标题： 从不变分解到旋量 显示英文标题

标题： From Invariant Decomposition to Spinors

摘要： 基于平面的几何代数（PGA）揭示了在$d$维伪欧几里得空间$\mathbb{R}_{p,q,1}$中的点可以通过$d$- blades 而不是向量来表示。 这一发现使得点可以分解为$d$个正交超平面，从而将点作为局部几何代数$\mathbb{R}_{pq}$的伪标量。 令人惊讶的是，这种分解的非唯一性揭示了每个点处存在一个局部的$\text{Spin}(p,q)$几何规范群。 此外，点也可以被分解为$\mathfrak{spin}(p,q)$的卡坦子代数元素的乘积，这些元素传统上用于标记旋波表示。 因此，点揭示了量子场论某些奥秘之前隐藏的几何基础。 这项工作概述了 PGA 对任何维度中旋波表示研究的影响，并是探索这一见解后果的研究计划的第一步。 显示英文摘要

摘要： Plane-based Geometric Algebra (PGA) has revealed points in a $d$-dimensional pseudo-Euclidean space $\mathbb{R}_{p,q,1}$ to be represented by $d$-blades rather than vectors. This discovery allows points to be factored into $d$ orthogonal hyperplanes, establishing points as pseudoscalars of a local geometric algebra $\mathbb{R}_{pq}$. Astonishingly, the non-uniqueness of this factorization reveals the existence of a local $\text{Spin}(p,q)$ geometric gauge group at each point. Moreover, a point can alternatively be factored into a product of the elements of the Cartan subalgebra of $\mathfrak{spin}(p,q)$, which are traditionally used to label spinor representations. Therefore, points reveal previously hidden geometric foundations for some of quantum field theory's mysteries. This work outlines the impact of PGA on the study of spinor representations in any number of dimensions, and is the first in a research programme exploring the consequences of this insight.