On the spinor representation

J M Hoff Da Silva,C H Coronado Villalobos,R J Bueno Rogerio,Roldão Da Rocha

doi:10.1140/epjc/s10052-017-5035-y

Abstract

A systematic study of the spinor representation by means of the fermionic physical space is accomplished and implemented. The spinor representation space is shown to be constrained by the Fierz–Pauli–Kofink identities among the spinor bilinear covariants. A robust geometric and topological structure can be manifested from the spinor space, wherein the first and second homotopy groups play prominent roles on the underlying physical properties, associated to fermionic fields. The mapping that changes spinor fields classes is then exemplified, in an Einstein–Dirac system that provides the spacetime generated by a fermion.

Highlights

The very definition of a spinor in dealing with physics may be treated as a matter of some importance in itself whatsoever
When represented as a section of a bundle comprised by the S L(2, C) group and C4, it is possible to understand several spinor properties by inspecting the multivector part constructed out specific S L(2, C) objects. These objects are nothing but the bilinear covariants associated to the regarded spinor [3,4]
Its trace is given by Tr(ρ(ψ)) = 4 ψ 0, where this notation is used to indicate the projection of a multivector onto its scalar part. This correspondence provides an immediate identification between ψ and the classical Dirac spinor field

Summary

Introduction

The very definition of a spinor in dealing with physics may be treated as a matter of some importance in itself whatsoever. From simple quaternionic compositions revealing a definite rotation [1] to the fermionic quantum internal structure [2], the spinorial approach reveals its richness. Among these possible systematizations concerning spinors, there is a relevant one that encodes all the algebraically necessary information and the important relativistic construction as well, namely, the multivector spinor representation. When represented as a section of a bundle comprised by the S L(2, C) group and C4, it is possible to understand several spinor properties by inspecting the multivector part constructed out specific S L(2, C) objects.

The three equivalent definitions of spinors

The representation space

Passing through spinor classes: a natural dynamical interplay

Concluding remarks