Complex structures and the Elie Cartan approach to the theory of spinors

TL;DR

This paper explores the relationship between complex structures on Euclidean spaces, Clifford algebras, and spinors, highlighting their connections to fermionic degrees of freedom and Fock states within the framework of Elie Cartan's approach.

Contribution

It provides a novel perspective on the correspondence between complex structures, Clifford algebras, and spinors using Elie Cartan's methodology, clarifying their physical and mathematical significance.

Findings

Complex structures correspond to Clifford algebra identifications.

Spinors are related to vacua and Fock states in fermionic systems.

The terminology of simple and pure spinors is inadequate for describing these states.

Abstract

Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good.