Some Remarks on dual helicity flag-dipole spinors

TL;DR

This paper introduces a mathematical mapping procedure that transforms single helicity spinors into dual helicity spinors, revealing a new class within Lounesto's classification and exploring their algebraic and physical properties.

Contribution

It presents a novel algebraic construction method for flag-dipole spinors and identifies their placement in Lounesto's classification, expanding understanding of spinor classes.

Findings

Identification of a new class of spinors within Lounesto classification

Development of a mapping procedure from single to dual helicity spinors

Exploration of algebraic and physical properties of flag-dipole spinors

Abstract

In this report we advance into a mapping procedure transmuting a single helicity spinor to a dual helicity spinor. Such a mathematical mechanism reveal us a class of spinor which fits into fourth class within Lounesto classification. The focus of the present manuscript is to expose the algebraic construction of flag-dipole spinors and briefly explore its underlying physical and mathematical contents.

Equations

$$P=m^{-1}{\gamma }_{\mu }{p}^{\mu },$$

$$\psi (\mathbf{p})=e^{i\kappa .\phi }\psi (0),$$

$$\psi(\boldsymbol{0})=\left(\begin{array}[]{c}\phi_{R}(\boldsymbol{0})\\ \phi_{L}(\boldsymbol{0})\end{array}\right).$$

$$\phi_{R}^{+}(\boldsymbol{0})=\phi_{L}^{+}(\boldsymbol{0})=\sqrt{m}\left(\begin{array}[]{c}\cos(\theta/2)e^{-i\phi/2}\\ \sin(\theta/2)e^{i\phi/2}\end{array}\right),$$

$$\phi_{R}^{-}(\boldsymbol{0})=\phi_{L}^{-}(\boldsymbol{0})=\sqrt{m}\left(\begin{array}[]{c}-\sin(\theta/2)e^{-i\phi/2}\\ \cos(\theta/2)e^{i\phi/2}\end{array}\right).$$

$$\vec{\sigma }\cdot \hat{p}{\varphi }^{\pm }(0)=\pm {\varphi }^{\pm }(0).$$

$$\displaystyle e^{i\kappa.\varphi}=\sqrt{\frac{E+m}{2m}}\left(\begin{array}[]{cc}\mathbbm{1}+\frac{\vec{\sigma}.\hat{p}}{E+m}&0\\ 0&\mathbbm{1}-\frac{\vec{\sigma}.\hat{p}}{E+m}\end{array}\right),$$

$$M_1{\psi }_{\{+,+\}}={\Lambda }_{\{+,-\}},$$

$$M_2{\psi }_{\{+,+\}}={\Lambda }_{\{-,+\}},$$

$$M_3{\psi }_{\{-,-\}}={\Lambda }_{\{+,-\}},$$

$$M_4{\psi }_{\{-,-\}}={\Lambda }_{\{-,+\}},$$

$$\vec{\sigma }\cdot \hat{p}{\mathcal{O}}_{(+-)}{\varphi }^{+}=-{\varphi }^{-},$$

$$\vec{\sigma }\cdot \hat{p}{\mathcal{O}}_{(-+)}{\varphi }^{-}=+{\varphi }^{+}.$$

$$M^{L}=\left(\begin{array}[]{cc}\mathbbm{1}&0\\ 0&\mathcal{O}\end{array}\right),$$

$$M^{R}=\left(\begin{array}[]{cc}\mathcal{O}&0\\ 0&\mathbbm{1}\end{array}\right),$$

$$M^{L}_{(+-)}=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&0&-e^{-i\phi}\\ 0&0&e^{i\phi}&0\end{array}\right),$$

$$M^{R}_{(+-)}=\left(\begin{array}[]{cccc}0&-e^{-i\phi}&0&0\\ e^{i\phi}&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right),$$

$$M^{L}_{(-+)}=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&0&e^{-i\phi}\\ 0&0&-e^{i\phi}&0\end{array}\right),$$

$$M^{R}_{(-+)}=\left(\begin{array}[]{cccc}0&e^{-i\phi}&0&0\\ -e^{i\phi}&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right),$$

$$\Lambda_{\{+,-\}}(\boldsymbol{0})=\sqrt{m}\left(\begin{array}[]{c}\alpha_{+}\cos(\theta/2)e^{-i\phi/2}\\ \alpha_{+}\sin(\theta/2)e^{i\phi/2}\\ -\beta_{+}\sin(\theta/2)e^{-i\phi/2}\\ \beta_{+}\cos(\theta/2)e^{i\phi/2}\\ \end{array}\right),$$

$$\Lambda_{\{-,+\}}(\boldsymbol{0})=\sqrt{m}\left(\begin{array}[]{c}-\alpha_{-}\sin(\theta/2)e^{-i\phi/2}\\ \alpha_{-}\cos(\theta/2)e^{i\phi/2}\\ \beta_{-}\cos(\theta/2)e^{-i\phi/2}\\ \beta_{-}\sin(\theta/2)e^{i\phi/2}\end{array}\right),$$

$${\Lambda }_{\{\pm ,\mp \}}(\mathbf{p})=e^{i\kappa .\phi }{\Lambda }_{\{\pm ,\mp \}}(0),$$

$$\Lambda_{\{+,-\}}(\boldsymbol{p})=\mathcal{B}_{+}\left(\begin{array}[]{c}-\alpha_{+}\Theta\phi_{L}^{-*}(\boldsymbol{0})\\ \beta_{+}\phi_{L}^{-}(\boldsymbol{0})\end{array}\right),$$

$$\Lambda_{\{-,+\}}(\boldsymbol{p})=\mathcal{B}_{-}\left(\begin{array}[]{c}\alpha_{-}\Theta\phi_{L}^{+*}(\boldsymbol{0})\\ \beta_{-}\phi_{L}^{+}(\boldsymbol{0})\end{array}\right),$$

$${\gamma }_{\mu }{p}^{\mu }\Lambda (\mathbf{p})\ne m\Lambda (\mathbf{p}),$$

$$(\square +m^2)\Lambda (\mathbf{p})=0,$$

Full text

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Flag-dipole; Type-4 spinors; Dual helicity.

I Introduction

Spinors may be defined in several different fashions. In the context of Clifford 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 beyondlounesto . Physically, a spinor represents the quantum wave function of a spin-$1/2$ particle roldaorevisiting , and as such the physical observables are obtained from real quadratic functional of these fields crawford1 . According to carmeli , spinors are used extensively in physics; it is widely, accepted that they are more fundamental than tensors (when the spacetime itself is represented by a manifold endowed with a Riemannian - or Lorenzian - metric structure) and the easy way to see this fact is the results obtained in general relativity theory by using spinors, results that could not have been obtained by using tensor methods only introducingspinors .

The Dirac equation is one of the most impressive successes in all of physics conceived from the purely theoretical reason to be a covariant first-order derivative field equation, it turned out to account for spin and matter/antimatter duality fabbrigeneral . It is known that the full Lorentz group is composed by rotation generators ($\vec{J}$), boost generators ($\vec{K}$) and discrete symmetries: parity ($P$) and time reversion ($T$) weinberg1 , we also have the charge conjugation symmetry ($C$), nevertheless, it does not stand for a Lorentz transformation.

Dirac spinors, which are responsible to describe the electron and its corresponding antiparticle the positron, are built within the full Lorentz group, keeping, usually, as the transformation law between the spinorial components, left-hand and right-hand components, the parity symmetry. In other words, this symmetry is exactly the link element between both parts of the representation space, i.e., it connects the subspaces $(j,0)$ and $(0,j)$ diracpauli .

In reference speranca , the author derive in a new fashion the parity operator ($P$), in the spin $1/2$ representation, thus, it reads

[TABLE]

which allows us to evoke the Dirac equation from the space-time symmetries alone. For Dirac spinors parity is used, and as an inexorable and direct consequence the Dirac dynamics is reached. This relation is literal: in acting on spinors, the parity operator is the Dirac operator and vice-versa. As a consequence, being the different sectors of the representation space related by means of another procedure (where parity plays no role), the Dirac dynamics is no longer expected elkograviton . Nevertheless, there are cases where the representation spaces are not connected by discrete symmetries, like in the Elko spinors case, proposed in reference jcap . In this case the transformation law between the spinorial still related by the Wigner Time-Reversal operator, $\Theta$, such observation was first made by Ramond in ramond , in such a way, that the above mentioned spinors do not belong to the Full Lorentz group any more, belonging, then, to the proper orthochronous Lorentz subgroup, denoted by $\mathcal{L}^{\uparrow}_{+}$.

According to the Lounesto classification there are six disjoint classes of spinors lounestolivro . The first three classes stand for regular spinors fields for spin $1/2$ fermions (covering three types of Dirac spinors) and the remaining three classes stand for singular spinors, encompassing type-4 (flag-dipole), Majorana (flag-pole) and Weyl spinors fields.

Up to the present moment, all the above mentioned spinor fields (and also the associated quantum field operators) are well established, except, then, for the type-4 spinor chengflagdipole . Type-4 spinors composes a set of very rare spinors in the literature. Their first appearance, in a very specific scenario, can be seen in esk and posteriorly roldaomeert , these kind of singular spinors work appropriately as a source of matter in ESK gravity, being the first explicit construction in Physics of flag-dipole spinors roldaonewspinor . The aforementioned spinor fields were shown to be solution of the Dirac field equation in a torsional framework in fabbrifr ; vignolofr . Although there is no quantum field operator constructed based on type-4 spinors, it is expected that it does not respect the full Lorentz symmetries. This spinors are potential candidates to describe dark matter, dark energy and to construct mass-dimension-one fermions, as remarked in cavalcanti4 . To the best of our knowledge, there is compelling evidence in astrophysics and cosmology that most of the mass of the Universe is composed of a new form of non baryonic dark matter, as it can be seen in Pereira2019 ; pereira2018evolution ; pereira2017lambda , there is a lack of evidence of the existence of new physics at LHC (Large Hadron Collider) and other particle physics experiments. On the theory side, many specific models with new particles and interactions beyond the standard model have been proposed to account for dark matter mirco .

The paper is organized as follows: Sect.II we briefly expose some basic conceptions about spinors transformation rules. Sect.III, we provide the main details of the programme of mapping single helicity spinors into dual helicity spinors, highlighting the main output that we reach. Then, in Sect.IV, we weave some comments about type-4 spinors behaviour under Dirac and also parity operator, we show that the introduced spinors are not ruled by the Dirac equation but rather the Klein-Gordon equation. Such results provides a hint towards mass dimensionality of the spinors at hand. Based on the Wigner classification, we also focus in establish some details about $C$, $P$ and $T$ discrete symmetries and also the observance of $(CPT)^{2}$. Finally, in Sect.V we conclude and present some outlooks.

II Foreword: Main concepts and setting up the notation

To obtain an explicit form of a given $\psi(\boldsymbol{p})$ spinor we call for the rest spinors, $\psi(\boldsymbol{0})$. For an arbitrary momentum, we have the following condition

[TABLE]

where the $\psi(\boldsymbol{0})$ is a direct sum of the $(1/2,0)$ and $(0,1/2)$ Weyl spinors spinor, defined as

[TABLE]

Thus, in spherically coordinate system the right-hand and left-hand components, in the rest-frame referential are defined as

[TABLE]

and

[TABLE]

the reader is cautioned to do not naively neglect the relation $\phi_{R}^{\pm}(\boldsymbol{0})=-\phi_{L}^{\pm}(\boldsymbol{0})$ when dealing antiparticles, as highlighted in nogo ; nondirac . An inspection of the helicity operator furnish

[TABLE]

Thus, the boost operator is defined as follows

[TABLE]

such programme allow one to write a boosted spinor in terms of a rest frame spinor, as we started our explanation in (2).

III Mapping the spinors: Introducing type-4 spinors

This section is reserved for a temptation on a mapping procedure between single helity spinors into dual helicity spinors. The path to be followed here, is based on defining a mapping matrix ($M$) responsible to transmute a single helicity spinor into a dual helicity one. To start this task, we employ the following relations

[TABLE]

where the single helicity spinors $\psi_{\{+,+\}}$ and $\psi_{\{-,-\}}$ are displayed in diracpauli ; nondirac , and the $M$ matrix commutes with the boost operators given in (9). Please, note that the relations (10) and (12) provide the same spinor $\Lambda$ (unless a global minus sign) and the same can be inferred to (11) and (13). The mapping task is accomplished imposing the following relations, first imposing to a component which carry positive eigenvalue transmute to a component which carry negative eigenvalue, given task is accomplished by defining the $\mathcal{O}$ operator

[TABLE]

finally, we impose the opposite

[TABLE]

Where the $\mathcal{O}$ operator stand for a $2\times 2$ matrix which changes the component’s helicity. Note that the lower index ($+,-$) and ($-,+$) indicates that the operator $\mathcal{O}$ takes a component which carry positive eigenvalue, to one which carry negative one and vice-versa. Such operator composes a more general operator $M$, which acts over the spinors, changing its component helicity. The $M$ matrix can be displayed as follows

[TABLE]

which keeps the right-hand helicity unchanged and acts only on the right-hand component, and we also may define

[TABLE]

where it acts only on the right-hand component, where $\mathbbm{1}$ stand for $2\times 2$ identity matrix. After a direct calculation, given matrix is explicitly given by

[TABLE]

we also may construct

[TABLE]

another possibility is given by

[TABLE]

and

[TABLE]

where the upper index $R$ and $L$ stand for the component (right-hand or left-hand) where $M$ acts, all the above $M$ matrix hold inverse, ensuring the possibility of an invertible mapping procedure. With the spinor given in nondirac at hand, we can explicit built the following set of dual helicity spinors in the rest frame referential

[TABLE]

and

[TABLE]

where the parameters $\alpha$ and $\beta$ are constant phases to be further determined. Note that the $(0,1/2)$ and $(1/2,0)$ representation spaces are now connected by the Wigner Time-Reversal operator, $\Theta$, similarly as Elko do aaca . Taking advantage of the last above feature and then taking a rest frame spinor to a momentum arbitrary referential, using the boost operator presented in (9), it follows

[TABLE]

then, such relation allow one to write111In order to make the notation clear, we have defined the Lorentz boost parameters as \mathcal{B}_{\pm}\equiv\sqrt{\frac{E+m}{2m}}\big{(}1\pm\frac{p}{E+m}\big{)}.

[TABLE]

and

[TABLE]

given spinors fits into fourth class within Lounesto classification lounestolivro if one constrain the parameters $|\alpha|^{2}\neq|\beta|^{2}$. In such a way, we have explicitly defined a type-4 spinor field. A quick inspection on the flag-dipole spinors relations, presented in Ref cavalcanticlassification , reveal us that is also possible to type-4 spinors carrying dual helicity feature, so, on this essay what we have is a matter standing for a particular case of the two presented in the above Reference. A parenthetic remark, for the Elko spinors the modulus of the phases are equal, leading then, such spinors to fit fifth class222The crucial difference between fourth and fifth Lounesto classes, reside in the axial-vector, the bilinear $\boldsymbol{K}=K_{\mu}\gamma^{\mu}$, standing non-null (null) for the first case (second case). within Lounesto classification whereareelko ; hoffdirac , thus, the only way to reconstruct Elko spinors from (25) and (26), is accomplished by a convenient choice of phases. Remarkably enough, Elko spinors have the following constraint of phases $\alpha=\pm i$ and $\beta=1$, holding the relation $|\alpha|^{2}=|\beta|^{2}$, thus, if a same akin reasoning is taking into account, type-4 spinors obey the charge-conjugation relation $C\Lambda=\pm\Lambda$.

IV Some comments related with dynamic and discrete symmetries

Regarding to the current literature about spinors, it is stated, and first stressed by Lounesto, that type-4 spinors can not satisfy the Dirac equation lounestolivro . Guided by the Lounesto statement, the first point that we should emphasize concerns to the connection between the representation spaces, for type-4 spinors we found that $(0,1/2)$ and $(1/2,0)$ representation spaces are connected by the Wigner time-reversal operator whereas for the Dirac case such connection occurs via parity relation ryder ; diracpauli . However, as expected for all physical fields, type-4 spinors obey the Klein-Gordon equation. As defined in speranca , we invoke the parity operator, $P=m^{-1}\gamma_{\mu}p^{\mu}$, looking towards obtain some information about the related dynamic, thus,

[TABLE]

such unambiguous calculation, presented in relation (27), hold for both spinors displayed in (25) and (26), i.e., the momentum space Dirac operator does not annihilate the introduced spinors (or any dual helicity spinor), without any phase constraint between $\alpha_{\pm}$ and $\beta_{\pm}$, one can not stablish any relation between the flag-dipole spinors presented above. Thus, such a direct observance provide a concrete conclusion about the Lounesto’s previous observations about type-4 spinors dynamic, as presented in his preprint lounestolivro . The aforementioned remark is a hint towards the canonical mass dimension related to the type-4 spinors, which allow us to assert that it is not $3/2$, as conventionally was expected for fermions. Acting again with the Dirac operator $\gamma_{\mu}p^{\mu}$, in (27), it yields

[TABLE]

holding the Klein-Gordon equation. Regarding to the discrete symmetries we have the following results $C^{2}=\mathbbm{1}$, $P^{2}=\mathbbm{1}$ and $T^{2}=-\mathbbm{1}$, as expected for the fermions which belong to the Standard Model Wigner1 , while the unconventional output comes from $(CPT)^{2}=+\mathbbm{1}$ rather than $(CPT)^{2}=-\mathbbm{1}$, nevertheless, corresponding to the third Wigner class of spin one-half particles Wigner1 .

It is worth to emphasize that up to now the introduced spinors are not eigenspinors of parity and neither charge conjugation operator. In this vein, we are not able to defined which spinor describe particle or antiparticle, due to the aforementioned reason, it precludes us to proceed with the computation of the spin sums and the quantum field operator. Guided by the dynamic that such spinors are ruled and also due to the dual helicity feature, in a speculative way, we may say that the associated quantum field operator may hold similarity with Elko spinor fields.

We restrict ourselves in the present essay addressing essentially the algebraic construction of the flag-dipole spinors, confining our attention to the rudimentary aspects of it, looking towards the accordance with what was stated in the previous literature. Perhaps, this the first time that it explicitly emerges in the literature. Regarding to phenomenological implications, once such spinors do not hold charge conjugacy feature, we may expect that it does not hold suppressed interaction with electromagnetic field, as example, however, at this very moment it is too early to advance into such assumption. Details concerning other subjects, e.g., couplings and interactions, cosmological implications, definition of an appropriated quantum field operator and etc, deserves our efforts to be soon developed.

V Final Remarks

The direct observance of type-4 (flag-dipole) fermions in physics is extremely rare. Besides the previous examples, cited in the scope of this manuscript, that arise in the literature as solutions of the derived equations of motion of specific frameworks, here we approach them by a different path; they are solutions of a given mathematical mapping procedure among single helicity and dual helicity spinors. All the solutions here developed are extremely important from a theoretical point of view, due to the fact that they evince that given spinor fields are not necessarily determined by their dynamics, however, a detailed discussion on their algebraic properties should be stressed.

In the present theoretical manuscript, we (unveiled) reached to an explicit set of dual helicity spinors, which representation spaces $(0,1/2)$ and $(1/2,0)$ are connected by the Wigner Time-Reversal operator rather than parity symmetry playing such role and it fits into the fourth class within Lounesto classification. Such result was possible by means a mathematical device which maps single helicity spinors into dual helicity spinors. To ensure the consistency of the results, we stated that under certain phases fixation, it becomes possible to recover the well-known Elko spinors, as a very particular case.

We also showed that type-4 spinors do not have an specific dynamic, and, as presented, the Dirac operator does not annihilate them. Thus, the only wave equation that type-4 spinors obey is the Klein-Gordon equation, as it is expected for all the physical fields. As a consequence, due to the last aforementioned features, it reveals us that the mass dimension of the field can not be $3/2$ as for the Dirac fermions, however, the correct path to ascertain the mass dimensionality makes necessary to built the quantum field operators, promoting the spinors to play the role of expansion coefficients of such physical amount.

We elucidate that it remains as open windows for further investigations, we look for define an adjoint spinor which provide a scalar, real and non-vanishing norm under Lorentz transformations, such a procedure, then, will allow us to look for an appropriate quantum field operator, a field propagator and to determine a positive definite Hamiltonian in terms of the creator and annihilator operators, yielding the Fermi-Dirac statistic for fermionic fields.

VI Acknowledgements

RJBR thanks CNPq Grant N*∘. 155675/2018-4 for the financial support and CHCV thanks CNPq PCI Grant N∘*. 300236/2019-0 for the financial support.