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Photonic Helicity in 3D Number Space Wave-Particle Duality as Helical Projection

Zhongqiang Liu

Published Aug 16, 2026• 0 citations
Publisher

Abstract

Abstract This paper proposes a geometric theory of wave-particle duality based on the algebra of three-dimensional numbers (3D Numbers). In the 3D number space T = σ + δe + ti (with e²=1, i²=-1, ei=0), the motion of a photon naturally takes the form of a helical trajectory T(t) = e^{δe}·e^{iωt}. The projection of this helix onto the (σ,i) plane yields the standard quantum mechanical wave function ψ(t) = cosh(δ)·e^{±iωt}, while the hyperbolic stretch along the e-direction corresponds to particle nature. Through five numerical experiments, we verify: (1) the helical structure is exact (radius deviation < 2×10⁻¹⁶); (2) projection to the complex plane gives the standard wave function; (3) all quantum mechanical quantities (E=ℏω, p=ℏk, λ=2π/k) are derived directly; (4) δ-parameter scan reveals a continuous wave-particle spectrum; (5) 3D numbers are the unique number system supporting both wave and particle nature. The central thesis: wave-particle duality is not a superposition of two attributes, but the projection decomposition of a single helical motion onto two orthogonal directions. Keywords: 3D numbers, helical motion, wave-particle duality, mid integration, foundations of quantum mechanics, non-associative algebra 摘要 本文提出一种基于三维数(3D Number)代数结构的光子波粒二象性几何理论。在三维数空间 T = σ + δe + ti(其中 e²=1, i²=-1, ei=0)中,光子的运动自然地表现为螺旋运动 T(t) = e^{δe}·e^{iωt}。该螺旋运动在 (σ,i) 平面上的投影给出标准量子力学波函数 ψ(t) = cosh(δ)·e^{±iωt},在 e 方向上的双曲伸缩对应粒子性。通过五组数值实验,我们验证了:(1) 螺旋结构严格成立(半径偏差 < 2×10⁻¹⁶);(2) 投影到复平面给出标准波函数;(3) 从螺旋运动可直接推导能量 E=ℏω、动量 p=ℏk、de Broglie 波长 λ=2π/k 等全部量子力学量;(4) δ 参数扫描揭示了波动-粒子连续谱;(5) 三维数是唯一同时支持波动性和粒子性的数系。本工作表明:波粒二象性的本质不是两种属性的叠加,而是同一螺旋运动在两个正交方向上的投影分解。 关键词:三维数、螺旋运动、波粒二象性、中道积分、量子力学基础、非结合代数

DOI
10.5281/zenodo.21965833

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Abstract This paper proposes a geometric theory of wave-particle duality based on the algebra of three-dimensional numbers (3D Numbers). In the 3D number space T = σ + δe + ti (with e²=1, i²=-1, ei=0), the motion of a photon naturally takes the form of a helical trajectory T(t) = e^{δe}·e^{iωt}. The projection of this helix onto the (σ,i) plane yields the standard quantum mechanical wave function ψ(t) = cosh(δ)·e^{±iωt}, while the hyperbolic stretch along the e-direction corresponds to particle nature. Through five numerical experiments, we verify: (1) the helical structure is exact (radius deviation < 2×10⁻¹⁶); (2) projection to the complex plane gives the standard wave function; (3) all quantum mechanical quantities (E=ℏω, p=ℏk, λ=2π/k) are derived directly; (4) δ-parameter scan reveals a continuous wave-particle spectrum; (5) 3D numbers are the unique number system supporting both wave and particle nature. The central thesis: wave-particle duality is not a superposition of two attributes, but the projection decomposition of a single helical motion onto two orthogonal directions. Keywords: 3D numbers, helical motion, wave-particle duality, mid integration, foundations of quantum mechanics, non-associative algebra 摘要 本文提出一种基于三维数(3D Number)代数结构的光子波粒二象性几何理论。在三维数空间 T = σ + δe + ti(其中 e²=1, i²=-1, ei=0)中,光子的运动自然地表现为螺旋运动 T(t) = e^{δe}·e^{iωt}。该螺旋运动在 (σ,i) 平面上的投影给出标准量子力学波函数 ψ(t) = cosh(δ)·e^{±iωt},在 e 方向上的双曲伸缩对应粒子性。通过五组数值实验,我们验证了:(1) 螺旋结构严格成立(半径偏差 < 2×10⁻¹⁶);(2) 投影到复平面给出标准波函数;(3) 从螺旋运动可直接推导能量 E=ℏω、动量 p=ℏk、de Broglie 波长 λ=2π/k 等全部量子力学量;(4) δ 参数扫描揭示了波动-粒子连续谱;(5) 三维数是唯一同时支持波动性和粒子性的数系。本工作表明:波粒二象性的本质不是两种属性的叠加,而是同一螺旋运动在两个正交方向上的投影分解。 关键词:三维数、螺旋运动、波粒二象性、中道积分、量子力学基础、非结合代数
Wave particle duality remains a central interpretational challenge in quantum theory. In this work, we develop a trajectory-based description of photon dynamics formulated in an extended complex space within the relativistic quantum Hamilton Jacobi framework. In this approach, photon motion is represented by complex trajectories whose real projections describe propagation, while imaginary components encode oscillatory structure. We show that momentum eigenstates correspond to straight line trajectories with uniform propagation at the speed of light, whereas superposition states lead to nontrivial quantum potentials and oscillatory motion in the complex plane. Extending the analysis to complex two dimensional space reveals richer dynamical behavior, including propagating wave like trajectories and standing wave like patterns in real projections. Energy momentum consistency is verified through an internal coherence analysis based on projected standing wave wavelengths. Rather than introducing new dynamical laws or additional physical dimensions, the complex space is employed as an interpretational framework that renders wave like and particle like aspects as complementary projections of a single underlying motion. The results suggest a unified geometric perspective on wave particle duality, while remaining fully compatible with standard quantum mechanics.
In this short note, the the key to unlocking the secrets of quantum physics will be elucidated by exploring the fundamentals of Schrodinger's wave mechanics approach to describing quantum phenomenon. We will show that de Broglie's wave-particle duality hypothesis which lies at the heart of Schrodinger's wave-function \psi produces a complex wave equation whose mathematical structure can be described by Euler's famous equation e^{i\theta}=cos(\theta)+isin(\theta) which basically describes a helical wave in 3D space. By comparing and contrasting the electromagnetic wave with that of a helical wave which Euler's equation represents, we may have discovered the geometric basis for spin and helicity and antimatter with negative energies that Dirac uncovered in his relativistic reformulation of Schrodinger's equation.
Abstract The standard formalisms of quantum mechanics and special relativity are extraordinarily successful, yet several foundational features are introduced to students as postulates or mathematical constructs without physical visualisation: why the wavefunction must be complex, what is physically associated with spin, why time dilation occurs in material particles, where the Lorentz-factor scaling of inertia comes from, why energy quantisation follows from a phase-closure condition, and what geometric account can be given of wave-particle duality. This paper presents a geometric picture—already implicit in Schrödinger’s Zitterbewegung and de Broglie’s internal periodicity—that may help students develop physical intuition for each of these features. The electron is represented as executing a phase-coherent helical motion at speed c , with transverse radius a 0 = ℏ / 2 m 0 c and axial pitch equal to the de Broglie wavelength. Within this picture the imaginary unit i admits an intuitive interpretation as the 90 ∘ rotation operator in physical transverse space; quantisation may be visualised as a phase-closure condition with the quantum number n counting helical nodes; time dilation and the Lorentz-factor scaling of inertia both have a transparent geometric reading from a single pythagorean speed constraint; and interference may be understood as a Moiré-type geometric phase effect. The model is fully consistent with standard quantum and relativistic predictions. Its contribution is pedagogical and interpretational: it offers a unified geometric visualisation that may help students develop intuition for these foundational concepts, rather than a derivation of quantum mechanics or special relativity from first principles. The paper is self-contained and addressed to advanced undergraduates and graduate students.
A theory employing the vortex shape of the electron was presented to resolve the enigma of the wave-particle duality. Conventions such as “particle” and “wave” were used to describe the behavior of quantum objects such as electrons. A superfluid vacuum formed the base to describe the basic vortex structure and properties of the electron, whereas various formulations derived from hydrodynamic laws described the electron vortex circumference, radius, angular velocity and angular frequency, angular momentum (spin) and magnetic momentum. A vortex electron fully explained the associations between momentum and wave, and hydrodynamic laws were essential in deriving the energy and angular frequency of the electron. In general, an electron traveling in space possesses internal and external motions. To derive the angular frequency of its internal motion, the Compton wavelength was used to represent the length of one cycle of the internal motion that is equal to the circumference of the electron vortex. The angular frequency of the electron vortex was calculated to obtain the same value according to Planck’s theory. A traveling vortex electron has internal and external motions that create a three-dimensional helix trajectory. The magnitude of the instantaneous velocity of the electron is the resultant of its internal and external velocities, being equal to the internal velocity reduced by the Lorentz factor (whose essence is presented in a detailed formulation). The wavelength of the helix trajectory represents the distance traveled by a particle along its axis during one period of revolution around the axis, resulting in the same de Broglie wavelength that corresponds to the helix pitch of the helix. Mathematical formulations were presented to demonstrate the relation between the energy of the vortex and its angular frequency and de Broglie’s wavelength; furthermore, Compton’s and de Broglie’s wavelengths were also differentiated.
This paper summarizes the current state of the helical worldline model developed across eleven previous publications. The model posits that all particles follow helical paths through spacetime with invariant speed v_path = √[v² + (c/γ)²] = c, where v is spatial velocity and c/γ represents motion through time. This single principle unifies special relativity, de Broglie matter waves (λ = h/p as spatial projection of the tilted helix), the Compton effect (λ_f - λ_i = (h/m₀c)(1-cos θ) from helical phase matching), and the photoelectric effect (work function φ derived from helical compression by crystal lattices). Spin ħ/2 emerges from the 4π symmetry of the helix, and Bohr quantization follows from helical standing wave conditions in atoms. The model makes testable predictions including tunneling time ≈ ħ/(2m₀c²) and work function scaling φ ∝ 1/L² with crystal structure. The author presents the current mathematical framework, summarizes confirmed connections to established physics, identifies open questions, and outlines further study needed to extend the model toward a full unification.
This work proposes a geometric representation of quantum phase fields motivated by the long-standing tension between localized particle cores and dispersive wave propagation in quantum mechanics. Following the original insight of Louis de Broglie's phase harmony, the quantum phase is reinterpreted as a helical structure surrounding a localized core. The construction is developed using an inverse methodology: instead of deriving the structure dynamically from first principles, a candidate geometry is postulated and tested for internal consistency with the kinematic constraints of quantum mechanics. At the matching boundary defined by the Compton radius Rc = ħ/(mc), the phase field acquires a quantized azimuthal winding, forming a helical structure that connects the localized core with the extended navigation wave. Within this framework, the de Broglie wavelength λ = h/p emerges naturally as the geometric pitch of the helical phase surface. The analysis demonstrates that this helical configuration satisfies three essential requirements for a double-solution structure:topological phase quantization, velocity matching between wave and particle, and the correct decomposition of rest and kinetic energy. The work is presented as a proof-of-existence for a geometrically consistent phase structure that could support a localized vortex-like particle guided by a continuous phase field. It forms part of a broader research program exploring the mechanical and topological origins of quantum phenomena.