The Extended Fock Basis of Clifford Algebra

We investigate the properties of the Extended Fock Basis (EFB) of Clifford algebras [1] with which one can replace the traditional multivector expansion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C} \ell(g)}$$\end{document} with an expansion in terms of simple (also: pure) spinors. We show that a Clifford algebra with 2m generators is the direct sum of 2m spinor subspaces S characterized as being left eigenvectors of Γ; furthermore we prove that the well known isomorphism between simple spinors and totally null planes holds only within one of these spinor subspaces. We also show a new symmetry between spinor and vector spaces: similarly to a vector space of dimension 2m that contains totally null planes of maximal dimension m, also a spinor space of dimension 2m contains “totally simple planes”, subspaces made entirely of simple spinors, of maximal dimension m.