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On the Spinor Representation Eur. Phys. J. C (2017) 77:487 DOI 10.1140/epjc/s10052-017-5035-y Regular Article - Theoretical Physics 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 homotopy 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 fermions from a different and useful perspective. provides the spacetime generated by a fermion. 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 rotation [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 Dirac equation 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 multivector 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) group 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]. tangent bundle 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 Clifford algebra 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 basis, 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 Lorentz group, which can be thought of as being sections Returning to Eq. (3), and using for instance the standard 4 of the vector bundle P e (M) ×τ C [5,6]. representation, the complex matrix 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 spacetime algebra, 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 ideal 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 Dirac spinor 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 trace 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 vector space 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.
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