On the spinor representation
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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.
1 Introduction
The very definition of a spinor in dealing with physics may be treated as a matter of some importance in itself whatsoever. In fact, 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 particularly 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 \(SL(2,\mathbb {C})\) group and \(\mathbb {C}^4\), it is possible to understand several spinor properties by inspecting the multivector part constructed out specific \(SL(2,\mathbb {C})\) objects. These objects are nothing but the bilinear covariants associated to the regarded spinor [3, 4].
Following this reasoning, the usefulness of such a representation is not surprising, since the bilinear covariants are, at least in principle, related to a set of fermionic observables. Our aim in this paper is to delineate the importance of the representation space, by studying its properties, and then relating them to their physical consequences. As one will realize, the representation space is quite complicated due to the constraints coming out the Fierz–Pauli–Kofink identities. However, a systematic study of the space properties ends up being useful to relate different domains (subspaces) to the corresponding physics. Moreover, this study permits the study of fermions from a different and useful perspective.
This paper is organized as follows: in the next section the standard framework and the three equivalent definitions of spinors are revisited for the Minkowski spacetime, emphasizing the most relevant aspects concerning our purposes. Section 3 is devoted to our approach to the Lounesto spinor classification and related issues. In Sect. 4 we construct and study the spinor representation space and explore the topological and physical consequences. Section 5 is devoted to a derivation of the spacetime around a selfinteracting spinor field that satisfies the Dirac equation coupled to the Einstein equations with cosmological constant. We show that, for periodic values of the time variable, regular spinors are led into flagdipoles, dynamically implementing the algebraic mapping proposed in Sect. 6. In the final section we conclude.
2 The three equivalent definitions of spinors
Consider the Minkowski spacetime \((M\simeq \mathbb {R}^4, \eta _{\mu \nu })\) and its tangent bundle TM, where \(\eta \) denotes the Minkowski metric and Greek (spacetime) indices run from 0 to 3. Denoting sections of the exterior bundle by \(\sec \Omega (M)\), the spacetime Clifford algebra shall be denoted by \(\mathcal {C}\ell _{1,3}\). The set \(\{{e}_{\mu }\}\) represents sections of the frame bundle \(\mathbf {P}_{\mathrm {SO}_{1,3}^{e}}(M)\), whereas the set \(\{\gamma ^{\mu }\}\) can be further thought of as being the dual basis, \(\gamma ^{\mu }({e}_{\nu })=\delta ^\mu _{\;\nu }\). Classical spinors are objects of the space that carry the usual \(\tau =(1/2,0)\oplus (0,1/2)\) representations of the Lorentz group, which can be thought of as being sections of the vector bundle \(\mathbf {P}_{\mathrm {Spin}_{1,3}^{e}}(M)\times _{\tau }\mathbb {C}^{4}\) [5, 6].
Given a representation \(\rho : \mathbb {C}\otimes \mathcal {C}\ell _{1,3}\rightarrow \mathcal {M}(4,\mathbb {C})\), the adjoint of \(A\in \mathbb {C}\otimes \mathcal {C}\ell _{1,3}\), defined by \(A^\dagger = \rho ^{1}(\rho (A)^\dagger )\) (where \(\rho (A)^\dagger \) denotes the standard Hermitian conjugation in \(\mathcal {M}(4,\mathbb {C})\)), reads \(A^\dagger = e_0\tilde{A}^* e_0\), where \(\tilde{A}\) stands for the reversion of A and \((\,\cdot \,)^*\) denotes the complex conjugation. Besides, its trace is given by Tr\((\rho (\psi )) = 4\langle \psi \rangle _0\), where this notation is used to indicate the projection of a multivector onto its scalar part.
This correspondence provides an immediate identification between \(\psi \) and the classical Dirac spinor field. Having recovered the equivalence between these current spinor definitions, we pass to the building blocks of the spinorial representation space, namely the Fierz aggregate and the bilinear identities, after which we define the space itself, allowing for the connection of its points to a physical spinor, regardless the chosen classical definition.
3 Lounesto’s spinor classification, Pauli–Fierz–Kofink identities, and the Fierz aggregate
Lounesto’s spinor field classification
Class  \(\sigma \)  \(\omega \)  \(\mathbf{K}\)  \(\mathbf{S}\)  \(\mathbf{J}\) 

1  \(\ne 0\)  \(\ne 0\)  \(\ne 0\)  \(\ne 0\)  \(\ne 0\) 
2  \(\ne 0\)  0  \(\ne 0\)  \(\ne 0\)  \(\ne 0\) 
3  0  \(\ne 0\)  \(\ne 0\)  \(\ne 0\)  \(\ne 0\) 
4  0  0  \(\ne 0\)  \(\ne 0\)  \(\ne 0\) 
5  0  0  0  \(\ne 0\)  \(\ne 0\) 
6  0  0  \(\ne 0\)  0  \(\ne 0\) 
The three first classes are composed of regular spinors that comprise the standard textbook Dirac spinor. As stated in the literature, the representation spaces for the mentioned spinors are linked by the parity symmetry, however, quite recently regular spinors have been shown to be built without reference to this symmetry [17]. The elements of the fifth class are also called flagpole spinors, represented by particular cases as Majorana and Elko spinors, whereas the sixth class comprises Weyl spinors. The fourth class, the flagdipole, has had its first physical example discovered recently [18]. For later reference we stress that \(\mathbf {J}\) is always nonnull within this context. The Lounesto classification has been explored in a comprehensive range of contexts, comprising field theory [22, 23], cosmology [24], gravitation [25] and formal aspects as well [16, 26, 27]. The general form of the spinors in each one of the above classes was derived in Refs. [18, 28], and a classification that encodes gauge aspects was established in Ref. [29].
4 The representation space
Bearing in mind that a given spinor can be written as a section of the bundle \(\mathbf {P}_{\mathrm {Spin}_{1,3}^{e}}(M)\times _{\tau }\mathbb {C}^{4}\) we shall envisage the spinor space structure adopting a bottom–up, and somewhat pragmatic, approach by defining the regarded manifolds and spaces with respect to their points and elements. Notice that, as emphasized throughout Sect. 2, the understanding of spinors as sections of the aforementioned bundle are not strictly necessary, although highly convenient as we shall see.
In what follows let us denote by \(\mathring{N}\) the 5dimensional manifold whose points are in sec \(\Omega ^a(M)\), with \(a=0,\ldots ,4\). The space \(\mathring{N}\) is isomorphic to the exterior bundle \(\Omega (M)=\oplus _{a=0}^4\Omega (M)\). Let us denote by \(P=(p^0,p^1,p^2,p^3,p^4)\) an arbitrary point of the manifold \(\mathring{N}\), and the function Z that establishes such a canonical isomorphism \(\mathring{N}\overset{Z}{\simeq }\Omega (M)\). Obviously \(Z(P)\in \Omega (M)\).
Definition 1
\(\mathring{\Sigma }\) is the space whose elements are given by \(Z\upeta \), where \(\upeta \in \mathbb {C}^4\).
Notice that as long as Z is restricted to the bilinear covariants, namely, we impose the requirement that it acts only upon points of N satisfying the FPK identities, then the Fierz aggregate is straightforwardly obtained. Equivalently, however, more generally, we proceed with the following direct construction.
Definition 2
N is a submanifold of \(\mathring{N}\) whose points are such that Z(P) obeys the FPK identities.
Definition 3
The representation space \(\Sigma (N)\) is performed by elements given by \(Z(N)\eta \), where Z(N) stands for Z(P) with \(P\in N\) only. Therefore \(Z\upeta \cong \Psi \in \Sigma (N)\) and the elements of \(\Sigma \) are, thus, physical spinors.
It is worth emphasizing that since the bilinears are invariant with respect to Lorentz transformation, the elements of \(\Sigma (N)\) are relativistically covariant. Clearly \(\Sigma (N)\subset \mathring{\Sigma }\), i.e., the representation space is contained in the broader spinorial space. Therefore, the complement space \(\mathring{\Sigma }{\setminus }\Sigma (N)\) comprises points corresponding to spinors which do not obey the FPK identities, the socalled anomalous spinors.
The underlying idea to this construction regards the possibility to change from one physical spinor configuration to another one, by covering a given continuous path in the representation space. Differently from what happens to \(\mathring{N}\), however, the submanifold N must have a quite constrained topology inherited from the validity of the FPK identities.
It is worth to emphasize that the space \(\xi _\mathrm{reg}(N)\) has a rich underlying geometric structure. Indeed, it consists not merely a of submanifold, but furthermore it manifests an intriguing structure arising from the monopole construction of the Hopf fibration \(S^1\cdots S^3\rightarrow S^2\), where \(S^1\) is homeomorphic to the Lie gauge group U(1) of the electromagnetism [7]. Using a similar construction, the instanton is related to a principal bundle with structure Lie group SU(2), homeomorphic to the 3sphere \(S^3\). The instanton was described in Refs. [7, 8] using the Hopf fibration \(S^3\cdots S^7\rightarrow S^4\), in the context of the Witten monopole equations, by means of the bilinear covariants associated with regular spinor fields, under the Lounesto spinor field classification [16].
Let us make this point clear, working with a slightly different \(\xi _\mathrm{reg}(N)\) space, after which the general case shall be regarded. Regular spinor fields in either class 1 or class 2 in Lounesto’s classification can be thought of as satisfying \(\sigma =1\) without loss of generality, defining the manifold \(S^7\), when the Dirac spinor field is classically described by an element of \(\mathbb {C}^4\simeq \mathbb {H}^2\). Considering \(\mathcal {C}\ell _4\) to be the Clifford algebra of the 4dimensional Euclidean vector space \(\mathbb {R}^4\), i.e., the algebra generated by the set of vectors \(\lbrace \mathrm{e}_{\mu }\rbrace \), subject to the relations \( \mathrm{e}_{\mu }^2 = 1\), and \(\mathrm{e}_{\mu }\mathrm{e}_{\nu } +\mathrm{e}_{\nu }\mathrm{e}_{\mu }=0\), with \( \mu = 0,1,2,3\).
Proposition 1
Proof

\(\dim \xi _\mathrm{reg}(N)=3\), and the spinor belongs to class 1 of Lounesto’s classification. In this case the point \((0,0,0)\in \xi _\mathrm{reg}(N)\) cannot be attained, implying that \(\pi _2(\xi _\mathrm{reg}(N))=n\in \mathbb {Z}\);

\(\dim \xi _\mathrm{reg}(N)=2\), and thus the spinor belongs to either class 2 or class 3 of Lounesto’s classification. This is the case when \(\pi _1(\xi _\mathrm{reg}(N))=n\in \mathbb {Z}\).
Nevertheless, one shall not be so optimistic just by looking at the example just studied, since we were dealing only with a projection. The \(\Sigma \) space, where not only regular spinors are taken into account, is certainly very difficult to analyze. There are, however, some interesting points that we shall report on the study of \(\Sigma \) in its general form. In fact, Lounesto’s classification provides six classes of spinors, wherein a continuous path in the representation space allows access to different configuration states. Let us make this idea clearer and more precise.
All spinors in an arbitrary class are connected by a simple rescaling. From the point of view of elements in \(\Sigma (N)\), two different elements \(\Psi '\) and \(\Psi \) are connected by an usual transformation along the same class by \(\Psi '=S\Psi \). In this context, it is possible to assert, in a manner akin to Wigner [30, 31], the following proposition.
Proposition 2
Let \(D_{\lambda }\) be a 1parameter infinitesimal operator acting on the spinor space of a given class according to Lounesto’s classification. Suppose that it is an homomorphism, \(D_\lambda D_{\lambda '}=D_{\lambda +\lambda '}\), with \(\lambda \in \mathbb {R}\). If there exists a physical state on which the application of \(D_\lambda \) is well defined, then there exists a dense set of such states in the respective class, with respect to Lounesto’s classification.
Proof
If the application of \(D_\lambda \) is well defined for a given state, there exists the limit \( \lim _{\lambda \rightarrow 0} \lambda ^{1}(D_{\lambda }1)\Psi ,\) which implies that \(\lim _{\lambda \rightarrow 0} \lambda ^{1}(D_{\lambda }S^{1}SS^{1}S)\Psi \). Since the rescaling commutes with \(D_\lambda \), it follows that there exists the limit \(\lim _{\lambda \rightarrow 0} \lambda ^{1}(D_{\lambda }1)S\Psi .\) Hence, within an arbitrary but fixed class in Lounesto’s classification, it is possible to operate with infinitesimal operators in a rather usual way. Moreover, in view of the above result, physical spinors are indeed points of \(\Sigma (N)\). \(\square \)
It is important to remark that an arbitrary class in Lounesto’s classification is invariant under S. It is furthermore possible, however, to connect two different classes by algebraic transformation. More specifically, it was shown in Ref. [32] that there exists a subset of spinors in class 1, 2, and 3 which can be mapped into a subclass of class 5 spinors. Let us denote this transformation, between different classes, by \(S_{C}\). It turns out that \(\det S_C\ne 0\) [32]. Hence, the alluded algebraic bridge, in a manner of speaking, is also dense. In fact, as far as we restrict ourselves to the subset of states which can be mapped, the proof of Proposition 2 holds, in this switching class case.
A given algebraic bridge, however, is not always necessarily well behaved. In the sequel we give a (counter) example, presenting a mapping between a subset of spinors in class 1, 2, and 3, and a subset in class 4, which is neither Hermitian nor invertible. As one shall see, this example is quite severe as regards the constraints it imposes, but the possibility of a more manageable mapping is not discarded. It is worth to stressing that the mapping, from regular spinors to class 4 spinors, is chosen as a particular case. Class 4 spinors are understood as the most unvoiced class in Lounesto’s classification, having just a rare single example in the literature [18] as a physical solution of the Dirac equation in a Riemann–Cartan BianchiI, f(R), background. Lounesto describes such class as the only one that, at that time, had not corresponded to any type of spinor already found in Nature [3]. Except for such solution, neither other types of flagdipole spinors nor their respective dynamics as well have been found, yet. The algebraic mapping between regular and class 4 spinors can be parenthetically seen, then, as an attempt to put forward a bottom–up approach, embracing flagdipoles spinors into the standard setup of high energy physics.
It is significant to stress that the above mapping was performed using class 1 regular spinors, as it is clear from (22) and (23). Nevertheless, as far as we implement the additional constraints coming from class 2 and 3 Diraclike spinors, the final form of the bilinear invariants are slightly modified, but the net result is the same. While property one is useful in the study of possible information as regards the representation space (in the context of Proposition 2), the study of the hermiticity property may be relevant in a quantum mechanical context.
The counterexample just studied indicates an elaborated representation space, whose nontriviality deserves further exploration. It must be once again emphasized, however, that the constraints (24) are too restrictive, since they extend the kernel of the transformation to the whole \((\sigma , \omega )\)plane.
We would like to finalize this section by pointing out three new classes of spinors, beyond the Lounesto’s classification in Table 1, which also reside in the spinor representation space. These spinors were obtained in the operatorial and algebraic form in Ref. [33], having, by construction, \(\mathbf {J}=0\). Therefore, their dynamics cannot be described by the Dirac operator. The nontrivial topology of the representation space, as already remarked, is inherited from the constraints imposed by the FPK identities. For the sector of \(\Sigma (N)\) comprised by regular spinors, \(\mathbf {J}\) is the generator of cohomology and cannot vanish. The sector of \(\Sigma (N)\) encompassing singular spinors, nevertheless, may also accommodate the spinors of [33]. Notice that a vanishing \(\mathbf {J}\) does not lead to a contradiction, and the FPK identities still hold in this case. Hence, these spinors are also physical in the sense previously discussed. The spinors found in [33], assuming that \(\mathbf{J}=0\), may be called pole (only \(\mathbf {K}\ne 0\)), flag (only \(\mathbf {S}\ne 0\)), or flagpole.^{2} They live in a special subspace of \(\Sigma (N)\) whose topology also deserves further attention.
5 Passing through spinor classes: a natural dynamical interplay
Apart from the mentioned algebraic bridges, and the counterexample previously examined, we shall study a physical system whose dynamics provides an interesting interface to the abstract idea of a path changing classes in the representation space. This section is somewhat disconnected from the mathematical scope we intended to attribute to this formal paper, nevertheless the physical discussion demands such a change.
The current density of the spinor field, \(J^\mu =\bar{\psi }\gamma ^\mu \psi \) implies \(D_\mu J^\mu = 0\). Turning off the spinor selfinteraction hereon, for the sake of simplicity, the metric (34) with components (42) yields an approximate timelike Killing vector \(\xi =\partial _t\frac{1}{r}\partial _t y_1\partial _r\) at the spatial infinity [19]. Besides, the Ricci scalar, computed with respect to the metric components Eq. (42), reads \(R=\frac{2\mathrm{A}^2k\sin [2(\upomega t+\vartheta )]\sin \theta }{m r^2}\), resulting in a curvature that oscillates with a frequency of \(2\upomega \), whose sign varies with such a frequency. Moreover, given \(\epsilon _{\mu \nu \rho \sigma }\) the LeviCivita tensor multiplied by \(\sqrt{g}/2\), the dual of the exterior product between the Killing vector and the spinor intrinsic angular momentum, \(S^\mu \equiv {i}\epsilon ^{\mu \nu \sigma \rho } S_{\sigma \rho }\xi _\nu =0\), vanishes.
6 Concluding remarks
The formalization of a spinor representation space, whose points can be faced as physical spinors, has been constructed. These spinors have been shown to behave as elements of dense paths of the representations space which, in view of the FPK identities, perform highly topologically constrained subsets. Some of these subsets have topological properties intrinsically connected to physical relevant quantities. The representation space shows itself as an adequate tool to explore dynamics and interactions usually by means of using infinitesimal operators.
It should be emphasized that along this work we took advantage of dealing with spinors as elements of \(\mathcal {C}\ell _{1,3}\frac{1}{2}(1+{e}_0)\) in Sect. 2. Similar constraints in the representation space, coming from the FPK identities in the \(\mathcal {C}l_4\) isomorphic case, are expected. However, our main interest here is the study of the representation space taking into account the Clifford algebra constructed upon the Minkowski space.
In showing that type 4 spinors cannot be led into regular ones, we asserted that the mapping connecting different physical spinors—spinors of different sectors of \(\Sigma (N)\), belonging to different classes. However, no reference has been made to quantum mechanics. It is time to elaborate this a little further. The very possibility of crossing over different classes, by means of a welldefined algebraic transformation connecting different sectors of \(\Sigma (N)\) could, in principle, be related to some type of swapping spinor class due to a specific physical process. In fact, bearing in mind the existence of a dense set in between different classes in the light of proposition two, this switching could be performed by a specific (unknown) scattering matrix modeling the physical process. Apart from unitarity concerns,^{3} it is difficult to envisage how this proposed process can duplicate the helicity states in going from regular spinors to type 5 spinors, these last spinors with known dual property helicity. Perhaps, and here we are entering the fancy ground of speculation, a comprehensive transformation performed in the quantum operator as a whole may give rise to more precision to and enlighten the formal aspect of this possible swapping. It turns out, however, that the physical process would still be lacking. On the other hand, we have provided a mapping which is neither invertible nor Hermitian, evincing the high degree of topological constraint presented on the representation space. Further investigation on the algebraic/topological relationship concerning singular spinors are under current investigation.
From the physical point of view, changing spinor classes is essentially a change of physical observables. The dynamical setup as a procedure to change spinor classes was implemented for an axisymmetric spacetime that is generated by the selfgravity of the spinor here studied, satisfying the equations of motion derived from an Einstein–Hilbert action with cosmological constant coupled to a Dirac system with selfinteraction.
Footnotes
 1.
A simple, but tedious calculation shows that the other bilinears behave in such a way that the mapping (28) works well, ensuring a final class 4 spinor.
 2.
It is worth mentioning that these flagpoles are essentially different of the standard flagpoles characterized by the flag \(\mathbf {S}\ne 0\)) and the pole (\(\mathbf {J}\ne 0\)), since in this case \((\mathbf {K}= 0)\) [3].
 3.
Typically, the \(S_C\) matrix have enough symmetry to be recast into a specific form allowing for an unitary scattering process.
Notes
Acknowledgements
JMHS thanks to CNPq (304629/20154; 445385/20146) for partial financial support. CHCV thanks to PNPDCAPES for financial support. RJBR thanks to CAPES for financial support, and RdR is grateful to CNPq (Grant No. 303293/20152), and to FAPESP (Grant No. 2015/102700), for partial financial support.
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