Fock space description of simple spinors

Abstract

Cartan’s simple—often called pure—spinors corresponding to even-dimensional complex vector spaces are defined in terms of the associated maximal totally null planes. Their geometrical properties are derived and described using notions familiar to physicists: Dirac and Weyl spinors, gamma matrices, tensors formed bilinearly from pairs of spinors, and creation and annihilation operators of Fermi states. A new theorem characterizes a simple spinor φ by the properties of the vector tψBγμφ, where ψ is an arbitrary spinor and B is the matrix connecting the gamma matrices with their transposes. The Cartan constraint equations on the components of simple spinors are given a new, geometrically transparent derivation based on the action on simple spinors of a maximal Abelian subgroup of the group Spin.