N ov 2 01 9 Subliminal aspects concerning the Lounesto ’ s classification

Abstract. In the present communication we employ a split programme applied to spinors belonging to the regular and singular sectors of the Lounesto’s classification [1], looking towards to unveil how it can be built or defined upon two spinors arrangement. We separate the spinors into two distinct parts and investigate to which class within the Lounesto’s classification each part belong. The machinery here developed open up the possibility to better understand how spinors behave under such classification. As we shall see, the resulting spinor from the arrangement of other spinors (belonging to a distinct class or not) does not necessarily inherit the characteristics of the spinors that compose them, as example, such characteristics stands for the class, dynamic or the encoded physical information.


I. INTRODUCTION
Spinors are objects widely used in physics and also in mathematics. Firstly, their importance comes from the fact that they carry a rich information about the space-time where they are built beyond the fact that they play the central role in describing fermionic matter fields. Such a mathematical objects were firstly defined byÉlie Cartan [2] where he provided the following definition to a spinor "A spinor is thus a sort of "directed" or "polarised" isotropic vector; a rotation about an axis through an angle 2π changes the polarisation of this isotropic vector ". Such entities may be defined without reference to the theory of representations of groups [3]. Spinors can be used without reference to relativity, but they arise naturally in discussions of the Lorentz group. One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed [4].
Due the importance of spinors, in a foundational work, Lounesto had the very idea to classify them [1]. The principle that he followed was an extensive and tedious algebraic analysis relating to spinors due to its bilinear covariants (physical information). Given this mathematical procedure, Lounesto assures the existence of only six classes of spinors, from which he denominates a group of three classes called "Dirac spinors of the electron" and the other remaining three spinors on a group denominated as "Singular spinors with a light-like pole" or only "Singular spinors". Having said that, the Lounesto's classification is taken as an algebraic classification. Several studies related to spinor and Lounesto's classification have been developed in recent times [5-10, and references therein].
The idea behind the present essay is based on the recent development [7], where it is shown a systematization/categorization on how to ascertain the spinor's class due to its phases factor, without the necessity to evaluate all the bilinear forms. Here our focus is look towards analyse how spinors may be constituted. The mechanism that we use for the present development, comes from the assumption that a spinor can be constructed from a arrangement of other spinors. In this vein, we investigate what are the conditions or the constraints that allow a spinor's composition, also we analyse what kind of combinations are possible and what result can be reached. As we shall see, such a mechanism shows the impossibility of a generalization for spinor construction. Some classes are built given some algebraic constraints and other classes, as class 6, for example, do not allow this procedure to be applied. As highlighted in [7,9], class 6 hold a very special case of spinors.
The paper is organized as it follows: in the next section we show the spinor's separation method to be used along the article. In Sect.II A we take advantage of the method and investigate the how to compose spinors belonging to the regular sector of the Lounesto's classification. Thus, in Sect.II B we perform the very same analyse but now for singular spinors. Finally, in Sect.III, we conclude.

II. DEFINING THE SPINORIAL DETACHMENT PROGRAMME
The programme to be employed here, is based on a spinor "division". Impose to the phase factors the following requirement -actually, the most general case -α, β ∈ C. The last mentioned feature allow one to write the following α = Re α + iIm α and β = Re β + iIm β. Now, suppose the following spinor split in other words, it may be expressed as it follows note that we now have two distinct spinors, where the label j, k and l stands for the corresponding Lounesto's classes of each spinor and it runs j, k, l = 1, . . . , 6. For the purpose of simplifying notation we omitted the spinor's momentum. The relation presented in (2) is just a way to separate a spinor, now we have a spinor which carry the real part of the phases (Re) and other carrying the imaginary part of the phases (Im). What we will check here is whether the combination (⊕) of spinors preserves the class and what is the result of adding different spinor classes. As we shall also observe is that the above procedure consists in a union of the physical information, i.e., the bilinear forms. Let Γ be a set of bilinear forms of a given spinor [11,12] where j stands for the spinor's class. When performing the procedure defined in (2), automatically a superposition of the spinor's physical information is observed -thus, such a feature translate into Note that (4) tell us that a given set of bilinear amounts can be obtained from an union of two distinct (or even equal) sets of bilinear amounts. What should be clear to the reader is that the above definition is not exactly a summation of each one of the bilinear forms separately but rather that the main outcome is a combination of bilinear forms that, when added together, provide a new set.

A. On the regular spinors framework
A single-helicity spinor in the rest-frame referential is defined as follows [7,13] in which we have defined the k µ rest-frame momentum as Commonly, the spinorial components in the rest-frame referential reads and where the phases factors α and β ∈ and the only requirement under the such factors, comes from the orthonormal relation, it stands for the regular spinor's case αβ * + α * β ∝ m, in which m stands for the mass of a particle. The upper indexes ± refers to the corresponding helicity of each component, for more details the Reader is cautioned to check [14]. Now, if one wish to define such rest spinors in an momentum arbitrary referential, such task is accomplished under action of the Lorentz boosts operator, which reads as usually defined cosh ϕ = E/m, sinh ϕ = p/m, andφ =p, yielding the following relation Now, consider the Dirac equation acting on (5) the Dirac operator acting over the ψ(p) spinor provide where the operator σ ·p stands for the helicity operator, in which σ stands for the Pauli matrices andp is the unit momentum vector. In order to follow through with the calculations, one impose to the components to carry positive helicity and, then, we obtain the following relation 1 up to uor knowledge, (11) is only fulfilled if α = β, otherwise the Dirac dynamic is not reached. The last result combined with Table 1 in [7] lead to the observation that only spinors belonging to class 2 within Lounesto's classification, under the requirement α = β, satisfy the Dirac equation. This brief calculation is intended to show that dynamics are not necessarily maintained when combining different spinors; that is, the sum of two spinors that satisfy Dirac's dinamic do not necessarily provide a spinor that satisfies Dirac's equation. Accordingly to the Table 1 of [7] we may perform the following analysis: 1) α, β ∈ C with α = β (class 1): In view of the protocol introduced above, we start by looking at this first case, which lead to thus, such mechanism brings to the light the following relation Note that a class 1 spinor may be built upon two spinors belonging to class 2. However, although class 1 spinors may be composed by class 2 spinors, they do not satisfy the Dirac's dynamic.
2) α ∈ C and β ∈ IR (class 1): Now, note that leading to highlighting a new possibility to write a spinor which belong to class 1.
3) α ∈ C and β ∈ Im (class 1): For this case we have and the only possibility stands for 4) α, β ∈ C with α = β (class 2): Such constraints leads to the above calculations allow one to write Given the less restrictive requirements for a spinor to belong to class 2, meantime, this is the only possibility to write them.

5) α ∈ Im and β ∈ IR (class 3):
Here we find two quite peculiar situations, the above requirements provides the following relation which translates into 6) α ∈ IR and β ∈ Im (class 3): It allows one to define leading to Interestingly enough, class 3 spinors can only be defined as a combination of two class 6 spinors. Thus, the above results can be summarized as it follows: Class Spinorial combination Phases constraints 1 ψ2 ⊕ ψ2 ∀α, β ∈ IR or ∀α, β ∈ Im 1 ψ2 ⊕ ψ6 ∀α ∈ C and ∀β ∈ IR or ∀α ∈ C and ∀β ∈ Im 2 ψ2 ⊕ ψ2 α, β ∈ C|α = β 3 ψ6 ⊕ ψ6 ∀α ∈ Im and ∀β ∈ IR or ∀α ∈ Im and ∀β ∈ Im We highlight that to obtain certain classes, we face some restrictions, e.g., the impossibility to construct a spinor belonging to class 6, it do not admit to be written as a combination of two distinct spinors.

B. On the singular spinors framework
In this section, we look towards apply the previous algorithm on dual-helicity spinors. Dual-helicity spinors can be defined as [7,9,13,14] where Θ it the well-known Wigner Time-reversal operator Taking advantage of Table 2 presented in [7], and the spinor defined in (26), one is able to define the following 1) α, β ∈ C with |α| 2 = |β| 2 (class 4): For this case we have which provide the following relations showing a wide variety of combinations.
2) α, β ∈ C with |α| 2 = |β| 2 (class 5): Furnishing the following detachment which can be divided into Note that the Majorana spinor (which describes the neutrino) can be written in terms of two non-neutral spinors.
Moreover, driven by the programme developed here, it is easy to see that when the spinor detach protocol is applied, the physical information do not necessarily is carried through the resulting spinor, as example, it does not hold the class, dynamic or even physical information.
Interesting enough, from an inspection of Table II, when dealing with the singular sector of the Lounesto's classification, we may construct neutral spinors from a combination of non-neutral spinors, as the case presented in the rows 7, 9 and 10. Notice that the same akin reasoning can be extended for the case of class 4 spinors, which can be built upon two neutral spinors, as the case in the rows 2 and 5. We emphasize that a similar connection between both Lounesto's sections, as shown in [7], can be performed here, however, no relevant physical information is disclosed.