface velocity is 274 times the speed of light. But the classical radius is not the electron's physical rotation radius. The correct rotation radius, derived from the Sagnac closure condition, is 285.5 fm — 101 times larger. At that radius, the rotation velocity is 0.822c, the bare magnetic moment is 0.608 μB, and Lorentz's objection does not arise. Both values are computed explicitly with no free parameters. The second wrong turn was a conflation. The closure geometry of the electron involves two independent numerical facts, both related to the number ½: (1) the least-work oscillation path that closes the Sagnac phase condition is a two-lobe ellipse, giving a winding number of n = ½ in the mass equation; (2) the electron's S³ closure surface presents two poles to the external field, giving a topological factor of 2 in the g-factor. In 1925, both facts were numerically present in the spectroscopic data, but they were collapsed into a single postulate: intrinsic angular momentum ℏ/2. That collapse created a half-integer angular momentum with no classical analogue, forced Dirac to construct the SU(2) spinor formalism to accommodate it, and caused the spinor to appear as a new kind of mathematical object requiring new physics — when it was always a description of closure surface topology. Every experimental result attributed to spin-½ as an intrinsic quantum property — two spots in Stern–Gerlach, the electron magnetic moment, Pauli exclusion, and fine structure — is reproduced by real rotation with two stable orientations, without superluminal motion, without abstract mathematical structure disconnected from physical content, and without the hundred-year admission of ignorance that has accompanied the standard treatment. The mathematical apparatus of quantum mechanics — the Dirac equation, the Pauli matrices, the SU(2) structure — remains valid. What changes is the interpretation. The spinor is not a mysterious new object. It is the natural representation of the two-pole topology of the S³ closure surface. The conflation of 1925 prevented this from being seen. Related work: The Sagnac closure condition and the γcause invariant used in this paper are derived in: D. J. Hallman, "The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure," Zenodo (2026). doi:10.5281/zenodo.20193279
[5 Application] This paper presents the first systematic application of the geometric framework introduced in A Model of an Electron Including Two Perfect Black Bodies (DOI: 10.5281/zenodo.16759284). It marks a conceptual step toward linking the 0-Sphere electron model with stochastic processes in quantum mechanics. The analysis builds on the closed geometric equationE₀ = E₀ ( cos⁴(θ/2) + sin⁴(θ/2) + 1/2 sin²(θ) ),assigning it a quantum-mechanical interpretation. This allows the two-kernel model to account algebraically for random walk and Brownian motion phenomena. A key insight comes from the temporal phase dynamics between kernels A and B. During inter-kernel transitions, the photon sphere oscillates harmonically. At the π time phase, all electron energy transfers to kernel B, giving sin⁴(θ/2) = 1. At this instant, the photon sphere’s energy drops to zero, creating a unique geometric state in which motion becomes differentiable across all temporal domains. The next kernel is then chosen probabilistically, forming the stochastic basis of the model. The study advances from one-dimensional random walks (drunkard’s walk) to two-dimensional lattice configurations. This dimensional progression shows how instantaneous velocity nullification enables the coexistence of continuous motion with genuine randomness, requiring probabilistic landing points to maintain mathematical continuity. As the first exploration of stochastic applications within the internal-structure model, this work establishes groundwork for explaining how deterministic geometric equations can produce apparently random quantum behaviors, opening a new research direction in the 0-Sphere theoretical program. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers
We've been describing spin with equations. What if we could *see* it instead? For a century, electron spin has been a mathematical abstraction: a two-valued quantum property with no classical picture, described by SU(2) algebra and 4π4\pi4π rotations. This work turns the problem inside out. It asks: *What minimal geometry could generate such behavior?* The answer is deceptively simple: a Möbius strip coupling the electron's internal cycle to its observable boundary. But not just one strip—two. A loop (G0G_0G0​) and a figure-eight (G8G_8G8​), energetically degenerate yet topologically distinct. Their choice corresponds to spin +12+\tfrac{1}{2}+21​ or −12-\tfrac{1}{2}−21​. From this single geometric postulate, the model *mechanically* explains: - **Why there are exactly two spin states** (no continuum)- **Why spin "remembers" its state** after a magnetic field is turned off (topological memory)- **How the g≃2g \simeq 2g≃2 relation** finds a striking qualitative parallel in a fundamental 2:12{:}12:1 geometric ratio between the electron's internal cycle and its boundary- **How spinorial 4π4\pi4π rotation** emerges from Möbius topology This is **"pencil-case physics"**: no Lagrangians, no operators, just visualizable construction. It's a *sonar*—a conceptual probe mapping possible shapes beneath quantum formalism. The model doesn't replace quantum mechanics; it asks what geometry might be whispering beneath its equations. For physicists tired of "because the math says so," this paper offers a "here's how it *could* work." It's an invitation to imagine spin not as a vector in Hilbert space, but as a twist in the fabric of the electron itself.
[4 Application] This paper offers a pivotal theoretical step in the development of the 0-Sphere electron model. It applies the previously established oscillatory two-point energy model (DOI: 10.5281/zenodo.16759284) to a longstanding puzzle in quantum mechanics: Tonomura’s observation that perfect 100% contrast is never achieved in electron double-slit experiments. This limitation had resisted theoretical explanation, leaving open whether it reflected a fundamental physical law or experimental imperfection. Building on the geometric framework of earlier works, the study proposes that the electron’s internal structure—two distinct kernels arranged along a 0-sphere geometry—explains this contrast limit. These kernels, which radiate and absorb Thermal Potential Energy (TPE), undergo temporal phase variations that alter the electron’s behavior during slit passage. When TPE condenses fully into a single kernel, the electron behaves as a point particle, and the interference pattern disappears entirely. This work forms a bridge between the initial geometric concepts of prior papers and the more developed internal-structure theories that followed. While the precise mathematical formulation of kernel dynamics was still under exploration, the paper demonstrates how the 0-Sphere model can resolve concrete experimental anomalies. The proposal that perfect contrast is fundamentally unattainable marks the first instance in which the framework directly explains puzzling laboratory observations—an important early step toward a complete internal-structure interpretation of quantum phenomena. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers