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Eur. Phys. J. C (2017) 77:487 DOI 10.1140/epjc/s10052-017-5035-y

Regular Article - Theoretical

On the spinor representation

J. M. Hoff da Silva1,a, C. H. Coronado Villalobos1,2,b, Roldão da Rocha3,c, R. J. Bueno Rogerio1,d 1 Departamento de Física e Química, Universidade Estadual Paulista, Guaratinguetá, SP, Brazil 2 Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Gragoatá, Niterói, RJ 24210-346, Brazil 3 Centro de Matemática, Computação e Cognição, Universidade Federal do ABC-UFABC, Santo André 09210-580, Brazil

Received: 16 February 2017 / Accepted: 30 June 2017 / Published online: 21 July 2017 © The Author(s) 2017. This article is an open access publication

Abstract A systematic study of the spinor representation at least in principle, related to a set of fermionic observ- by means of the fermionic physical space is accomplished ables. Our aim in this paper is to delineate the importance and implemented. The spinor representation space is shown of the representation space, by studying its properties, and to be constrained by the Fierz–Pauli–Kofink identities among then relating them to their physical consequences. As one the spinor bilinear covariants. A robust geometric and topo- will realize, the representation space is quite complicated due logical structure can be manifested from the spinor space, to the constraints coming out the Fierz–Pauli–Kofink iden- wherein the first and second groups play promi- tities. However, a systematic study of the space properties nent roles on the underlying physical properties, associated ends up being useful to relate different domains (subspaces) to fermionic fields. The mapping that changes spinor fields to the corresponding physics. Moreover, this study permits classes is then exemplified, in an Einstein–Dirac system that the study of from a different and useful perspective. provides the generated by a . 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, empha- 1 Introduction sizing the most relevant aspects concerning our purposes. Section 3 is devoted to our approach to the Lounesto spinor The very definition of a spinor in dealing with physics may classification and related issues. In Sect. 4 we construct and be treated as a matter of some importance in itself whatso- study the spinor representation space and explore the topo- ever. In fact, from simple quaternionic compositions reveal- logical and physical consequences. Section 5 is devoted to a ing a definite [1] to the fermionic quantum inter- derivation of the spacetime around a self-interacting spinor nal structure [2], the spinorial approach reveals its richness. field that satisfies the coupled to the Einstein Among these possible systematizations concerning spinors, equations with cosmological constant. We show that, for peri- there is a particularly relevant one that encodes all the alge- odic values of the time variable, regular spinors are led into braically necessary information and the important relativistic flag-dipoles, dynamically implementing the algebraic map- construction as well, namely, the spinor represen- ping proposed in Sect. 6. In the final section we conclude. tation. When represented as a section of a bundle comprised by the SL(2, C) and C4, it is possible to understand several spinor properties by inspecting the multivector part 2 The three equivalent definitions of spinors constructed out specific SL(2, C) objects. These objects are nothing but the bilinear covariants associated to the regarded Consider the Minkowski spacetime (M  R4,ημν) and its spinor [3,4]. TM, where η denotes the Minkowski met- Following this reasoning, the usefulness of such a repre- ric and Greek (spacetime) indices run from 0 to 3. Denoting sentation is not surprising, since the bilinear covariants are, sections of the exterior bundle by sec (M), the spacetime shall be denoted by C1,3.Theset{eμ} rep- a e-mail: [email protected] resents sections of the frame bundle P e (M), whereas the SO1,3 b e-mails: [email protected]; [email protected] set {γ μ} can be further thought of as being the dual , μ μ c e-mail: [email protected] γ (eν) = δ ν. Classical spinors are objects of the space that d e-mail: [email protected] carry the usual τ = (1/2, 0) ⊕ (0, 1/2) representations of 123 487 Page 2 of 10 Eur. Phys. J. C (2017) 77 :487 the , which can be thought of as being sections Returning to Eq. (3), and using for instance the standard 4 of the P e (M) ×τ C [5,6]. representation, the complex associated to the spinor Spin1,3 The underlying idea that can join the three definitions operator in (3) reads of spinors relies on a quite straightforward root, and it   was inspired by the , whose elements sat- A −B ψ −ψ∗ [ ]= , A = 1 2 , isfy eμeν + eνeμ = 2ημν1. Indeed, any arbitrary element BA for ψ ψ∗ μ μν μντ 2 1 = s + s eμ + s eμeν + s eμeνeτ + pe e e e ∈  0 1 2 3 ψ −ψ∗ C , has a quaternionic representation. By denoting H B = 3 4 , 1 3 ψ ψ∗ (7) the quaternionic ring, a spinor representation of the Clif- 4 3 ford algebra C1,3  M(2, H) can be derived. A prim- = 1 ( + ) where itive idempotent f 2 1 e0 defines a minimal left C1,3 f , whose arbitrary element can be expressed as 23 13 12 [7,8] ψ1 = s + s i,ψ2 = s + s i,     ψ = + 10 ,ψ= 02 + 30 . ξ = s + s0 + s23 + s023 e e 3 p s i 4 s s i (8)    2 3   − s13 + s013 e e + s12 + s012 e e f   3 1  1 2 ψ 123 1 01 The standard was identified, e.g., in Ref. [3]as + p − s + s − s e2e3     an element of the minimal left ideal (C ⊗ C1,3)f associated 2 02 3 03 + s − s e3e1 + s − s e0e1e2e3 f, (1) to the complexified spacetime algebra (C⊗C1,3), generated by the primitive idempotent [3] constituting an algebraic spinor ξ ∈ C1,3 f . The set com- prised by the units i = e2e3, j = e3e1, and k = e1e2 settles ⎛ ⎞ a basis for the quaternionic algebra H. 1000 1 ⎜0000⎟ Representations of the {eμ} in M(2, H) read [9] f = (1 + e )(1 + ie e ) = ⎜ ⎟ , (9) 0 1 2 ⎝0000⎠     4   0 i   0 j 0000 e = , e = , 1 i 0 2 j 0         1 0 k 10 yielding ψ = (1 + iγ1γ2) ∈ (C ⊗ C1,3) f , with the e3 = , e0 = . (2) 2 k 0 0 −1 identification eμ → γμ. It yields the bijection between the algebraic spinor [5,6,12] Then the elements f and e e e e f have, respectively, the  0 1 2 3   10 00 ⎛ ⎞ representations [ f ]= and [e0e1e2e3 f ]= [10]. 00 10 ψ1 000 ∈ C+ ⎜ψ ⎟ Hence, an arbitrary element 1,3, in the even subalge- ⎜ 2 000⎟ ψ = ⎝ ⎠ ∈ (C ⊗ C1,3) f  M(4, C), (10) bra, corresponds to the so-called spinor operator [3] ψ3 000   ψ4 000 μν q1 −q2 = s + s eμeν + pe0e1e2e3  ∈ M(2, H), q2 q1 ψ = (ψ ,ψ ,ψ ,ψ ) ∈ C4 (3) and the classical one 1 2 3 4 . Given a representation ρ : C ⊗ C1,3 → M(4, C),the † −1 † where, according to Eq. (1), we have adjoint of A ∈ C ⊗ C1,3, defined by A = ρ (ρ(A) ) ρ( )† 23 31 12 (where A denotes the standard Hermitian conjugation q1 = s + s i + s j + s k, (4) † ˜∗ ˜ in M(4, C)), reads A = e0 A e0, where A stands for the reversion of A and ( · )∗ denotes the complex conjugation. 01 02 03 q2 =−p + s i + s j + s k. (5) Besides, its is given by Tr(ρ(ψ)) = 4 ψ 0, where this notation is used to indicate the projection of a multivector H2  C+  C4  C The isomorphisms 1,3 1,3 f onto its scalar part. constitute the landmark for the relationship among the spinor This correspondence provides an immediate identification operator, the algebraic, and the classical definitions of a between ψ and the classical Dirac spinor field. Having recov- spinor [6,11]. Hence, it is possible to alternatively  write the ered the equivalence between these current spinor definitions, q Dirac algebraic spinor field as an element 1 of the ring we pass to the building blocks of the spinorial representation q 2 space, namely the Fierz aggregate and the bilinear identi- H ⊕ H,as[7–9]  ties, after which we define the space itself, allowing for the q1 −q2 connection of its points to a physical spinor, regardless the [ f ]∈C1,3 f. (6) q2 q1 chosen classical definition. 123 Eur. Phys. J. C (2017) 77 :487 Page 3 of 10 487

1 2 1 1 3 Lounesto’s spinor classification, Pauli–Fierz–Kofink Z = σ Z, ZγμZ = JμZ, Zi[γμ,γν]Z = SμνZ, identities, and the Fierz aggregate 4 4 4 (15a) 4 1 1 Any spinor field ψ ∈ sec PSpine (M)×τ C can be employed 1,3 Ziγ0γ1γ2γ3γμZ = KμZ, − Zγ0γ1γ2γ3Z = ωZ. to construct its bilinear covariants as section of bundle (M), 4 4 reading [3,4,13] (15b)

σ = ψψ¯ ∈ 0(M), (11a) These conditions are satisfied also by regular spinors. Such relations are called more general Fierz–Pauli–Kofink identi- ω =−ψγ¯ γ γ γ ψ ∈ 4(M), (11b) 0 1 2 3 ties, however, they are written based on the Fierz aggregate. = (ψγ¯ ψ) γμ ∈ 1( ), J μ M (11c) Moreover, the inversion theorem (to be further studied in ¯ μ 1 K = i(ψγ γ γ γ γμψ) γ ∈  (M), (11d)  0 1 2 3  the next section) is inspired by this spinor representation. ¯ μ ν 2 S = ψ [γμ,γν] ψ γ ∧ γ ∈  (M). (11e) More significantly here, the aggregate plays a central role within the Lounesto classification since, in order to com- Equivalently, the components of the bilinear covariants are, plete the classification itself, Z has to be promoted to a respectively, denoted by boomerang, satisfying Z2 = 4σ Z. Obviously, for regular spinors the above condition is satisfied and Z is automati- = ψγ¯ ψ, Jμ μ (12a) cally a boomerang. However, for the singular spinor case it ¯ Kμ = iψγ0γ1γ2γ3γμψ, (12b) is not so direct. Indeed, for singular spinors we must envisage ¯ Sμν = ψ [γμ,γν] ψ. (12c) the underlying geometric structure to the multivector. From the geometric point of view, the following relations between The bilinear covariants in the Dirac theory are interpreted the bilinear covariants must be fulfilled, in order to ensure respectively as the mass term (or invariant length) in the that the aggregate is a boomerang: J field must be parallel to Lagrangian of the (σ), the pseudo-scalar (ω)rel- K and both are elements in the plane formed by the bivector evant for -coupling, the current of probability density field S. Hence, using the Eq. (14) and taking into account sin- (J), the chiral current density (K), and the probability density gular spinors, it is straightforward to see that the aggregate of the (intrinsic) electromagnetic moment (S)[3,4]. A promi- can be recast as [3] nent requirement for the Lounesto spinor classification is that Z = J(1 + is + ihγ γ γ γ ), (16) the bilinear covariants satisfy quadratic algebraic relations, 0 1 2 3 namely, the so-called Fierz–Pauli–Kofink (FPK) identities, where s is a space-like vector orthogonal to J, and h is a real which read scalar that is related to the spinor helicity. The multivector as expressed in Eq. (16) is a boomerang [16]. The condition 2 μ 2 2 J ≡ J Jμ = ω + σ , (13a) Z2 = 4σ Z yields Z2 = 0, for singular spinors have the prop- 2 μ 2 K ≡ K Kμ =−J , (13b) erty that, namely, the Fierz aggregate is . However, μ J · K ≡ J Kμ = 0, (13c) for the FPK identities to hold, the vector field J must be light-like (isotropic) and the multivector hγ γ γ γ +s must J ∧ K =−(ω + σγ γ γ γ )S. (13d) 0 1 2 3 0 1 2 3 be a pure imaginary [3,16]. † With these ingredients, it is possible to envisage six dif- When an arbitrary spinor ξ satisfies γ0ξ γ0ψ = 0, the orig- inal spinor ψ = 0 can be reconstructed, using the aggregate ferent classes of spinors, according to the classification in Table 1. Z = σ + J + iS + iKγ0γ1γ2γ3 + ωγ0γ1γ2γ3 (14) The three first classes are composed of regular spinors that comprise the standard textbook Dirac spinor. As stated in the by (a version) of the inversion theorem, ψ = √ 1 e−iθ ξ †γ ξ  2  0Z ξ, θ = (ξ †γ ψξ†γ ξ)−1/2 Z where i log 2 0 0Z is [4,14, Table 1 Lounesto’s spinor field classification 15]. Moreover, when σ, ω, J, S, K satisfy the Fierz iden- Class σωKS J tities, then the complex multivector operator Z is called a Fierz aggregate, which can be self-adjoint, being called 1 = 0 = 0 = 0 = 0 = 0 a “boomerang” in this case [3]. The regular spinors are 2 = 00 = 0 = 0 = 0 those whose at least one of the bilinear covariants σ and 30 = 0 = 0 = 0 = 0 ω do not vanish. On the other hand, singular spinors present 400 = 0 = 0 = 0 σ = = ω 0 , and, in this case, the Fierz identities, given in Eq. 5000 = 0 = 0 (13a), are in general replaced by the most general conditions 600 = 00 = 0 [4]: 123 487 Page 4 of 10 Eur. Phys. J. C (2017) 77 :487 literature, the representation spaces for the mentioned spinors Definition 3 The representation space (N) is performed are linked by the parity symmetry, however, quite recently by elements given by Z(N)η, where Z(N) stands for Z(P) regular spinors have been shown to be built without refer- with P ∈ N only. Therefore Zη =∼ ∈ (N) and the ence to this symmetry [17]. The elements of the fifth class elements of  are, thus, physical spinors. are also called flag-pole spinors, represented by particular cases as Majorana and Elko spinors, whereas the sixth class It is worth emphasizing that since the bilinears are invari- comprises Weyl spinors. The fourth class, the flag-dipole, ant with respect to , the elements of has had its first physical example discovered recently [18]. (N) are relativistically covariant. Clearly (N) ⊂ ˚ , i.e., For later reference we stress that J is always non-null within the representation space is contained in the broader spinorial this context. The Lounesto classification has been explored in space. Therefore, the complement space ˚ \(N) comprises a comprehensive range of contexts, comprising field theory points corresponding to spinors which do not obey the FPK [22,23], cosmology [24], [25] and formal aspects identities, the so-called anomalous spinors. as well [16,26,27]. The general form of the spinors in each The underlying idea to this construction regards the pos- one of the above classes was derived in Refs. [18,28], and a sibility to change from one physical spinor configuration to classification that encodes gauge aspects was established in another one, by covering a given continuous path in the rep- Ref. [29]. resentation space. Differently from what happens to N˚ ,how- ever, the submanifold N must have a quite constrained topol- ogy inherited from the validity of the FPK identities. 4 The representation space Let us make this point clearer by considering merely reg- ular spinors for a moment. In this case, it is possible to attain Bearing in mind that a given spinor can be written as a the appropriate subspace of (N) by defining the following 4 section of the bundle P e (M) ×τ C we shall envisage ξ Spin1,3 canonical projector reg: the spinor space structure adopting a bottom–up, and some- what pragmatic, approach by defining the regarded ξreg : N → (M) and spaces with respect to their points and elements. Notice (σ, J, K, S,ω) → (σ, J, 0, 0,ω), that, as emphasized throughout Sect. 2, the understanding 0 1 4 of spinors as sections of the aforementioned bundle are not with image  (M) ⊕  (M) ⊕  (M) = ξreg(N). Within strictly necessary, although highly convenient as we shall see. the space ξreg(N) ⊂ , taking into account the identity In what follows let us denote by N˚ the 5-dimensional man- J2 = σ 2 + ω2, which holds for regular spinors, it is always ifold whose points are in sec a(M), with a = 0,...,4. possible to associate a topological invariant for every regular The space N˚ is isomorphic to the exterior bundle (M) = state. Moreover, taking into account the usual mass dimen- ⊕4 ( ) = ( 0, 1, 2, 3, 4) / a=0 M . Let us denote by P p p p p p an sion 3 2 fermion, for which J represents the conserved cur- arbitrary point of the N˚ , and the function Z that rent, the regarded topological invariant must be related to the Z electric . Before proceeding, let us make two paren- establishes such a canonical isomorphism N˚  (M).Obvi- thetical remarks. First, it is straightforward to realize that ξ ously Z(P) ∈ (M). reg may be naturally adapted for a lower projection Definition 1 ˚ is the space whose elements are given by leading to elements as either (σ, J, 0, 0, 0) or (0, J, 0, 0,ω). Zη, where η ∈ C4. Second, when we refer to the mass dimension of a given Notice that as long as Z is restricted to the bilinear covari- spinor, we mean the canonical mass dimension which shall be ants, namely, we impose the requirement that it acts only inherited by the quantum field from the dynamics respected upon points of N satisfying the FPK identities, then the Fierz by the expansion coefficients. Particular cases of the expan- aggregate is straightforwardly obtained. Equivalently, how- sion coefficients are the objects treated here. ξ ( ) ever, more generally, we proceed with the following direct It is worth to emphasize that the space reg N has a rich construction. underlying geometric structure. Indeed, it consists not merely a of submanifold, but furthermore it manifests an intriguing ˚ Definition 2 N is a submanifold of N whose points are such structure arising from the monopole construction of the Hopf ( ) that Z P obeys the FPK identities. fibration S1 ···S3 → S2, where S1 is homeomorphic to the When acting upon elements of M it is convenient to write Lie gauge group U(1) of the [7]. Using a Z(P) as similar construction, the instanton is related to a principal bundle with structure SU(2), homeomorphic to Z(P) = σ + J + K + S + ω, the 3-sphere S3. The instanton was described in Refs. [7,8] making explicit the multivector structure in terms of the bilin- using the Hopf fibration S3 ···S7 → S4, in the context of ear covariants, just as to express P = (σ, J, K, S,ω). the Witten monopole equations, by means of the bilinear 123 Eur. Phys. J. C (2017) 77 :487 Page 5 of 10 487

th covariants associated with regular spinor fields, under the where πi (R) stands for the i -homotopy group of a given R Lounesto spinor field classification [16]. space. In the case of a mass dimension 3/2 fermion described Let us make this point clear, working with a slightly dif- by the regular spinor, the topological invariant is associated ferent ξreg(N) space, after which the general case shall be to the electric charge. regarded. Regular spinor fields in either class 1 or class 2 in Lounesto’s classification can be thought of as satisfying Proof The first assertion directly follows from the above dis- ξ ( ) σ = 1 without loss of generality, defining the manifold S7, cussion. In fact, a regular spinor is an element of reg N and when the Dirac spinor field is classically described by an therefore there exist just two possibilities: 4 2 element of C  H . Considering C4 to be the Clifford • ξ ( ) = algebra of the 4-dimensional space R4, i.e., dim reg N 3, and the spinor belongs to class 1 of ( , , ) ∈ the algebra generated by the set of vectors {eμ}, subject to the Lounesto’s classification. In this case the point 0 0 0 2 ξ ( ) π (ξ ( )) = relations eμ = 1, and eμeν + eνeμ = 0, with μ = 0, 1, 2, 3. reg N cannot be attained, implying that 2 reg N The quaternionic representation can be constructed by n ∈ Z; • ξ (N) = using Eq. (2), by ei →[ei ][e0]. One defines the observ- dim reg 2, and thus the spinor belongs to either ables of ψ ∈ C4 in a 4-dimensional by the class 2 or class 3 of Lounesto’s classification. This is the π (ξ ( )) = ∈ Z following expressions: case when 1 reg N n . σ = ψψ,¯ ω = ψ¯ ψ, = ψ¯ ψ, e5 Jμ eμ In general, J is not necessary related to the conserved current ¯ ¯ Kμ = iψe5eμψ, Sμν = ψ[eμ, eν]ψ, (17) associated to a given fermion. Nevertheless, when it does— which is the case for mass dimension 3/2 fermions—then where e = e e e e . Note that the only different relations 5 0 1 2 3 the conserved charge is the electric charge itself. Since a due to the case is the expression for ω (see vanishing charge is forbidden in these situations, one has Eq. (11a)). In fact, in the C , algebra one has e2 =−1, 1 3 5 the physical counterpart of the above topological constraint. whereas considering C yields e2 = 1. In this way, the 4 5 Indeed the (0, 0, 0) point for regular spinors can never be Fierz identities must be modified [7,15], yielding reached.  J2 = σ 2 − ω2, J2 = K2, μ As readily verified, in the case of the 3-dimensional J ∧ K = (σ − e5ω)S, J Kμ = J · K = 0. (18) ξreg(N) space the conserved current may also be seen as the 1 1 2 In this context, the first Fierz identity (18) provides the generator of cohomology, for H (ξreg(N))  H (R \{0}), expression J2 + ω2 = 1 defining the 4-sphere S4, with coor- and the usual closed form dinates (Jμ,ω). σ dω − ωdσ , (21) Now, taking into account Eqs. (6) and (11a) yields [7,8] σ 2 + ω2 σ = · + · ,ω= ( ∗ ), q1 q1 q2 q2 2 q1 q2 is not everywhere exact. The usefulness of the representation jk ∗ space construction is now evident. By treating physical spinor J0 = q1 · q1 −q2 · q2, Ji = 2  (q e j ek q2), (19) i 1 (states) as points of a given space, constrained by the alge- , , = , ,  jk for i j k 1 2 3 and i is the Levi-Civita symbol. Hence, braic bilinear relations, it is possible to work in the interplay the representation in Eq. (17) reads [7,8] of , multivector algebra, and physics. As a matter of σ =|ψ | 2 +|ψ |2 +|ψ |2 +|ψ |2, fact, the very existence of a relativistic spinor is related to a 1 2 3 4 topological invariant in the representation space. ω = (ψ ψ∗ + ψ ψ∗), 2 1 3 2 4 Nevertheless, one shall not be so optimistic just by looking 2 2 2 2 J0 =|ψ1 | +|ψ2 | −|ψ3 | −|ψ4 | , at the example just studied, since we were dealing only with = (ψ ψ∗ + ψ ψ∗), a projection. The  space, where not only regular spinors are J1 2 1 4 2 3 ∗ ∗ taken into account, is certainly very difficult to analyze. There J = 2(ψ ψ − ψ ψ ), 2 2 3 1 4 are, however, some interesting points that we shall report on = (ψ ψ∗ + ψ ψ∗). J3 2 3 1 2 4 (20) the study of  in its general form. In fact, Lounesto’s classifi- This important geometric feature reveals that the underly- cation provides six classes of spinors, wherein a continuous ing geometry induced by spinors classes can further point path in the representation space allows access to different to more structures. Indeed, the following proposition regards configuration states. Let us make this idea clearer and more the topological invariants associated to regular spinors. precise. All spinors in an arbitrary class are connected by a simple ξ ( ) Proposition 1 Let reg N be the space consisting of regular rescaling. From the point of view of elements in (N),two spinors. Then different elements  and are connected by an usual trans- π (ξ ( )) = ∈ Z,  = [dim(ξreg(N))−1] reg N n formation along the same class by S . In this context, 123 487 Page 6 of 10 Eur. Phys. J. C (2017) 77 :487  it is possible to assert, in a manner akin to Wigner [30,31], M11 M12 We start by introducing a matrix M = ∈ the following proposition. M21 M22 M(4, C), defining the mapping: Proposition 2 Let Dλ be a 1-parameter infinitesimal oper- : D → T ator acting on the spinor space of a given class according M 4 to Lounesto’s classification. Suppose that it is an homomor- D → 4 = MD, (22) phism, Dλ Dλ = Dλ+λ , with λ ∈ R. If there exists a physical where D and T4 stands for the sets comprising regular and state on which the application of Dλ is well defined, then there flag-dipole spinors, respectively. The formalism is clearly exists a dense set of such states in the respective class, with representation-independent,  however, the Weyl representa- respect to Lounesto’s classification. OI O σk tion γ0 = , γk = shall be used, σk being IO −σk O Proof If the application of Dλ is well defined for a given −1 the usual , to fix the notation hereby. state, there exists the limit limλ→0 λ (Dλ − 1) , which − − − According to the Lounesto spinor classification, using the implies that limλ→ λ 1(Dλ S 1 S − S 1 S) . Since the 0 mapping defined in (22), class 4 spinors satisfy rescaling commutes with Dλ, it follows that there exists the λ−1( − ) . σ = † †γ  = ,ω=−† †γ  = . limit limλ→0 Dλ 1 S Hence, within an arbitrary D M 0 M D 0 D M 123 M D 0 but fixed class in Lounesto’s classification, it is possible to (23) operate with infinitesimal operators in a rather usual way. Moreover, in view of the above result, physical spinors are Let us investigate here the constraints exclusively on the M indeed points of (N).  matrix. The conditions (23)imply M†γ M = , M†γ M = , It is important to remark that an arbitrary class in 0 0 123 0 (24) Lounesto’s classification is invariant under S. It is further- which, in the Weyl representation, yield more possible, however, to connect two different classes by ± M† M + M† M = O =±M† M + M† M , algebraic transformation. More specifically, it was shown in 11 21 21 11 11 22 21 12 Ref. [32] that there exists a subset of spinors in class 1, 2, (25) and 3 which can be mapped into a subclass of class 5 spinors. ±M† M + M† M = O =±M† M + M† M . 12 21 22 11 12 22 22 12 Let us denote this transformation, between different classes, (26) by SC . It turns out that det SC = 0[32]. Hence, the alluded algebraic bridge, in a manner of speaking, is also dense. In The system (25)–(26) is satisfied when fact, as far as we restrict ourselves to the subset of states † † † † M M21 = M M22 = M M21 = M M22 = O. (27) which can be mapped, the proof of Proposition 2 holds, in 11 11 12 12 this switching class case. Therefore, by considering a general representation,   A given algebraic bridge, however, is not always necessar- M = m11 m12 , M = m13 m14 , ily well behaved. In the sequel we give a (counter-) example, 11 12 m21 m22 m23 m24 presenting a mapping between a subset of spinors in class   m31 m32 m33 m34 1, 2, and 3, and a subset in class 4, which is neither Her- M21 = , M22 = , mitian nor invertible. As one shall see, this example is quite m41 m42 m43 m44 severe as regards the constraints it imposes, but the possi- it is possible to rewrite bility of a more manageable mapping is not discarded. It is ⎛ ⎞ m11 m12 m13 m14 worth to stressing that the mapping, from regular spinors to ⎜ m11m22 m13m22 m14m22 ⎟ ⎜ m m22 m m ⎟ class 4 spinors, is chosen as a particular case. Class 4 spinors = ∗12 ∗ ∗12 ∗12 . M ⎜ −m m41 −m m42 −m m43 −m m44 ⎟ (28) ⎝ 22∗ 22∗ 22∗ 22∗ ⎠ are understood as the most unvoiced class in Lounesto’s m12 m12 m12 m12 classification, having just a rare single example in the lit- m41 m42 m43 m44 erature [18] as a physical solution of the Dirac equation in The M matrix is responsible for performing the mapping a Riemann–Cartan Bianchi-I, f (R), background. Lounesto between regular and class 4 spinors.1 There are, however, describes such class as the only one that, at that time, had not important restrictions on such a mapping which must be high- corresponded to any type of spinor already found in Nature lighted. [3]. Except for such solution, neither other types of flag- Firstly, the mapping performed cannot occur from class dipole spinors nor their respective dynamics as well have 4 to regular spinors. In fact, supposing the existence of M˚ been found, yet. The algebraic mapping between regular and class 4 spinors can be parenthetically seen, then, as an attempt 1 A simple, but tedious calculation shows that the other bilinears behave to put forward a bottom–up approach, embracing flag-dipoles in such a way that the mapping (28) works well, ensuring a final class spinors into the standard setup of high energy physics. 4 spinor. 123 Eur. Phys. J. C (2017) 77 :487 Page 7 of 10 487 such that M˚ 4 = D, then M M˚ 4 = M 4, leading to 5 Passing through spinor classes: a natural dynamical M˚ = M−1. Nevertheless, as it can be explicitly calculated interplay from (28), det M = 0 and there is no such a M˚ matrix. Secondly, the mapping (28) cannot be Hermitian. Indeed, Apart from the mentioned algebraic bridges, and the counter- the requirement M = M† yields example previously examined, we shall study a physical sys- tem whose dynamics provides an interesting interface to the = ∗ , = ∗ , = ∗ , = ∗ , abstract idea of a path changing classes in the representation m11 m11 m12 m12 m22 m22 m14 m41 − − space. This section is somewhat disconnected from the math- = m22m14 =− ∗ , = m12m43 . m13 m42 m44 (29) ematical scope we intended to attribute to this formal paper, m12 m22 nevertheless the physical discussion demands such a change. Nevertheless, the other bilinear invariants do not behave as Coupling a self-interacting to gravity is the ones for class 4 spinors. In fact, it can be verified that ruled by an action related to the Einstein–Dirac system, which reads K = 0 = S for this case, rendering a spinor different from a   √ class 4 one and then the hermiticity is forbidden. = 4 − 2( − ) + i (( ψ)γ¯ μ S d x g Mp 2R Dμ It is significant to stress that the above mapping was per- 2  λ formed using class 1 regular spinors, as it is clear from (22) ¯ μ ¯ ¯ 2 − ψγ Dμ)ψ − mψψ + (ψψ) , (30) and (23). Nevertheless, as far as we implement the additional 2 constraints coming from class 2 and 3 Dirac-like spinors, the where  denotes the cosmological constant, m represents the final form of the bilinear invariants are slightly modified, but mass of the spinor field and the Planck mass M−2 = 8πG the net result is the same. While property one is useful in the p shall be 2, setting 16πG = 1; λ denotes the fermion (effec- study of possible information as regards the representation tive) self-interaction coupling constant. In fact such a term space (in the context of Proposition 2), the study of the her- is not perturbatively renormalizable, and therefore it must miticity property may be relevant in a quantum mechanical be understood as an effective coupling, being suppressed by context. powers of a fundamental scale. Besides, the covariant deriva- The counter-example just studied indicates an elaborated tive read [18,19] representation space, whose non-triviality deserves further  exploration. It must be once again emphasized, however, that i ρ ν ab Dμψ = ∂μ + (μνeaρ − ∂μeaν)e γ ψ, (31) the constraints (24) are too restrictive, since they extend the 2 b of the transformation to the whole (σ, ω)-plane. ρ where the μν are the and the tetrad eaμ We would like to finalize this section by pointing out μ μ satisfy the relations gμν = ηabe μe ν and γ = γ ae .The three new classes of spinors, beyond the Lounesto’s clas- a b a equations of motion that can be derived from the action (30) sification in Table 1, which also reside in the spinor repre- are sentation space. These spinors were obtained in the operato- μ ¯ γ Dμψ + mψ = λψψ, (32) rial and algebraic form in Ref. [33], having, by construction,   = ¯ ¯ J 0. Therefore, their dynamics cannot be described by the −4i(2Rμν − gμν) = ψγ(μ Dν)ψ − D(μψγν)ψ . The nontrivial topology of the representation  ¯ 1 ¯ 2 space, as already remarked, is inherited from the constraints + gμν mψψ − λ(ψψ) . (33) 2 imposed by the FPK identities. For the sector of (N) com- prised by regular spinors, J is the generator of cohomology Using spherical coordinates, axisymmetric M and cannot vanish. The sector of (N) encompassing singu- can have an ansatz of type [20] lar spinors, nevertheless, may also accommodate the spinors ds2 = (g˚2 − )dt2 + g2dr 2 of [33]. Notice that a vanishing J does not lead to a contra- z t r − ( + ) φ + ( 2 − 2 2) φ2 + 2 θ 2. diction, and the FPK identities still hold in this case. Hence, 2 gt gz gpg˚z dtd gp gt gz d gyd these spinors are also physical in the sense previously dis- (34) = cussed. The spinors found in [33], assuming that J 0, may Besides, closed time-like curves are precluded when the con- = = 2 be called pole (only K 0), flag (only S 0), or flag-pole. dition (g2 − g2g2)>0 is imposed. The vierbeins can be ( ) p t z They live in a special subspace of N whose topology also expressed as deserves further attention. 0 1 e = gt (dt + gzdφ), e = gr dr, 2 3 e = gydθ, e = gpdφ − g˚zdt. (35) 2   It is worth mentioning that these flag-poles are essentially different ψ = ψ ( ), ψ ( ), ψ ( ), ψ ( )  of the standard flag-poles characterized by the flag S = 0) and the pole The spinor field 1 x 2 x 3 x 4 x , where (J = 0), since in this case (K = 0) [3]. ψi = ψa1 + iψa2 : M → C, can be used to solve (32) 123 487 Page 8 of 10 Eur. Phys. J. C (2017) 77 :487 and (33), for asymptotically flat spacetimes. The ansätzes A6(4k2 + ω2) + λ sin[2(ωt − ϑ)] sin 3θ, are Maclaurin series in the variable r − j , for the metric com- 64k4ωm3r 3  ponents, given by A2 g = ω ( kr) 2 θ z 2 2 2 6 cos 2 sin ∞ ∞ ∞ 8k m r  t j z j r j g = 1 + , g = , g = 1 + , + k sin(2kr) sin[2(ωt − ϑ)] sin 2θ t r j z r j r r j j=1 j=1 j=1 A2ω sin(2kr) sin2 θ (36) − , (42) ⎛ ⎞ 2km2r ∞ ∞ y j ⎝ p j ⎠ = θ gy = r + , gp = sin θ r + , (37) and gp sin gy. This metric can be led to the case where r j r j −3 j=0 j=0 there is no self-interaction, considering terms to order r , ∞ as in Ref. [19]. Hence, the spinor field reads aij + ibij ψ = , for i = 1,...,4. (38)  i r j A+ j=1 ψ = A+ 1 − cos(ωt + kr + ϑ) 1a r  The involved functions are assumed to be functions ofr,θand t and are expanded with respect to r,uptoorderO(r −4), con- + A˚ +(1 + λA˚ +) sin(ωt − kr + ϑ) , sidering one more order than Ref. [19] and the self-interaction  A− in Eq. (30). Such expansions in Eqs. (36)–(38) ensure that ψ = A− 1 − cos(ωt − kr + ϑ) 1b r the limit yields an asymptotically flat metric.  Given (37) and (38), the leading contribution to (32)isof + ( − λ ) (ω + + ϑ) , O( 1 ) ± = A˚ − 1 A˚ − sin t kr the order r . Assuming light-cone like coordinates r r ± ct, the equations are given by  A+ ψ a = A+ 1 − cos(ωt − kr + ϑ) − ∂ = , + ∂ = , 2 r mb31 −a11 0 ma31 −b11 0  mb + ∂+a = 0, ma − ∂+b = 0, (39) 41 21 41 21 + A˚ +(1 + λA˚ +) sin(ωt + kr + ϑ) , + ∂ = , − ∂ = , mb11 +a31 0 ma11 +b31 0  − ∂− = , + ∂− = . A− mb21 a41 0 ma21 b41 0 (40) ψ = A− 1 − cos(ωt + kr + ϑ) 2b r These equations can be completely solved by    + A˚ −(1 − λA˚ −) sin(ωt − kr + ϑ) , a = a (θ) cos(kr) + a (θ) sin(kr) cos(ωt) i1 i1a i1b   + a (θ) cos(kr) + a (θ) sin(kr) sin(ωt), (41) A+ i1c i1d ψ =−A+ 1 − sin(ωt − kr + ϑ) 3a r 2 2 2  where ω = k + m .Thebi1 coefficients in Eq. (38)are solved in terms of ai1. The leading contribution to (33)is + A˚ +(1 + λA˚ +) cos(ωt + kr + ϑ) , of the order O(r). The equations are wave equations for y0  and p0, whose solutions read functions that can be generally A− ψ3b = A− 1 − sin(ωt − kr + ϑ) expressed as the wave c = c (θ)+c (θ, r −t)+c (θ, r + r 0 0 01 02  t), for the two first coefficients y0 and p0 in the series (37) [19]. The metric coefficients can be solved as + A˚ −(1 − λA˚ −) cos(ωt − kr + ϑ) ,  A2ω cos θ A+ gt = 1 − , ψ =−A+ 1 − sin(ωt − kr + ϑ) 4kmr 4a r  A2ω3 cos θ A2k sin[2(ωt − ϑ)] sin θ gr = 1 − + + ˚ ( + λ ˚ ) (ω − + ϑ) , 4k3mr 4ω2mr2 A+ 1 A+ cos t kr A2k2 sin(3ωt − 3ϑ)sin 2θ  − λ , A− 16ω4m2r 3b ψ4b = A− 1 − sin(ωt − kr + ϑ) r A2  g = r + ω( k2 + ω2) θ y 3 2 cos 4k m + A˚ −(1 − λA˚ −) cos(ωt + kr + ϑ) , (43) A2k − [ (ωt − ϑ)] θ √ 2 sin 2 sin 4ω mr A sin θ A2(4k∓ω) cos θ where A± = √ ,A± = , and A˚ ± = 4( 2 − ω2) r ω∓k 8k2mr A 2k ( ±ω) θ + sin[2(ωt − ϑ)] sin 2θ, 2k cot , where A is a constant and ϑ is a phase. 16k2ωm2r 2 4k(ω±k)r 123 Eur. Phys. J. C (2017) 77 :487 Page 9 of 10 487

The current density of the spinor field, J μ = ψγ¯ μψ 6 Concluding remarks μ implies Dμ J = 0. Turning off the spinor self-interaction hereon, for the sake of simplicity, the metric (34) with com- The formalization of a spinor representation space, whose ponents (42) yields an approximate time-like Killing vector points can be faced as physical spinors, has been constructed. ξ = ∂ − 1 ∂ ∂ t r t y1 r at the spatial infinity [19]. Besides, the Ricci These spinors have been shown to behave as elements of scalar, computed with respect to the metric components Eq. dense paths of the representations space which, in view of 2 [ (ω +ϑ)] θ R = 2A k sin 2 t sin the FPK identities, perform highly topologically constrained (42), reads mr2 , resulting in a curva- ture that oscillates with a frequency of 2ω, whose sign varies subsets. Some of these subsets have topological properties  with such a frequency.√ Moreover, given μνρσ the Levi-Civita intrinsically connected to physical relevant quantities. The multiplied by |g|/2, the dual of the exterior product representation space shows itself as an adequate tool to between the Killing vector and the spinor intrinsic angular explore dynamics and interactions usually by means of using μ μνσρ momentum, S ≡ i Sσρξν = 0, vanishes. infinitesimal operators. Moreover, for values of the time variable It should be emphasized that along this work we took C 1 ( +   advantage of dealing with spinors as elements of 1,3 2 1 − − D e ) in Sect. 2. Similar constraints in the representation space, t ∼ ω 1 kr+ϑ −tan 1 0 − ω coming from the FPK identities in the Cl isomorphic case, 4k 2 4 2km(2k−ω) cot θ are expected. However, our main interest here is the study − , c ∈ Z, (44) B 1 of the representation space taking into account the Clifford algebra constructed upon the Minkowski space. where In showing that type 4 spinors cannot be led into reg-  ular ones, we asserted that the mapping connecting differ- − B2 = 16C 1 r 2m2k6 −8r 2m2ωk5 −4A2rmω cos θk5 ent physical spinors—spinors of different sectors of (N), belonging to different classes. However, no reference has + 4r 2m2ω2k4 +A4ω2 cos2 θk4 +m2 cot2 θk4 been made to . It is time to elaborate this − 4ω3 2 θ 3/ − 2ω 2 θ 3 3A cos k 2 m cot k a little further. The very possibility of crossing over different − 2A2rmω3 cos θk3 + A4ω4 cos2 θk2 classes, by means of a well-defined algebraic transformation +4m2ω2 cot2 θk2 − A2rmω4 cos θk2 connecting different sectors of (N) could, in principle, be related to some type of swapping spinor class due to a spe- +3A4ω5 cos2 θk/5 + A4ω6 cos2 θ  cific physical process. In fact, bearing in mind the existence + 2 ω2 θ 4 − π / , 7A rm cos k c1 8 (45) of a dense set in between different classes in the light of proposition two, this switching could be performed by a spe- ∈ Z where c1 and cific (unknown) scattering matrix modeling the physical pro-  cess. Apart from unitarity concerns,3 it is difficult to envisage D = 32rmk4 − 48rmωk3 + 16rmω2k2  how this proposed process can duplicate the helicity states in + 20A2ω2 cos θk2 − 2A2ω3 cos θk − 2A2ω4 cos θ going from regular spinors to type 5 spinors, these last spinors (46) with known dual property helicity. Perhaps, and here we are entering the fancy ground of speculation, a comprehensive C2 = 4(r 2m2k6 − 8r 2m2ωk5 − 4A2rmω cos θk5 transformation performed in the quantum operator as a whole 2 2 2 4 4 2 2 4 2 2 4 + 4r m ω k + A ω cos θk + m cot θk may give rise to more precision to and enlighten the formal + 7A2rmω2 cos θk4 − 3A4ω3 cos2 θk3/2 aspect of this possible swapping. It turns out, however, that − m2ω cot2 θk3 − 2A2rmω3 cos θk3) the physical process would still be lacking. On the other hand, we have provided a mapping which is neither invertible nor + A4ω4 cos2 θk2 +4m2ω2 cot2 θk2 Hermitian, evincing the high degree of topological constraint − 2 ω4 θ 2 + 4ω5 2 θ 16A rm cos k 6A cos k presented on the representation space. Further investigation +A4ω6 cos2 θ, (47) on the algebraic/topological relationship concerning singular spinors are under current investigation. the spinor field ψ is a flag-dipole one, since the components From the physical point of view, changing spinor classes is ¯ μ ν of the intrinsic Sμν = ψ[γ ,γ ]ψ van- essentially a change of physical observables. The dynamical ¯ ¯ ish, together also with σ = ψψ and ω = ψγ5ψ. It dynami- setup as a procedure to change spinor classes was imple- cally implements the formalism constructed in the previous mented for an axisymmetric spacetime that is generated by section. In fact, such a dynamical setup there are changes of 3 the regular spinors of type 1 into singular spinors of type 4, Typically, the SC matrix have enough symmetry to be recast into a resulting in a physical couterpart to the mapping (22). specific form allowing for an unitary scattering process. 123 487 Page 10 of 10 Eur. Phys. J. C (2017) 77 :487 the self-gravity of the spinor here studied, satisfying the equa- 9. R. Abłamowicz, I. Gonçalves, R. da Rocha, J. Math. Phys. 55, tions of motion derived from an Einstein–Hilbert action with 103501 (2014) cosmological constant coupled to a Dirac system with self- 10. J. Vaz, R. da Rocha, An Introduction to Clifford Algebras and Spinors (Oxford University Press, Oxford, 2016) interaction. 11. V.L. Figueiredo, E. Capelas de Oliveira, W.A. Rodrigues, Int. J. Theor. Phys. 29, 371 (1990) Acknowledgements JMHS thanks to CNPq (304629/2015-4; 445385/ 12. D. Hestenes, J. Math. Phys. 8, 798 (1967) 2014-6) for partial financial support. CHCV thanks to PNPD-CAPES 13. J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics for financial support. RJBR thanks to CAPES for financial support, and (McGraw Hill, New York, 1964) RdR is grateful to CNPq (Grant No. 303293/2015-2), and to FAPESP 14. P.R. Holland, Found. Phys. 16, 701 (1986) (Grant No. 2015/10270-0), for partial financial support. 15. R.A. Mosna, J. Vaz, Phys. Lett. A 315, 418 (2003) 16. R. da Rocha, J.M. Hoff da Silva, Adv. Appl. Clifford Algebras 20, Open Access This article is distributed under the terms of the Creative 847 (2010) Commons Attribution 4.0 International License (http://creativecomm 17. C.H. Coronado Villalobos, R.J. Bueno Rogerio, EPL 116, 60007 ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, (2016) and reproduction in any medium, provided you give appropriate credit 18. R. da Rocha, L. Fabbri, J. Hoff da Silva, R. Cavalcanti, J. Neto, J. to the original author(s) and the source, provide a link to the Creative Math. 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