On spinors and null vectors**This paper is dedicated to the memory of my father Paolo Budinich who passed away in November 2013 not before transferring to me his enthusiasm for simple spinors.

Abstract

We investigate the relations between spinors and null vectors in Clifford algebra of any dimension 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 subspaces 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 which becomes a corollary of this result. We also show how to write down easily the most general spinor with a given associated totally null plane.