Transformation of Dirac Spinor under Boosts & 3-Rotations

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1 January 016 Volume 7 Issue 1 pp Article Transformation of Dirac Spinor under Boosts P. Lam-Estrada 1, M. R. Maldonado-Ramírez 1, J. López-Bonilla * & R. López-Vázquez 1 Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional (IPN), Edif. 9, Col. Lindavista CP 07738, México DF ESIME-Zacatenco, IPN, Edif. 5, 1er. Piso, Lindavista 07738, México DF Abstract We exhibit the transformation rule for the 4-spinor of Dirac under 3-rotations and boosts. Keywords: Dirac equation, 4-spinor, homogeneous Lorentz group, Weyl equations. 1. Introduction In the Dirac equation for spin-1/ particles [1-3] [(x μ ) = (t, x, y, z), ħ = c = 1]: (iγ μ μ m 0 )ψ = 0, i = 1, μ =, (1) μ ψ is a 4-spinor with the γ μ matrices verifying the anticommutator [4-6]: {γ μ, γ ν } = g μν I 4x4, (g μν ) = Diag(1, 1, 1, 1). () Here we consider the options: a) Dirac-Pauli (or standard) representation [7]. γ 0 = ( I 0 0 I ), γj = ( 0 σ j σ j 0 ), j = 1,, 3, γ5 i γ 0 γ 1 γ γ 3 = ( 0 I ), (3) I 0 with the Cayley [8]-Sylvester [9]-Pauli [10] matrices: σ 1 = ( ), σ = ( 0 i i 0 ), σ 3 = ( 1 0 ), (4) 0 1 b) Weyl (or chiral) representation [3, 11-13]. x * Correspondence: J. López-Bonilla, ESIME-Zacatenco-IPN, Edif. 5, Col. Lindavista CP 07738, México DF jlopezb@ipn.mx ISSN:

2 January 016 Volume 7 Issue 1 pp γ 0 = ( 0 I I 0 ), γj = ( 0 σ j σ j 0 ), j = 1,, 3, γ5 = ( I 0 ), (5) 0 I to study the transformation law of ψ under the homogeneous Lorentz group [14-16]: x μ = L μ ν x ν, (6) which implies the existence [4, 17] of a non-singular matrix S such that: L μα S γ α = γ μ S, (7) and we obtain the relativistic invariance of (1) if the Dirac spinor obeys the transformation rule: ψ = S ψ. (8) In this work we deduce the structure of S for boosts and 3-rotations, working with the representations (3) and (5).. Construction of S First we consider infinitesimal Lorentz transformations, besides L is an orthogonal matrix (L μ αl να = g μν ) [14], then it differs infinitesimally from the unit matrix by a skewsymmetric matrix [18]: L μα = g μα + ε F μα, F βν = F νβ, S = I + ε Q, ε 1, (9) and we must determine Q with the constraint required by (7) via the commutator: [γ μ, Q] = F μβ γ β = 1 F αβ(δ μ α γ β δ μ β γ α ) = 1 4 F αβ[γ μ, γ α γ β ] = [γ μ, 1 4 F αβ γ α γ β ], that is, Q = 1 4 F αβ γ α γ β, hence for a finite Lorentz transformation [we take ε = 1 N ]: S = lim N (I + ε 4 F μν γ μ γ ν ) N = exp ( 1 4 F μν γ μ γ ν ). (10) Therefore, given L we have F μν, then (8) and (10) allow to construct the new 4-spinor. ISSN:

3 January 016 Volume 7 Issue 1 pp Rotations in three dimensions For rotations around of the axes X, Y, Z, the matrix (F μν ) is given by [3, 19]: ij 1 θ 1 = θ 1 ( ), ij θ = θ ( ), ij 3 θ 3 = θ 3 ( ), (11) respectively, then in the Dirac-Pauli and Weyl representations the matrix (10) takes the form: S = ( exp (i σ θ ) 0 0 exp ( i σ θ ). (1) ) 4. Boosts In this case, for boosts in the directions X, Y, Z, the matrix (F μ ν) has the structure [3, 1]: ik 1 φ 1 = φ 1 ( ), ik φ = φ ( ), ik 3 φ 3 = φ 3 ( ), (13) respectively, where tanh φ k = v k. We note that the full matrix of Lorentz transformations of rotations and boosts adopts the expression: 0 φ 1 φ φ 3 (L μ φ ν) = exp(ij θ + ik φ ) = exp ( 1 0 θ 3 θ ), (14) φ θ 3 0 θ 1 φ 3 θ θ 1 0 where only one rotation or one boost angle can be applied at any one time [1]. In (14) we see six parameters for the homogeneous Lorentz group. ISSN:

4 January 016 Volume 7 Issue 1 pp We employ (13) into (10) to obtain: S = ( cosh (φ k ) I sinh (φ k )σ k sinh ( φ k )σ k cosh ( φ ) Dirac-Pauli scheme, (15) k ) I = ( exp ( 1 σ φ ) 0 0 exp ( 1 σ φ ) Weyl scheme, (16) ) hence, in the representation (5), for rotations and boosts the matrix S acquires the general structure [3, 1]: S = ( exp [1 σ (iθ φ )] 0 0 exp ([ 1 σ (iθ + φ )] ). (17) 5. Weyl spinors We write the Dirac spinor in the form: ψ 1 ψ ψ 3 ψ = ( ) = ( ), (18) ψ L ψ 4 where ψ R and ψ L are called Weyl spinors [1, 0], then with (8) and (17) it is immediate to deduce their transformation laws (in the chiral scheme) under an arbitrary Lorentz mapping (14) [3]: ψ R ψ R = exp [ 1 σ (iθ φ )] ψ R, ψ L = exp [ 1 σ (iθ + φ )] ψ L, (19) and they do not preserve parity (they are not invariant with respect to the change x x), hence they were assumed to represent neutrinos, which are all left-handed (described by ψ L ) while antineutrinos are all right-handed (described by ψ R ). The Dirac spinor, being composed of both spinors, is fully parity-preserving [1]. The standard representation necessarily mixes the Weyl spinors under Lorentz transformations, so their distinction is not noticeable; ψ R and ψ L are ISSN:

5 January 016 Volume 7 Issue 1 pp dotted and undotted -spinors, respectively, that is, they correspond to the representations ( 1, 0) and (0, 1 ) of the Lorentz group [3]. In fact [1]: ξ1 ψ R = ( ξ ), ψ L = ( η 1 η ), (0) and we can consider (19) with θ 1 0, therefore: ( η 1 η ) = U 1 ( η1 η ), U 1 = ( cos ( θ 1 ) i sin(θ 1 ) i sin( θ 1 ) cos (θ 1 ) ), det U 1 = 1, (1) because [] η 1 = ε A1 η A = η and η = ε A η A = η 1 ; besides: (ξ 1 ξ ) = (ξ 1 ξ )U 1. () Similarly, from (19) for φ 1 0: T ψ R = T ψr U η 1, ( η ) = U ( η1 η ), U = ( cosh ( φ 1 ) sinh (φ 1 ) sinh ( φ 1 ) cosh (φ 1 ) ), det U = 1, (3) thus (1), () and (3) show [] the undotted and dotted character of ψ L and ψ R, respectively. The matrix exp ( 1 σ φ ) is not unitary, hence we have above a finite-dimensional and nonunitary representation of the non-compact Lorentz group, however, it has infinite-dimensional unitary representations [3]. If we employ (5) and (18) in the Dirac equation (1) for massless particles, we obtain the Weyl equations [3, 11, 1] ( 0 σ j j )ψ L = 0 and ( 0 + σ j j )ψ R = 0, that is: (p 0 + σ p )ψ L = 0, (p 0 σ p )ψ R = 0, (4) thus ψ L (ψ R ) are eigenstates of negative (positive) helicity, which tells us how closely aligned the spin of a particle is with its direction of motion. ISSN:

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