2024/01/02 by Martin Roelfs, Roelfs, Martin, David Eelbode +3 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2401.01142
openalex publication_date 2024/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Plane-based Geometric Algebra (PGA) has revealed points in a d-dimensional pseudo-Euclidean space ℝp,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 ℝpq. Astonishingly, the non-uniqueness of this factorization reveals the existence of a local 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 \mathfrakspin(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.