波数：离散序列代数、筛投影子与动力学几何
Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry
浏览论文内容
中文总结 AI 辅助
本文对离散格上的波闭包空间进行了代数、几何和物理分类，证明了波群结构，并建立了叠加定理、筛投影子以及时空和洛伦兹群的涌现，为离散量子模拟提供了数学基础。
中文摘要 AI 辅助
我们建立了由离散空间格 $\mathbb{Z}$ 上的本原平面波序列生成的波闭包空间的全面代数、几何和物理分类。通过展开相空间解决格简并，我们证明了在逐点乘积、逆和根提取下的线性波群 $\mathcal{G}$ 是一个无限可除阿贝尔挠群，同构于 $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$，具有到 Prüfer $p$-群和极大分圆值域 $\mathbb{Q}^{\mathrm{ab}}$ 的典范分解。扩展到坐标幂得到多项式相位群 $\mathcal{G}_{\mathrm{poly}}$，由整值多项式 $\operatorname{Int}(\mathbb{Z})$ 分类。加上加法得到群代数 $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$，我们为其建立了置换对称相量叠加定理，将叠加分解为集体重心载波和封闭的 Born 概率包络。利用单位根上的代数筛，我们构造了幂等投影子和互补的 Not-筛陷波滤波器，实现了动量和局域 Kronecker 位置基的精确代数合成。推广到非阿贝尔四元数 $\mathcal{V}_\mathbb{H}$，我们证明了极分解为标量包络和 $\mathrm{SU}(2)$ 旋量转子。将坐标推进等同于时间，双四元数行列式内在地产生闵可夫斯基时空 $s^2 = c^2 t^2 - \\\\|\mathbf{x}\\\\|^2$ 和洛伦兹群 $\mathrm{SO}^+(1,3)$。我们展示了涌现的真空零点抖动、有理频率的拓扑选择以及与离散 qudit 的对应关系。
英文摘要
We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\mathbb{Q}^{\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\mathcal{G}_{\mathrm{poly}}$, classified by integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. Adjoining addition yields the group algebra $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\mathcal{V}_\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \|\mathbf{x}\|^2$ and the Lorentz group $\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.
