发表机构
Ulm University of Applied Sciences（乌尔姆应用技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出量子力学标量代数的最小双复扩展，引入理想值扇区，证明其唯一性并探讨其数学结构，但物理意义尚待探究。
AI 中文摘要
标准量子力学以复数作为其标量代数。我们探究这一复结构在代数上是否根本，抑或它可能构成一个更大的交换结合标量代数中一个特殊的嵌入扇区。要求普通复扇区保持完整，同时允许一个额外的稳定的类复解析扇区，后者成为一个真理想，使得在有限维中零因子不可避免，并暗示实数维数的下界为四。双复代数实现了这一最小值，且在额外的相容性假设下，四维实现在同构意义下是唯一的。所得代数展现出两种不同的分解：正交幂等理想的正则分解，以及适应于嵌入复扇区和额外理想值扇区的分解。因此，普通复标量位于正则幂等分解的对角线上。理想扇区支持泰勒级数、振荡指数、傅里叶型核以及薛定谔型表达式，并承载形式化的理想值特征结构。将玻恩规则限制于嵌入复扇区可恢复其标准形式。这种双标量架构在代数框架之外是否具有物理意义，仍是一个开放问题。
英文摘要
Standard quantum mechanics takes the complex numbers as its scalar algebra. We ask whether this complex structure is algebraically fundamental or may instead form a distinguished embedded sector of a larger commutative and associative scalar algebra. Requiring the ordinary complex sector to remain intact while admitting an additional stable complex-like analytic sector turns the latter into a proper ideal, makes zero divisors unavoidable in finite dimension, and implies a lower bound of four real dimensions. The bicomplex algebra realizes this minimum, and under an additional compatibility assumption the four-dimensional realization is unique up to isomorphism. The resulting algebra exhibits two distinct decompositions: the canonical decomposition into orthogonal idempotent ideals and a decomposition adapted to the embedded complex sector and the additional ideal-valued sector. The ordinary complex scalars therefore lie diagonally across the canonical idempotent decomposition. The ideal sector supports Taylor series, oscillatory exponentials, Fourier-type kernels, and Schrödinger-type expressions, and carries a formal ideal-valued eigenvalue structure. Restricting the Born rule to the embedded complex sector recovers its standard form. Whether this dual scalar architecture has physical significance beyond the algebraic framework remains an open question.
Comments46 pages
