Non-Unitary Behavior of Three-Dimensional Complex Numbers in Quantum Evolution: Numerical Experiments and Controlled Verification 三维复数在量子演化中的非幺正行为:数值实验与对照验证
Zhongqiang Liu
Abstract This report systematically investigates the numerical behavior of the three-dimensional complex algebra under the standard quantum mechanical evolution framework. The three-dimensional complex number is a commutative and associative real algebra spanned by , satisfying . By replacing the conventional complex-valued wave function with a -valued wave function while preserving the formal structure of the Schrödinger equation, numerical evolution under the explicit Euler scheme exhibits continuous norm drift and a directional component transfer from to . To isolate algebraic effects from numerical artifacts, we design controlled experiments with standard two-dimensional complex numbers under identical computational conditions. The results show that both 2D complex and 3D complex systems produce nearly identical norm drift magnitudes and growth patterns. Both systems exhibit exponential numerical divergence after approximately 600 iteration steps under the explicit Euler method. We conclude that the observed non-unitary drift primarily originates from the inherent instability of the explicit Euler discretization, rather than from the algebraic structure of . Further high-precision verification using unitary-preserving algorithms such as the Crank–Nicolson scheme is required to confirm whether possesses intrinsic non-unitary quantum evolution behavior. 摘要 本报告系统记录了三维复数 在标准量子力学演化框架下的数值行为。三维复数 是实数域上的三维交换结合代数,基底 满足 。将标准薛定谔方程中的波函数值域从复数替换为三维复数后,在显式欧拉格式下观察到范数的线性漂移及 的方向性分量转移。 为排除数值方法的影响,本文设计了标准二维复数对照组实验。结果表明:在相同显式欧拉格式下,二维复数与三维复数表现出几乎完全一致的范数漂移模式与量级;在本次对照轮次的仿真运行当中,两套系统均在约 600 步后发生指数级数值爆炸。本报告的核心结论是:在显式欧拉格式下观察到的非幺正行为,主要可归因于数值方法本身的缺陷,而非三维复数的代数结构。三维复数作为开放量子系统数学基础的定位,需要在更换保幺正演化算法(如 Crank‑Nicolson 或谱方法)后重新评估。
•Aug 19, 2026•0 citations