American International Journal of Research in Science, Technology, Engineering & Mathematics
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ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research)
Invariant Lagrangian for Neutrinos Field in Terms of Divisors Dr. Bulikunzira Sylvestre University of Rwanda University Avenue, B.P 117, Butare, Rwanda Abstract: In previous works, Weyl´s equation for neutrinos has been written in tensor form, in the form of ⃗ . Complex vector đ??š = non-linear Maxwell´s like equations, through isotropic complex vector⃗⃗⃗đ??š = đ??¸âƒ— + đ?‘–đ??ť 2 ⃗ satisfies non-linear condition⃗⃗⃗đ??š = 0, equivalent to two conditions for real quantities đ??¸âƒ— 2 − đ??ť ⃗2=0 đ??¸âƒ— + đ?‘–đ??ť and ⃗⃗⃗đ??¸ . đ??ť ⃗ = 0, obtained by equating to zero separately real and imaginary parts in the equality ⃗⃗⃗đ??š 2 = 0. In development of the above ideas, in this work, we wrote the Lagrange function for neutrinos field in terms of divisors. Keywords: Invariant Lagrangian, neutrinos field, divisors.
I. Introduction In previous works, Weyl´s equation for neutrinos has been written in tensor form, in the form of non-linear ⃗ =E ⃗ + iH ⃗⃗ . It has been proved, that complex vector Maxwell´s like equations through isotropic complex vector F ⃗F = ⃗E + iH ⃗⃗ satisfies non-linear condition ⃗F 2 = 0, equivalent to two conditions for real quantities ⃗E 2 − ⃗H ⃗2=0 2 ⃗ .H ⃗⃗ = 0, obtained by equating to zero separately real and imaginary parts in equality F ⃗ = 0. It has been and E ⃗ and H ⃗⃗ have the same properties as those of vectors E ⃗ and H ⃗⃗ , components of proved, that the vectors E electromagnetic field. For example, under Lorentz relativistic transformations, they are transformed as components of a second rank tensor FΟν . In addition, it has been proved that, the solution of these non-linear equations for free particle as well fulfils Maxwell´s equations for vacuum (with zero at the right side). In development of the above ideas, in this work, we shall write Lagrange function for neutrinos field in terms of divisors. II. Research Method In this work, we shall investigate the possibility of elaboration of Lagrange formalism for neutrinos field in tensor formalism, through divisors. Via Cartan map, spinor WeylËŠs equation for neutrinos has been written in ⃗ =E ⃗ + iH ⃗⃗ . Further, tensor form, in the form of non-linear Maxwell’s like equations, through isotropic vector F these non-linear equations have been written in covariant form, through divisors K Îź and mÎź . In this work, using the method of tensor analysis, we shall write the Lagrange function for neutrinos field in tensor form, through divisors K Îź and mÎź . Lagrange Function for Neutrinos Field through Divisors In previous works , Weyl´s equation for neutrinos (1) p0 Ξ=(p âƒ—Ďƒ ⃗ )Ξ, ⃗ ⃗ ⃗ ⃗ has been written in vector form, through isotropic complex vector F = E + iH as follows ⃗⃗ Ă— ⃗F − (D ⃗⃗ Fi )vi . (2) D0 ⃗F = iD Where D0 =
i ∂ 2 ∂t i
⃗⃗ =- ∇ ⃗ D ⃗ = v
2 ⃗ Ă—H ⃗⃗ E ⃗| |E
2
(3) .
Here we use the natural system of units in which c=Ń›=1. Separating real and imaginary parts in equation (2), we obtain
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Dr. Bulikunzira Sylvestre, American International Journal of Research in Science, Technology, Engineering & Mathematics, 12(1), September-November, 2015, pp. 06-07 ⃗⃗
⃗ + ∂H = vi (∇ ⃗ Hi ) rotE ⃗⃗ − rotH
∂t ⃗ ∂E ∂t
⃗ Ei ) = −vi (∇
(4)
⃗ = −vi ∂Ei divE ∂t
⃗⃗ = −vi ∂Hi { divH ∂t In terms of complex isotropic vectors, Maxwell's equations for vacuum take the form ⃗⃗ Ă— ⃗F D ⃗F = iD { 0 . (5) ⃗D ⃗ ⃗F = 0 ⃗ 2 = 0, whereas However, in general, the solution of Maxwell's equations does not satisfy isotropic condition F the solution of neutrinos equations always satisfies this condition. Let us introduce bivector field đ?”˝ÎźÎ˝ , satisfying isotropic condition ĚƒÎźÎ˝ =0 . đ?”˝ÎźÎ˝ đ?”˝ (6) Here đ?”˝ÎźÎ˝ is chosen in the following form đ?”˝ÎźÎ˝ = K Îź Λ mν . (7) Where K Îź is a real four vector and mÎź is a complex four vector. In formula (7) the sign "Λ" means antisymmetrisation over indices đ?œ‡ and ν. The four vectors K Îź and mÎź are called divisors and are expressed ⃗ and H ⃗⃗ as follows through E ⃗ Ă—H ⃗⃗ ⃗ E F ⃗ |, K Îź = (|E ) , mÎź = (0, ) . (8) ⃗| |E
⃗| |E
Then, equations (4) can be written through divisors in the following covariant form Dν đ?”˝ÎźÎ˝ = K ν âˆ‚Îź mν . (9) In spinor formalism, Weyl's equation (1) can be obtained using variation principle from Lagrange function đ?‘– L = (ÎžĎƒÎź âˆ‚Îź Ξ∗ − âˆ‚Îź ÎžĎƒÎź Ξ∗ ). (10) 2 Written through isotropic vectors, formula (10) takes the form â „ ⃗⃗∗ 1 2
đ?‘– ⃗⃗ Ă— ⃗F + vi (D ⃗⃗ Fi )]F ⃗ ∗ − [D0 ⃗F ∗ + iD ⃗⃗ Ă— ⃗F ∗ + vi (D ⃗⃗ )Fi∗ ]F ⃗ }/(FF ) L= {[D0 ⃗F − iD 2 2 Formula (11) can be written through divisors as follows đ?‘– L = {[Dν đ?”˝ÎźÎ˝ − K ν âˆ‚Îź mν ]mÎź ∗ − [Dν đ?”˝âˆ—Ον + K ν âˆ‚Îź mâˆ—Î˝ ]mÎź }. 2
.
(11) (12)
III. Discussion and Conclusion In this work, we solved the problem of elaboration of Lagrange formalism for neutrinos field in tensor formalism through divisors. Via Cartan map, spinor WeylËŠs equation for neutrinos has been written in tensor ⃗ . Further, these non-linear form, in the form of non-linear Maxwell’s like equations, through strengths ⃗E and ⃗H equations have been written in covariant form, through divisors K Îź and mÎź . In this work, we elaborated the Lagrange formalism for neutrinos field in tensor formalism. We found the Lagrange function for neutrinos field, written through divisors K Îź and mÎź . It is clear that, applying Noether's theorem, we can derive from the obtained Lagrange function, expressions for fundamental dynamical variables (energy, momentum, charge and spin) conserved in time. These expressions for fundamental dynamical variables have been obtained in previous works. References [1]. [2]. [3]. [4]. [5]. [6].
S.Bulikunzira, “Lagrange formalism for neutrino field in terms of complex isotropic vectors,� Asian Journal of Fuzzy and Applied Mathematics, vol.3, no4, Aug. 2015, pp.126-130. S.Bulikunzira, “Tensor formulation of Dirac equation through divisors,� Asian Journal of Fuzzy and Applied Mathematics, vol.2, no6, Dec. 2014, pp.195-197. S.Bulikunzira, “Tensor formulation of Dirac equation in standard representation,� Asian Journal of Fuzzy and Applied Mathematics, vol.2, no6, Dec. 2014, pp.203-208. S.Bulikunzira,� Formulation of conservation laws of current and energy for neutrino field in tensor formalism,� Asian Journal of Fuzzy and Applied Mathematics, vol.3, no1, Feb. 2015, pp.7-9. F.Reifler, “Vector wave equation for neutrinos,� Journal of mathematical physics, vol.25, no4, 1984, pp.1088-1092. P.Sommers, “Space spinors,� Journal of mathematical physics, vol.21, no10, 1980, pp.2567-2571.
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