Lagrangian Formalism for the New Dirac Equation

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1 Adv. Studies Theor. Phys., Vol. 7, 2013, o. 3, HIKARI Ltd, Lgrgi Formlism for the New irc Equtio OUSMANE MANGA Admou eprtmet of Physics, Fculty of Scieces Abdou Moumoui Uiversity of Nimey, P.O. Box 10662, Niger SAMSONENKO Nicoli Vldimirovich eprtmet of Theoreticl Physics, Russi Friedship Uiversity 3, Ordjookidze, Moscow, Russi MOUSSA Aboubcr eprtmet of Mthemtics d Computer Scieces, Fculty of Scieces Abdou Moumoui Uiversity of Nimey P.O. Box 10662, Niger msboubcr@yhoo.fr Abstrct The expressios for the eergy-impulse tesor d the spi opertors re obtied for the prticle described by the ew irc equtio i the frmework of the Lgrge formlism. Mthemtics Subject Clssifictio: 83C57, 7B15, 7B25 Keywords: irc equtio, lest ctio priciple, field fuctios, ctio vritio, reltivistic lgrgi, eergy-impulse tesor, spi opertor.

2 12 OUSMANE MANGA Admou et l. 1. Itroductio The ew reltivistic wve equtio proposed by irc i 1971 (see [3]) is ot symmetric i term of positive d egtive vlues of eergy. This equtio describes spiless prticle with positive eergy, iterl structure d o-zero rest mss. The equtio hs the followig form: x + α + mβ qψ = 0 r 0 xr (1) where α r (r = 1,2,3) re rel mtrices d β is tisymmetric mtrix give by β = (2) 2 Note tht β = 1. The qutity q is colo vector. The symbol q will deote lie vector (q 1,q 2,q 3,q ) where q,q 1 3 = p 1 d q,q 2 = p 2 re the dymic vribles of two hrmoic oscilltors describig the iterl structure of the prticle. The qutities q ( = 1,2,3, ) stisfy followig commutig lw [ ] q,q = q q q q = iβ (3) b b b b The wve fuctio ψ is oe-compoet d depeds o x 0,x r d two commutig qutities q (for exmple q 1 d q 2 ). The mtrices α r (r = 1,2,3) d β stisfy the Clifford-irc lgebr reltios: α α + α α = 2δ r s s r rs αβ+ βα = 0 r r (r,s= 1,2,3) ()

3 Lgrgi formlism for the ew irc equtio 13 Itroducig the ottios tkes the form: ( ) d α 0 = I the uity mtrix, the equtio (1) x α + mβ qψ = 0 ( = 0,1,2,3) (5) The multiplyig equtio (5) by mtrix β t the right we get: ( ) α β m qψ = 0 ( = 0,1,2,3) (6) 2. Lgrgi The ew irc equtio, like the old oe (see []) c be cosidered s equtio of some field. This field is described by fuctios qψ d ψ qb. With correspodig Lgrge desity fuctio we c obti the field equtio (1) usig vritiol method. From the Lgrge desity fuctio L (see [1]) give by: 1 1 L = ( ψ q αβ qψ ψq αβ qψ) + ψq mqψ (7) 2 we c obti the Lgrge equtio i term of the field fuctio ψ q s followig: L L 0 =, (8) x ( ψ q ) ψ q where L 1 1 = α q mq β ψ + ψ, ψ q 2 L 1 q = αβ ψ. x ( ψ q ) So we obti the equtio (6). 3. Bsic physicl qutities Accordig to Noether s theorem, to every fiite-prmeter (depedig o s costt prmeters) cotiuous trsformtio of the field fuctios d

4 1 OUSMANE MANGA Admou et l. coordites vishig the vritio of the ctio correspod s dymic ivrits, i.e. time-coserved combitios of field fuctios d their derivtives (see [2]). Cosider the followig ifiitesiml trsformtio of the field fuctios d coordites: x x = x + δ x, ϕ ϕ = ϕ + δϕ (9),,,, ϕ ϕ = ϕ + δϕ where ϕ = q ψ, = 1,2,3,; ϕ = ψq, = 5,6,7,8. The vritios δ x d δϕ re expressed i terms of the lierly idepedet ifiitesiml trsformtio prmeters δω usig the formuls: δ x X = () δω 1 s (10) δϕ = Ψ δω () 1 s, Note tht δϕ is ot derivte of δϕ, i.e. opertors / x d δ do ot commute. The fct is tht δϕ is vritio of the field fuctio s by chgig its shpe d by the rgumet. eote the vritios of the form of the field fuctios s: δϕ, (, = ϕ ϕ = δϕ τ ϕδx τ () X τ τ = Ψ ϕ ()) δω (11) The opertios δ d / x do commute. We ow defie the vritio of the ctio: δ I = Ldx Ldx, where, L = L ( ϕ, ϕ ) = L + δl. The totl vritio δ L is equl to: L L, dl δ L = δϕ + δϕ, = δ L + δ x. ϕ ϕ dx, Here δ L is the vritio of L due to the vritios of the forms ϕ d ϕ : L L, δ L = δϕ + δϕ, (12) ϕ ϕ

5 Lgrgi formlism for the ew irc equtio 15 I result we obti: dl δ x d δi = ( L + δ L + δx )(1 + )dx L dx = ( δ L + ( L δx ))dx dx x dx Usig the Lgrge equtios (8) we trsform δ L ito the followig form: L L L δ L =, δϕ + δϕ, = δϕ, x ϕ ϕ x x ϕ Substitutig this expressio for δ L i (13) we obti: L δi = δϕ, + Lδx dx x ϕ d L, τ τ =, ( () X ()) L X() dx Ψ ϕ δω + δω dx ϕ d L, τ τ =, ( Ψ () ϕ X() ) + L X() dxδω dx ϕ Sice δ I = 0, the by the lier idepedece of trsformtio prmeters δω d the rbitrriess of the itegrtio domi, we hve: d L, τ τ, ( Ψ () ϕ X ()) + L X() 0 = (1) dx ϕ We itroduce the ottios L, Θ () = ( Ψ, () ϕ τ X τ ()) + L X () (15) ϕ The (1) tkes the form: d Θ () = 0 (16) dx So d Θ () dx = 0 (17) dx Trsformig this itegrl by the Guss theorem, we c obti the coservtio lws of the correspodig surfce itegrls. Cosiderig tht the itegrtio is over volume, costtly expdig i spce-like directios d limited i the time-like directios by spce-like three-dimesiol surfces σ 1 d σ 2, d ssumig tht the sptil boudry of the field is zero, we get (13)

6 16 OUSMANE MANGA Admou et l. dσθ () dσθ () = 0 (18) σ1 σ2 Here dσ is the projectio of the elemet of surfce σ i 3-ple perpediculr to the xis x. This equtio shows tht the surfce itegrls C( σ ) = dxθ() σ do ot deped o the surfce σ. ) Eergy-impulse tesor Cosider ifiitesiml spce-time trsltio x = x + δ x. Choosig δ x s the trsformtio prmeters, we hve:, i.e. X δ x = X δω = X δ x =. Sice the field fuctios re ot coverted, the Ψ = 0. With i mid, d i this prticulr cse, from (15) we get secod order tesor: L L Θ() T = qψ ψqb + L δ qψ ψq δ ( ) ( b) 1 = ( ψq α β qψ ψq α βqψ) + L δ As for qψ d ψ q stisfyig the field equtios we hve L 0, the Hece 1 T = ( ψ q αβ qψ ψq αβqψ ) (19) 1 qψ ψq T00 = ψ q β βqψ t t i ψ ψ = ψ ψ 2 t t (20) I the sme wy we obti the expressios for the remiig compoets of the tesor: i ψ ψ T 0 = ψ ψ 2 x x, = 1,2,3 (21) The coserved qutity i this cse is: 3 P = T0d x Usig the solutio of the ew irc equtio for free prticle (1)

7 Lgrgi formlism for the ew irc equtio ψ ( x,q 1,q 2 ) = k exp q1 + q2 + ip 1( q1 q 2 ) 2ip2q1q 2 /( p0 + p 3 ) exp ip x 2 (22) d the ormlistio coditio 3 ψψ d x= 1 we obti the expressio for the eergy d 3-mometum P = T d x= pψψ d x= p P = T d x= pψψ d x= p 3 3 r r0 r r { } b) Agulr mometum tesor d spi tesor Cosider the ifiitesiml -rottio x x = x + x, (23) = +, where =. Thus, i this cse del with 6-prmeter trsformtios group. The ρσ σ δx = X = x = x δ = ρσ σ ρ< σ σ x δ σ< σ> ( δσ σδ) σ = + x δ = σ σ = x x σ< Cosequetly, we hve: σ Xρσ = xσδ ρ xρδσ (2) We ow fid expressio for Ψ ρσ. Sice ϕ = ϕ + δϕ, the the requiremet of reltivisticlly ivrit field equtios i coordite trsformtio (23), the field fuctios re coverted s followig [3]: 1 ρσ qψ = ( 1 βn) qψ, where N = α ρβασ, hece 1 ρσ dc eb δϕ = ( βn ) bϕb = βd αρ βceασ ϕb We lso hve ρσ δϕ = Ψ ρσ, (= 1,2,3,).

8 18 OUSMANE MANGA Admou et l. Comprig these expressios for δϕ, we get: 1 dc eb Ψ ρσ = βdαρ βceασ ϕb (25) Give the trsformtio lw for fuctios ϕ% b, we get similr expressio for Ψ bρσ : 1 m lk Ψ bρσ = % ϕα m ρ β lασ βkb (26) I this cse, the tesor Θ () trsforms ito the tesor M : ρσ L τ τ Θ() Mρσ = Ψ ρσ qψx ρσ + ( qψ ) (27) L τ τ + Ψ bρσ ψqx b ρσ + LX ρσ ( ψ q b ) where τ τ τ ρ σ qψ Xρσ = qψ ( xσ δρτ xρδστ ) = ( xσ xρ ) qψ. So L σ ρ L σ ρ Mρσ = xρ xσ q + xρ xσ ψqb + ( ) ( ) ( ) ( ) ψ qψ ψqb L L L( xσ δρ xρδσ ) Ψ ρσ Ψ bρσ ( qψ) ( ψqb) + + = = xt σ ρ xt ρ σ + ψ qαρβασβαβqψ + ψqαβ αρβασqψ = = xt xt + S σ ρ ρ σ ρσ Here xσtρ xρtσ is the orbitl gulr mometum of the prticle. The tesor S ρσ correspods to the spi gulr mometum of the prticle. Cosider the sptil prt of the spi gulr mometum: 0 1 Sρσ = ψ q αρβασ qψ 8 We c write the tisymmetric form i term of idices ρ d σ : 0 1 Sρσ = ψ q ( αρβασ ασ βαρ ) qψ 16 Itegrtig this expressio over the etire volume, we obti the spi gulr mometum tesor i the form: (28)

9 Lgrgi formlism for the ew irc equtio ( ) 3 Sρσ = ψq αρβασ ασ βαρ qψ d x From this we c defie the spi opertors s follows: 1 = ( α βα α βα ) = 16 1 i = q αρβασq+ gρσ 8 Ŝρσ q ρ σ σ ρ q (29). Coclusio Thus the Lgrge formlism llows us to obti expressios for ll physicl qutities, d these expressios re ideticl to those formuls obtied by irc without usig the vritiol method. The resultig formuls give the opportuity to geerlize the cosidered cse of clssicl field for more iterestig, from physicl poit of view, cse of qutized field, i.e. crry out the procedure of secod qutiztio. However, it should be emphsized tht i this pproch remis usolved problem of icludig the iterctio of the field with kow physicl fields (see, for exmple []). Refereces [1] N.N. Bogolubov,.V. Shirkov, Itroductio to the theory of qutized fields (i Russi), Huk, Moscow, 198. [2] J.E. Cstillo H. d A.H. Sls, A Covrit Reltivistic Formlism for the New irc Equtio, Adv. Studies Theor. Phys., Vol. 5, o. 8, (2011), [3] P.A.M. irc, A positive eergy reltivistic wve equtio, Proc. Roy. Soc., Lodo, A.322, issue 1551 (1971), 35-5.

10 150 OUSMANE MANGA Admou et l. [] P.A.M. irc, The Qutum Theory of the Electro, Proc. R. Soc. A.117 (1928), Received: November, 2012

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