Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 5, N. 1, 2017. Trabalho apresentado no CNMAC, Gramado - RS, 2016. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics A Geometric Motivated Approach of Lie Derivative of Spinor Fields Rafael de Freitas Le~ao1 Instituto de Matem´aticae Estat´ısticae Computa¸c~aoCient´ıfica,UNICAMP, Campinas, SP Waldyr Alves Rodrigues Jr.2 Instituto de Matem´aticae Estat´ısticae Computa¸c~aoCient´ıfica,UNICAMP, Campinas, SP Samuel Augusto Wainer3 Instituto de Matem´aticae Estat´ısticae Computa¸c~aoCient´ıfica,UNICAMP, Campinas, SP Abstract. In this paper using the Clifford bundle (C`(M; g)) and spin-Clifford bundle (C` e (M; )) formalism, which permit to give a meaningfull representative of a Dirac- Spin1;3 g Hestenes spinor field (even section of C` e (M; )) in the Clifford bundle, we give a geo- Spin1;3 g metrical motivated definition for the Lie derivative of spinor fields in a Lorentzian structure (M; g) where M is a manifold such that dim M = 4, g is Lorentzian of signature (1; 3). Our s Lie derivative, called the spinor Lie derivative (and denoted $ξ) is given by nice formulas s when applied to Clifford and spinor fields, and moreover $ξg = 0 for any vector field ξ. With this we compare our definitions and results in [11] with the many others appearing in literature on the subject. Keywords. Lie Derivative, Spinor Fields, Dirac-Hestenes Spinor Fields, Clifford Fiber Bundle, Spin-Clifford Fiber Bundle 1 Introduction Lie derivatives of tensor fields are defined once we give the concept of the push forward and pullback mappings (which serves the purpose of defining the image of the tensor field) associated to one-parameter groups of diffeomorphisms generated by vector fields. These concepts are well known and very important in the derivation of conserved currents in physical theories. It happens that physical theories need also the concept of spinor fields living on a Lorentzian manifold and the question arises as how to define a meaningful image for these objects under a diffeomorphism. There are a lot of different approaches to the subject, as the reader can learn consulting, e.g., [1{4, 6{10, 12, 15]. [email protected] [email protected] [email protected] DOI: 10.5540/03.2017.005.01.0206 010206-1 © 2017 SBMAC Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 5, N. 1, 2017. 2 4 We recall [14] that fixing a global spinor basis Ξ (x) = (x; u (x)) for P e (M; g), 0 0 Spin1;3 l and given an algebraic spinor Ψ 2 sec C` e (M; g), the associated Dirac-Hestenes Spinor Spin1;3 0l Field (DHSF) Ψ 2 sec C` e (M; g) can be represented in the Clifford bundle by the Spin1;3 object 0 Ξ0 2 sec C` (M; g): (1) ~ Remark 1.1. When Ξ0 Ξ0 6= 0 we can easily show that Ξ0 has the following represen- tation 1 τ gβ 2 − 2 Ξ0 = ρ e R; (2) where ρ, β 2 sec V0T ∗M,! sec C`0(M; g) and [13] F e 0 R = ±e 2 sec Spin1;3(M; g) ,! sec C` (M; g); (3) with F 2 sec V2T ∗M,! sec C`0(M; g): Let ξ 2 sec TM be a smooth vector field. For any x 2 M there exists an unique integral curve of ξ, given by t 7! h(t; x), with x = h(0; x). We recall that for (t; x) 2 I(x)×M (I(x) ⊂ R) the mapping h: (t; x) 7! h(t; x) is called the flow of ξ. We suppose 0 in what follows that the mappings ht := h(t; ): M ! M, x 7! x = ht(x) generate a one-parameter group of diffeomorphisms of M (i.e., I(x) = R). τ β 1 − g F(x) Thus, we see that there exists no difficulty in defining the pullback of ρ 2 e 2 e 0 under ht (or of more generally, for any Ξ0 2 C` (M; g)), which will be written as τ0 (x)β(x0(x)) 1 0 − gt F 0(x) ρ 2 (x (x))e 2 e t : (4) However, we immediately have a Problem: The object defined by Eq.(4) is of course, a representative in C`0(M; g) of some Dirac-Hestenes spinor field but there is no way to know to which the spinor frame that object is associated. Thus, we must find another way to define the Lie derivative for spinor fields. Our way, as we will see, is based in a geometric motivated definition for the concept of image of Clifford and spinor fields under diffeomorphisms generated by one-parameter group associated to an arbitrary vector field ξ. But, we need first to introduce some results proved in [11], starting with the Proposition 1.1. Let $ξ denotes the standard Lie derivative of tensor fields. If ξ is a Killing vector field then 1 $ γα = [L(ξ) + dξ; γα] (5) ξ 4 1 = D γα + [dξ; γα] (6) ξ 4 4Such a basis must exists according to Geroch Theorem [5]. DOI: 10.5540/03.2017.005.01.0206 010206-2 © 2017 SBMAC Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 5, N. 1, 2017. 3 1 L(ξ) := (c + c + c )ξκγα ^ γι (7) 2 ακι και ιακ α·· α where c·κι are the structure coefficients of the basis feαg dual of fγ g. Remark 1.2. Moreover, one can easily show using the previous results that when C 2 sec C`(M; g) and ξ 2 sec TM is a Killing vector field then 1 $ C = d C+ [S(ξ); C]: (8) ξ ξ 4 Indeed, Eq.(8) follows trivially by induction and noting that $ξ(AB) = $ξ(A)B + A$ξ(B), where A; B 2 sec C`(M; g); when ξ 2 sec TM is a Killing vector field. This suggests that L(ξ) should be involved in the definition of the Lie derivative of spinor fields. Based on this, and recalling Eq.(3) we propose that the spinor lifting of an integral curve of a generic smooth vector field ξ 2 sec TM to PSpine1;3(M; g) in the parallelizabe manifold M equipped with the global orthonormal cobasis fγαg is given by the following Definition 1.1. Consider the integral curve ht : R ! M of an arbitrary smooth vector field ξ. The spinor lifiting h˘ of h to P e (M; ) is the curve t t Spin1;3 g h˘t(p) = (ht(π(p)); aut(ht(π(p))) (9) − 1 t(S(ξ)(x)) e ut(x) := e 4 2 Spin1;3; (10) S(ξ) = L(ξ) + dξ; (11) with π(p) = π((x; a)) = x. To see why the above definition is really important consider that for t << 1 it is 1 u = 1 − tS(ξ) + O(t2) + ··· (12) t 4 Then, we have for t << 1 that 1 1 u−1γαu = f1 + tS(ξ) + O(t2) + · · · gγαf1 − tS(ξ) + O(t2) + · · · g t t 4 4 1 = γα + t[S(ξ); γα] + O(t2) + ··· (13) 4 Deriving in t = 0 we obtain the expression of the previous proposition. 0α ∗ α Now, recall that the pullback γt = ht γ when ξ is an arbitrary vector field for t << 1 0α α α 2 γt (x) = γ (x) + t$ξγ (x) + O(t ) + ··· (14) Using the Proposition (1.1), comparing Eq.(14) with Eq.(13) and recalling Eq.(5), we see that up to the first order we have 0α −1 α γt (x) = ut (x)γ (x)ut(x) (15) α From Eq.(15), the Lie derivative $ξγ can be calculated in two ways, using the usual definition by pullback or by the action of ut. Note that the action of ut is always orthogonal, regardless of ξ be Killing. We will use this fact to give our geometric motivated concept of Lie derivatives for Clifford and spinor fields. DOI: 10.5540/03.2017.005.01.0206 010206-3 © 2017 SBMAC Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 5, N. 1, 2017. 4 s 2 The Spinor Lie Derivative $ξ 2.1 Spinor Images of Clifford and Spinor Fields α Given the spinorial frame Ξ (x) = (x; u ) in P e (M; ) we see that the basis fγˇ g ut t Spin1;3 g t of P e (M; ) such that SO1;3 g α −1 α α β γˇt (x) = ut (x)γ (x))ut(x) = Λtβ(x)γ (x); (16) s is always orthonormal relative to g. This suggests to define a mapping ht (associated with a one parameter group of diffeomorphisms ht generated by a vector field Vp ∗ l r 0 ξ) acting on sections T M; C`(M; g), C` e (M; g), C` e (M; g). With x = ht(x) Spin1;3 Spin1;3 we start giving Definition 2.1. s Vp ∗ Vp ∗ ht : sec C`(M; g) - sec T M ! sec T M,! sec C`(M; g); 1 P (x0) 7! Pˇ (x) = P (x0(x))γˇi1 (x) ··· γˇip (x) t p! i1···ip t t 1 = P (x0(x))u−1γi1 (x) ··· γip (x)u (17) p! i1···ip t t 1 P (x) = P (x)γi1 (x) ··· γip (x) 6= Pˇ (x) p! i1···ip t 1 P (x0) = P (x0)γi1 (x0) ··· γip (x0) (18) p! i1···ip Eq.(17) extends by linearity to all sections of C`(M; g).