On Spinors and Null Vectors

TL;DR

This paper explores the mathematical relationship between spinors and null vectors in Clifford algebra, establishing new properties and conditions for simple (pure) spinors, and providing methods to construct specific spinors with given null planes.

Contribution

It introduces new properties of null vectors, characterizes simple spinors as one-dimensional subspaces, and generalizes classical theorems to identify conditions for simple spinors.

Findings

Null vectors bisect spinor space into two equal parts

Simple spinors form one-dimensional subspaces

Generalized conditions for simple spinors based on classical theorems

Abstract

We investigate the relations between spinors and null vectors in Clifford algebra with particular emphasis on the conditions that a spinor must satisfy to be simple (also: pure). In particular we prove: i) a new property for null vectors: each of them bisects spinor space into two parts of equal size; ii) that simple spinors form one-dimensional subspaces of spinor space; iii) a necessary and sufficient condition for a spinor to be simple that generalizes a theorem of Cartan and Chevalley that appears now as a corollary of this result. We also show how to write down easily the most general spinor with a given associated totally null plane.