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

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) = eδe·eiωt. 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) = eδe·eiωt。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