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:
6 January 016 Volume 7 Issue 1 pp References 1. B. Thaller, The Dirac equation, Springer-Verlag, Berlin (199). S. Weinberg, The quantum theory of fields.i, Cambridge University Press (1995) Chap L. H. Ryder, Quantum field theory, Cambridge University Press (1996) Chap. 4. R. H. Good Jr., Properties of the Dirac matrices, Rev. Mod. Phys. 7, No. (1955) E. Piña G., Vector representation of interacting Dirac equation, Int. J. Theor. Phys. 40, No. 1 (001) J. López-Bonilla, L. Rosales, A. Zúñiga-Segundo, Dirac matrices via quaternions, J. Sci. Res. (India) 53 (009) J. Leite-Lopes, Introduction to quantum electrodynamics, Trillas, Mexico (1977) Chaps. 4, 7 and A. Cayley, A memoir on the theory of matrices, London Phil. Trans. 148 (1858) J. Sylvester, On quaternions, nonions and sedenions, John Hopkins Circ. 3 (1884) W. Pauli, Zur quantenmechanik des magnetischen electrons, Zeits. für Physik 43 (197) D. McMahon, Quantum field theory, McGraw-Hill, New York (008) 1. W. Straub, Lorentz transformation of Weyl spinors, Jan 11, M. D. Schwartz, Quantum field theory and the standard model, Cambridge University Press (014) Chaps. 10 and J. L. Synge, Relativity: the special theory, North-Holland, Amsterdam (1965) Chap Z. Ahsan, J. López-Bonilla, B. Man Tuladhar, Lorentz transformations via Pauli matrices, J. of Advances in Natural Sciences, No. 1 (014) B. Carvajal G., I. Guerrero M., J. López-Bonilla, Quaternions, x complex matrices and Lorentz transformations, Bibechana (Nepal) 1 (015) W. Pauli, Contributions mathématiques a la théorie de Dirac, Ann. Inst. Henri Poincaré 6 (1936) J. L. Synge, Classical dynamics, Handbuch der Physik 3, Part 1, Springer, Berlin (1960) p J. López-Bonilla, R. López-Vázquez, J. C. Prajapati, 3-rotations via the Olinde Rodrigues-Cartan and Hamilton-Cayley expressions, 6, No. 11 (015) J. López-Bonilla, R. López-Vázquez, Square root of an operator: Laplacian & Weyl and Dirac equations, Information Sciences and Computing (India) V013, No., October 1. M. Carmeli, S. Malin, Theory of spinors: An introduction, World Scientific, Singapore (000) Chap. 5. A. Hernández-Galeana, J. López-Bonilla, R. López-Vázquez, G. R. Pérez-Teruel, Faraday tensor and Maxwell spinor, 6, No. (015) ISSN:
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