# Constraints on mapping the Lounesto’s classes

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## Abstract

The so-called Lounesto’s classification engenders six distinct classes of spinors, divided into two sectors: one composed by regular spinors (single-helicity spinors) and the other composed by singular spinors (comprising dual-helicity spinors). In the present essay we develop a mechanism to fully define the right class within the Lounesto’s classification a spinor belongs to, without necessity to evaluate the 16 bilinear forms. The analysis lies in the following criteria: a judicious inspection of the phases factor present in both spinor’s components. Thus, the machinery developed here works for both regular and singular spinors. Taking advantage of the present algorithm, we analyse, under certain conditions, the possibility to transmute between the six classes.

## 1 Introduction

Spinors comprises of mathematical and also physical objects, responsible to describe matter, and may be defined in several different fashions. Within the context of the Clifford’s algebra, the spinors are defined to be elements of a left minimal ideal, whereas in the context of group theory we say that the spinors are carriers of the fundamental representation of the group [1, 2]. The first fundamental aspect related with spinors is that a single given spinor cannot be experienced or even detected. Surely, by the Lorentz group compositions laws related to spinors do not recover the original state when a \(2\pi \) rotation are performed on a single spinor. Hereupon, up to our knowledge, a given single spinor cannot be/or represent a physical observable. Nevertheless, even the composite of given mathematical quantities are directly associated to physical observables, thus, the importance of the bilinear covariants (bilinear forms). A algorithm to right define the bilinear forms, thus, recast into define an appropriate dual structure for a given spinor and, then, compute all the bilinear covariants associated to them.

The well-known Lounesto’s classification [2] is a comprehensive and exhaustive algorithm that based on the bilinears covariants associated to a given spinor, embraces the possibility of a vast diversity of spinors. Such classification is divided into two sectors: one comprises of regular and the other singular spinors. The Lounesto classification is categorized by six distinct classes and there is nothing that preclude that each class have subclasses. The possibility of finding more classes has not been ruled out in [3], however, to the best of our knowledge, the physical interpretation of the spinors that may occupy these new classes is still quite complicated, at least so far, no entity harbouring these classes was, in fact, found or even defined. However, mathematically this was shown to be valid as an extension of Lounesto’s classification.

The path followed by Lounesto is to classify spinors according with the physical information encoded in the spinors – not alluding to dynamic, helicity or any other attribute. In this sense, it is possible to accommodate in the aforementioned classification the Dirac, Majorana and Weyl spinors. The regular classes encompass the Dirac spinors and also single-helicity spinors, which do not necessarily satisfy the Dirac’s dynamic, whereas singular classes are conditioned to embrace flag-dipole spinors [4, 5], Majorana (flag-pole) spinors and Weyl spinors, filling in classes 4, 5 and 6 respectively. As we shall see in the scope of the present work, flag-dipole and flag-pole spinors are known to carry a very similar structure, the main difference between them lies on the phases parameters. A curious fact that should be mentioned is related to the Weyl’s spinors (class 6), due to its peculiarities [6], it is possible to map any class into class 6, however, the opposite is not true. The Weyl spinors are taken to belong to a univocal class within the Lounesto’s classification. The whole path to be followed in this manuscript is based on the canonical form, we not deal with the second quantization (quantum fields) and will not allude to them either.

The paper is organized as it follows: in the next section we set the spinors notation and introduce the Lounesto’s classification. In Sect. 3 we define a general single-helicity spinor and analyse all the possible phases combinations, displayed in a table, making explicit the restrictions to a spinor to belong to a determined class. Taking advantage of the previous results, in Sect. 4, we develop an analogous protocol, however, now for dual-helicity spinors. Thus, in Sect. 5 we look towards to establish a link between both sectors of the Lounesto’s classification besides analyse the possibility to map the classes. Finally, we conclude.

## 2 Elementary review

This section is reserved for a brief review on the introductory elements that are necessary for the study carried out in the scope of this paper.

### 2.1 Notation, spinors and spinorial components

^{1}

### 2.2 The Lounesto’s classification

In the Dirac’s theory, the above bilinear covariants are interpreted respectively as the mass of the particle (\(\sigma \)), the pseudo-scalar (\(\omega \)) relevant for parity-coupling, the current of probability (\(\mathbf {J}\)), the direction of the electron spin (\(\mathbf {K}\)), and the probability density of the intrinsic electromagnetic moment (\(\mathbf {S}\)) associated to the electron. In general grounds, it is always expected to associate such bilinear structures to physical observables.

^{2}), known as

*Lounesto’s Classification*[2]:

- 1.
\(\sigma \ne 0\), \(\quad \omega \ne 0\);

- 2.
\(\sigma \ne 0\), \(\quad \omega =0\);

- 3.
\(\sigma =0\), \(\quad \omega \ne 0\);

- 4.
\(\sigma =0=\omega ,\) \(\mathbf K \ne 0,\) \(\quad \mathbf S \ne 0\);

- 5.
\(\sigma =0=\omega ,\) \(\mathbf K =0,\) \(\quad \mathbf S \ne 0\);

- 6.
\(\sigma =0=\omega ,\) \(\mathbf K \ne 0,\) \(\quad \mathbf S =0\),

As it was shown in Ref. [13], due to the contrasting adjoint structure of the Elko spinors [9], it is extremely necessary to rethink the physical interpretation carried by Lounesto’s classification. Quite recently a more general treatment and interpretation for the bilinear forms carrying an adjoint structure, which differs from the Dirac’s one, was performed [14]. Such analyse takes into account both charged and uncharged particles, in this vein, it is possible to fully understand the bilinear forms as it follows: \(\sigma \) still standing for the invariant length. The four-vector \(\mathbf J \) represents the electric current density for charged particles, whereas for neutral particles it may be understood as the effective electromagnetic current four-vector [15], the bilinear form \(\mathbf K \) shall be related with the spin-alignment due to a coupling with matter- or electromagnetic-field. And finally, the bi-vector \(\mathbf S \) is related to the electromagnetic momentum density for charged particles, for neutral particles case it may correspond to the spin-density momentum or may represent spin-precession (spin-oscillation) in the presence of a matter or a electromagnetic field [16, 17, 18, and references therein]. The interpretation of the bilinear \(\omega \) is less clear, however, when combined into the FPK identity (16), it can be interpreted as probability density for regular spinors [2, 19, 20, 21, 22], also some authors claim that such amount gives us clues of spinor behavior under *CPT* symmetry [2]. Nonetheless, for the general classification as developed in [14], the physical interpretation of the bilinear \(\omega \) is not clear.

## 3 Part 1: further investigations on the regular spinors

*m*stands for the mass of a particle, being a real number.

^{3}

The phases constraints to classify regular spinors

Single-helicity spinors | |||
---|---|---|---|

\(\alpha \) | \(\beta \) | Class | Constraints |

\(\mathbb {R}\) | \(\mathbb {R}\) | 2 | – |

\(\mathbb {C}\) | \(\mathbb {C}\) | 2 | \(\alpha =\beta \) |

\(\mathbb {C}\) | \(\mathbb {C}\) | 1 | – |

\(\mathbb {C}\) | \(\mathbb {R}\) | 1 | – |

\(\mathbb {C}\) | \(\mathrm{Im}\) | 1 | – |

\(\mathrm{Im}\) | \(\mathrm{Im}\) | 2 | – |

\(\mathrm{Im}\) | \(\mathbb {R}\) | 3 | – |

0 | \(\mathbb {R}\), \(\mathbb {C}\) or \(\mathrm{Im}\) | 6 | – |

\(\mathbb {R}\), \(\mathbb {C}\) or \(\mathrm{Im}\) | 0 | 6 | – |

where \(\mathbb {R}\) denotes any real number, \(\mathbb {C}\) stands for any c-number, \(\mathrm{Im}\) stands for any purely imaginary number and the symbol “–” means “No Constraint”. Imposing to (26) to satisfy the Dirac’s equation, automatically the link that arises between the phases is \(\alpha =\beta \), thus, belonging to class 2.

## 4 Part 2: further investigations on the singular spinors

The phases constraints to classify singular spinors

Dual-helicity spinors | |||
---|---|---|---|

\(\alpha \) | \(\beta \) | Class | Constraints |

\(\mathbb {R}\) | \(\mathbb {R}\) | 4 | \(\alpha \ne \beta \) |

\(\mathbb {R}\) | \(\mathbb {R}\) | 5 | \(\alpha =\beta \) |

\(\mathbb {C}\) | \(\mathbb {C}\) | 4 | \(|\alpha |^2\ne |\beta |^2\) |

\(\mathbb {C}\) | \(\mathbb {C}\) | 5 | \(|\alpha |^2=|\beta |^2\) |

\(\mathbb {C}\) | \(\mathbb {R}\) | 4 | \(|\alpha |^2\ne |\beta |^2\) |

\(\mathbb {C}\) | \(\mathbb {R}\) | 5 | \(|\alpha |^2=|\beta |^2\) |

\(\mathbb {C}\) | \(\mathrm{Im}\) | 4 | \(|\alpha |^2\ne |\beta |^2\) |

\(\mathbb {C}\) | \(\mathrm{Im}\) | 5 | \(|\alpha |^2=|\beta |^2\) |

\(\mathrm{Im}\) | \(\mathrm{Im}\) | 5 | \(|\alpha |^2=|\beta |^2\) |

\(\mathrm{Im}\) | \(\mathrm{Im}\) | 4 | \(|\alpha |^2\ne |\beta |^2\) |

\(\mathrm{Im}\) | \(\mathbb {R}\) | 4 | \(|\alpha |^2\ne |\beta |^2\) |

\(\mathrm{Im}\) | \(\mathbb {R}\) | 5 | \(|\alpha |^2=|\beta |^2\) |

0 | \(\mathbb {R}\), \(\mathbb {C}\) or \(\mathrm{Im}\) | 6 | – |

\(\mathbb {R}\), \(\mathbb {C}\) or \(\mathrm{Im}\) | 0 | 6 | – |

Thus, we turn explicit the constraints among the phases to define a specific class. Regarding to the class 4 and 5, we lay emphasis on their difference. The distinction among the aforementioned spinors is present on the value of the bilinear form \(\varvec{K}\). Such bilinear form, in general grounds, depends on the subtraction between the modulo of the phases, thus, if \(|\alpha |^2=|\beta |^2\) it leads to a spinor to belong to class 5. On the other hand, for a spinor carrying \(|\alpha |^2\ne |\beta |^2\), it provides \(\varvec{K}\ne 0\), leading, then, to belong to class 4, as it can easily been checked for the spinors in Refs. [4, 5, 29].

We highlight an important fact here, dual-helicity spinors, independent of the phases, are automatically singular spinors and can not never be classified as regular spinors.

## 5 Connecting the Lounesto’s regular and singular sectors

Constraints to map the regular and singular sector of the Lounesto’s classification

Single-helicity spinors | Dual-helicity spinors | |||||
---|---|---|---|---|---|---|

\(\alpha \) | \(\beta \) | Class | \(\alpha \) | \(\beta \) | Class | Constraints |

\(\mathbb {R}\) | \(\mathbb {R}\) | 2 | \(\mathbb {R}\) | \(\mathbb {R}\) | 4 (5) | \(\alpha \ne \beta \) (\(\alpha =\beta \)) |

\(\mathbb {C}\) | \(\mathbb {C}\) | 2 | \(\mathbb {C}\) | \(\mathbb {C}\) | 5 | \(\alpha =\beta \) |

\(\mathbb {C}\) | \(\mathbb {C}\) | 1 | \(\mathbb {C}\) | \(\mathbb {C}\) | 4 | \(\alpha \ne \beta \) |

\(\mathbb {C}\) | \(\mathbb {C}\) | 1 | \(\mathbb {C}\) | \(\mathbb {C}\) | 5 | \(|\alpha |^2=|\beta |^2\) |

\(\mathbb {C}\) | \(\mathbb {R}\) | 1 | \(\mathbb {C}\) | \(\mathbb {R}\) | 4 (5) | \(|\alpha |^2\ne |\beta |^2\) (\(|\alpha |^2=|\beta |^2\)) |

\(\mathbb {C}\) | \(\mathrm{Im}\) | 1 | \(\mathbb {C}\) | \(\mathrm{Im}\) | 4 (5) | \(|\alpha |^2\ne |\beta |^2\) (\(|\alpha |^2=|\beta |^2\)) |

\(\mathrm{Im}\) | \(\mathrm{Im}\) | 2 | \(\mathrm{Im}\) | \(\mathrm{Im}\) | 4 (5) | \(|\alpha |^2\ne |\beta |^2\) (\(|\alpha |^2=|\beta |^2\)) |

\(\mathrm{Im}\) | \(\mathbb {R}\) | 3 | \(\mathrm{Im}\) | \(\mathbb {R}\) | 4 (5) | \(|\alpha |^2\ne |\beta |^2\) (\(|\alpha |^2=|\beta |^2\)) |

*R*and

*S*stands for regular and singular sectors. Obviously, the inverse procedure indeed exists. Note that, however, to map a single-helicity spinor into a dual-helicity spinor, one may make use of only one of the two relations (35) and (36).

Now, taking advantage of (37), if one act on the spinor introduced in (26) easily obtain the same relations presented in (33), connecting the Lounesto’s regular sector with the singular sector. Based on the two previous tables, one is able to define the following (Table 3)

Note, however, the possibility to map classes 1, 2, 3 into classes 4 and 5. As we shall see, the protocol developed here is in agreement with the previous one performed in [30], where the authors map Dirac spinors into Elko spinors (and vice-versa), showing the possibility to connect both sectors of the Lounesto’s classification. Further investigations, but now concerning the dynamics of Dirac and Elko field are taken into account, were performed in [31]. Starting with a fundamental action, representing a mass dimension-transmuting operator between Dirac and Elko spinor fields were performed on the Dirac Lagrangian, in order to lead it into the Elko Lagrangian. Such a programme allow Elko spinors to be incorporated in the Standard Model. Such a possible connection between both sectors show - up to our knowledge - an useful tool to understand the underlying properties of the spinors encompassed into the Lounesto’s classification.

We purposely omitted, but all classes can be mapped into class 6.

## 6 Final remarks

In the present report we introduced a set of single-helicity and dual-helicity spinors on its algebraic form. After defining the eigenstates of the helicity operator, we explicitly defined the spinors in terms of spherical coordinates. Thus, with such spinors at hands, we developed a mechanism which turns possible to right ascertain to what class the spinors belongs within Lounesto’s classification, without an exhaustive computation of the bilinear forms. Such a mathematical device is based on the analyses of the arbitrary phases factors, being them real numbers, complex numbers or purely imaginary numbers, thus, we displayed in two tables all the possible combinations among the phases, one table for regular spinors and the other one for singular spinors. As highlighted above, we emphasize again that Lounesto’s classification show a strong dichotomy between the regular sector and the singular sector.

By a judicious inspection of the structure of the bilinear forms, one can show the impossibility – given by mathematical inconsistencies – to a given single-helicity spinor belong to the singular sector and we also evinced that dual-helicity spinors can not belong to the regular sector.

We shall finalize making an allusive comment to the results found in Sect. 5, as we can see, the phases factors play the central role in connecting both sectors of the Lounesto’s classification, through the study that was developed here, we can also say that a portion of the physical information carried by the spinors is encoded in the phases factor.

## Footnotes

- 1.
We reserve the right to omit the right-hand or left-hand component label, as commonly presented in the textbooks, due to the fact that this title comes from the way such components are transformed by Lorentz transformations.

- 2.
Interesting enough, by construction, the vector \(\varvec{J}\) is always non-null due to the fact that at least one of its components depends exclusively of the sum of the square modulo of the phases factor – regardless of whether the class is regular or singular. Nonetheless, the same can not be asserted for the axial vector \(\varvec{K}\).

- 3.

## Notes

### Acknowledgements

RJBR thanks CNPq Grant no. 155675/2018-4 for the financial support.

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