Rozenfeld's Geometric Approach to Spinors

TL;DR

This work revisits Rosenfeld's geometric view of spinors, where spinor coordinates are identified with plane generators of the absolute in extended non-Euclidean spaces and spinor motions correspond to movements of these absolutes. It unifies this geometric perspective with the Clifford-algebraic treatment of spinors, detailing how Spin$(p,q)$ groups arise as double covers of $SO_0(p,q)$ via even subalgebras and how their coordinates encode rotation parameters across different mod-8 cases. Through explicit constructions, including the $(4,4)$ example and Karoubi-type factorizations, the paper demonstrates a continuous, two-way link between real spinors and plane-generator data, enabling a geometric interpretation of spinor representations in terms of absolute structures. The results illuminate the interplay between geometry (absolutes and plane generators) and algebra (minimal left ideals of $\ell_{p,q}$) and provide a framework for interpreting spinor transformations as motions of non-Euclidean spaces, with potential relevance to relativistic physics.

Abstract

Rosenfeld's geometric approach to spinors is considered, according to which the coordinates of spinors are represented by the coordinates of the plane generators of the maximal dimension of the absolutes of non-Euclidean spaces. As an example, non-Euclidean spaces with neutral signatures are investigated.