AI 中文总结
该研究为量子态空间建立严格统一几何框架，通过嵌入高阶对偶数代数，将量子态表示为非约化概型理论点，使控制刘维尔 - 冯·诺依曼动力学的换位子映射到代数流上，确立幂零对偶代数为高维量子运动学几何图景。
AI 中文摘要
本文通过构建到高阶对偶数代数\(\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2 - 1})\)的光滑、正则嵌入，为量子态空间建立了一个严格、统一的几何框架。每个量子态都被忠实地表示为一个非约化的概型理论点。在这个截断环的统一族下，控制刘维尔 - 冯·诺依曼动力学的非线性矩阵换位子全局映射到平坦、线性和刚性代数流上，确立幂零对偶代数作为高维量子运动学的原始几何图景。
英文摘要
This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras $\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2-1})$, wherein every quantum state is faithfully represented as a non-reduced scheme-theoretic point. We show that under this unified family of truncated rings, the non-linear matrix commutators governing the Liouville-von Neumann dynamics map globally onto flat, linear, and rigid algebraic flows, establishing nilpotent dual algebras as a pristine geometric landscape for higher-dimensional quantum kinematics. As $N \to \infty$, this family converges to a Cauchy-complete power series ring $\mathcal{Q}_\infty$, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.
CommentsPlaced the persistent scalar trace background 1/N which defines the exact baricentro of the configuration space representing the maximally mixed state across all dimensions N
