On the generalized spinor classification: beyond the Lounesto\u2019s classification

C H Coronado Villalobos,D Beghetto,R J Bueno Rogerio,A R Aguirre

doi:10.1140/epjc/s10052-020-7801-5

Abstract

In this paper we advance into a generalized spinor classification, based on the so-called Lounesto’s classification. The program developed here is based on an existing freedom on the spinorial dual structures definition, which, in certain simple physical and mathematical limit, allows us to recover the usual Lounesto’s classification. The protocol to be accomplished here gives full consideration in the understanding of the underlying mathematical structure, in order to satisfy the quadratic algebraic relations known as Fierz–Pauli–Kofink identities, and also to provide physical observables. As we will see, such identities impose restrictions on the number of possible spinorial classes allowed in the classification. We also expose a subsidiary mathematical device—a slight modification on the Clifford algebra basis—which ensures real spinorial densities and holds the Fierz–Pauli–Kofink quadratic relations.

Highlights

The well known Lounesto’s spinor classification is a comprehensive and exhaustive categorization based on the bilinear covariants that discloses the possibility of a large variety of spinors, comprising regular and singular spinors which includes the cases of Dirac, Weyl, and Majorana as very particular spinors [1].Elko spinor, proposed in [5], is a spin-1/2 fermionic field endowed with mass dimension one, built upon a complete set of eigenspinors of the charge conjugation operator, which has the property of being neutral with respect to gauge interactions
As Elko spinors have mass dimension one, there is nothing that precludes the appearance of mass dimension one spinor further in classes (4) and (6) in Lounesto’s classification [8]
For a more complete understanding of this subject the reader is referred to [12,13,14]. In this communication we look for the underlying mathematical approach that allows us to define a spinorial dual structure, by making correct use of the Crawford mechanism to evaluate the bilinear covariants, using the correct composition law of the basis vector of the Clifford algebra, and verifying if such forms satisfy the algebraic Fierz–Pauli–Kofink identities

Summary

Introduction

The well known Lounesto’s spinor classification is a comprehensive and exhaustive categorization based on the bilinear covariants that discloses the possibility of a large variety of spinors, comprising regular and singular spinors which includes the cases of Dirac, Weyl, and Majorana as very particular spinors [1]. For a more complete understanding of this subject the reader is referred to [12,13,14] In this communication we look for the underlying mathematical approach that allows us to define a spinorial dual structure, by making correct use of the Crawford mechanism to evaluate the bilinear covariants, using the correct composition law of the basis vector of the Clifford algebra ( taking into account the Dirac normalization), and verifying if such forms satisfy the algebraic Fierz–Pauli–Kofink identities (we will call FPK identities).

A brief overview on spinorial duals

Basic conceptions on the Lounesto’s classification and Spinorial densities

On the set-up of a generalized Spinor classification

A Detour on the Clifford algebra basis modification

Final remarks