Yang-Mills Families

I S S N 2 3 4 7 -‐ 3 4 8 7 V o l u m e 1 3 N u m b e r 6 J o u r n a l o f A d v a n c e s i n P h y s i c s

Yang-Mills Families

R. M. Doria1, S. Machado2 [1]Quarks, Petrópolis 25600, RJ, Brazil [2]Centro Brasileiro de Pesquisas Físicas, Rua Doutor Xavier Sigaud 150, RJ, Brazil

[email protected] [email protected] ABSTRACT The Yang-‐Mills theory structure is based on group theory. It rules the symmetry relationship where the number of potential fields transforming under a same group must be equal to the number of group generators. It defines the group valued expression Aµ

= Aµa ta from where the corresponding non-‐abelian symmetry properties are derived.

Green functions, one derives the physical Lagrangian L(GI ) written in

functions, one derives the physical Lagrangian L(GI ) written in Nevertheless Green based on different origins as Kaluza-‐Klein, the physical basis {GµI }. fibre bundles, supersymmetry, σ-‐model , BRST and anti-‐BRST algorithm, counting degrees of freedom leads to µI a }. Yang-‐Mills extension under the existence of different potential fields rotating the physical basis {G

A new physical Lagrangian associated to SU (N ) symmetry is genA new physical Lagrangian associated to SU (N −1 ) symmetry is −1geni The meaning of Yang-Mills families A symmetry under a same single group. They eerated. stablish for SU(N) the relationship where U =of A'µ I = U appears. A ∂µ U ⋅U eiωata and µ I U +appears. erated. The meaning of Yang-Mills families A symmetry of gI difference is realized. Where every quanta is distinguished from each difference is realized. Where every quanta is distinguished from each I is a flavor index, I = 1…N . other. Physically, says that different diversity Yang-‐Mills families can share a common symmetry group. They It it yields a quanta associated to corresponding Yangother.gauge It yields a quanta diversity associated to corresponding Yangdevelop a whole non-‐abelian t heory. Mills families. There are N-spin 1 and N-spin 0 quanta separated by Mills families. There are N-spin 1 and N-spin 0 quanta separated by quantum numbers a whole N-dynamics. An exten{Dµ , Xµi } The effort in this work different is to explore such non-‐abelian to build up it on extenthe so-‐called different quantum numbers throughextension. athrough wholeFirst, N-dynamics. An sion becomes tois m QCD becomes possible. sion tosymmetry QCD possible. constructor basis where gauge ore available for expressing the corresponding fields strengths, Lagrangian and classical

equations. After that, given that the physical fields are those associated to the poles of two-‐point Green functions, one derives the physical Lagrangian L (GI ) written in the physical basis {Gµ I } .

1

1 Introduction Introduction

A new physical Lagrangian associated to SU(N ) symmetry is generated. The meaning of Yang-‐Mills families appears. A

Since 1954 Yang-Mills have been physics guiding [1, physics [1, Until 2, 3, 4]. the twenty symmetry of difference is realized. Where every quanta is distinguished from each other. It yields a quanta diversity associated to Since 1954 Yang-Mills have been guiding 2, 3, 4]. theUntil twenty corresponding Yang-‐Mills fcentury amilies. There are Ntwo -‐spin 1decades, and N-‐spin 0physics quanta separated by different had numbers through a whole performances been determined by century first twofirst decades, physics performances had beenquantum determined by N-‐dynamics. An extension to QCD becomes possible.

geometry[5]. Weyl introduction group in theory in physics geometry[5]. However,However, the Weyltheintroduction of groupoftheory physics [6]. Quantum Mechanics theoryon bound changed changed the routethe [6].route Quantum Mechanics became became a theoryabound the on the Plank constant plus group Following theory. Following the Standard 1 Introduction Plank constant plus group theory. that, thethat, Standard Model [7,Model 8, [7, 8, 9], Supersymmetry [10], String Theory[11] and other models became physical Supersymmetry [10],guiding String Theory[11] other became Since 9], 1954 Yang-‐Mills have been physics [1, 2, 3, and 4]. Until the models twenty century first physical two decades, physics gauge symmetry. performances had been determined by geometry [5]. However, the Weyl introduction of group theory in physics changed the route insertionsinsertions based onbased gaugeonsymmetry. [6]. Quantum Mechanics became a t heory b ound o n t he to Plank constant on plus group theory. Following hat, the Standard Model [7, This work to intends enlarge non-abelian gauge tsymmetry. Beyond to8, This work intends enlarge on non-abelian gauge symmetry. Beyond to 9], Supersymmetry [10], String Theory [11] and other models became physical insertions based on gauge symmetry. Yang-Mills theory develop the sosymmetry called symmetry of difference. Introduce Yang-Mills theory develop the so called of difference. Introduce a This work intends t o e nlarge o n n on-‐abelian g auge s ymmetry. B eyond t o Y ang-‐Mills t heory d evelop t he s o c alled symmetry a common gauge symmetry field families, AµI ,the where the under a under common gauge symmetry differentdifferent field families, AaµI , where a I A of difference. Introduce under a common gauge symmetry different field families, , where the index means the inserted index Ithe means the inserted . I. .gives , N . It initial fields µ index I means inserted families, families, I = 1, . .I. ,= N 1, . It an gives initialanfields set transforming under a same gauge symmetry families, I = 1,…, . I t g ives a n i nitial f ields s et t ransforming u nder a s ame g auge s ymmetry N set transforming under a same gauge symmetry i 0 1 AµIiU 1 + =U @1µ, U · U 1 , (1) A0µI = U A AµI + @µ U · U (1) µI U g I gI

i! t i!a tU a = e a a , where ta are the group generators of SU (N ). The index a where where U = e , where ta are the group generators of SU (N ). The index a where U = e , where ta are the group generators of SU(N ) . The index a is an internal index and run according to the is an internal index and run according to the choice. group’s choice. is an internal index and run according to the group’s group's choice. Eq.(1) movesbeyond physicsYang-Mills. beyond Yang-Mills. It introduces mass term without Eq.(1) moves physics It introduces mass term without Eq.(1) moves physics beyond Yang-‐Mills. It introduces mass term without pursuing the ofGoldstone Theorem [12]. Its pursuing the Goldstone Theorem [12]. Its symmetry difference pursuing the Goldstone Theorem [12]. Its symmetry of difference offers anoffers an symmetry of difference offers an alternative to the spontaneous symmetry breaking [13, 14, 15] and dynamical symmetry breaking alternative to the spontaneous symmetry [13,and 14, dynamical 15] and dynamical alternative to the spontaneous symmetry breakingbreaking [13, 14, 15] symmetry breaking [16]. For this new symmetry approach to happen, differ symmetry breaking [16]. For this new symmetry approach to happen, different origins based on Kaluza-Klein fibre[19], bundle [19], supersymmetry ent origins based on Kaluza-Klein [17, 18], [17, fibre18], bundle supersymmetry [20, 21, 22], [23], -model [23],and BRST and anti-BRST algorithm[24, have been [20, 21, 22], -model BRST anti-BRST algorithm[24, 25] have 25] been studied 4927 | P a g e studied eq.(1) existence. in order in to order prove to onprove eq.(1)on existence. J u n e 2 0 1 7 Thus eq.(1) Thus provides eq.(1) provides a new performance a new performance for generating for generating physical physical laws. It laws. It w w w . c i r w o r l d . c o m introduces the antireductionist approach. The principle where physical laws introduces the antireductionist approach. The principle where physical laws iω ata

2

2

I S S N 2 3 4 7 -‐ 3 4 8 7 V o l u m e 1 3 N u m b e r 6 generated from of fields a setfrom J ounder u r n a al same o f A gauge dav same a n symmetry. c egauge s i n symmetry. P h y s i c s are generated a settransforming of fields transforming under

are are generated fromprimordials a set of fields transforming under gauge symmetry. A physics which whole as whole thea same fundamental generated from aare set in of the fields transforming under sameunity. gaugeIn symmetry. Aare physics which primordials are in the as thea fundamental unity. In A physics which primordials are in the whole as the fundamental unity. In other words, understands nature as a whole. A physics which primordials are in the whole as the fundamental unity. In other words, understands nature as a whole. are understands generated from a setas of afields transforming under a same gauge symmetry. [16]. For this new symmetry approach to happen, different origins based on Kaluza-‐Klein [17, 18], fibre bundle [19], supersymmetry other words, nature whole. are generated from a set of fields transforming under a same gauge symmetry. are A generated from a set of fields transforming under a same gauge symmetry. non-abelian whole gauge symmetry appears to be understood. Conother nature as a whole. And non-abelian whole gauge symmetry toofundamental be understood. Con[20, 21, 22], σ-‐model [23], BRST awords, nti-‐BRST aunderstands lgorithm [24, 25] have are been studied in oappears rder tas o prove n eq.(1) existence. unity. A aphysics which primordials in the whole the In A non-abelian whole gauge symmetry appears to beas understood. ConA physics which primordials are in the whole the fundamental unity. In A physics which primordials are in the whole as the fundamental unity. In sidering that these fields satisfy the Kamefuchi-O’Raifeartaigh-Salam theA non-abelian whole gauge symmetry appears to be understood. Considering that these fields satisfy the Kamefuchi-O’Raifeartaigh-Salam theother words, understands nature as a whole. Thus sidering eq.(1) provides a new performance for generating physical laws. It introduces the antireductionist approach. The that these fields satisfy thenature Kamefuchi-O’Raifeartaigh-Salam theother words, understands as a Kamefuchi-O’Raifeartaigh-Salam whole. other words, understands nature as atheir whole. orem [26], one can change preserving So, in order sidering that these fields satisfy the theorem [26], onefrom can change variables physics. So, in order principle where physical laws are generated a their set of variables fields transforming under preserving a physics. same gauge symmetry. A physics which A non-abelian whole gauge symmetry appears to be understood. Conorem one can change their variables preserving physics. So,understood. in order Aw[26], non-abelian whole gauge symmetry appears to be understood. Connon-abelian whole gauge symmetry appears to be Conget a to better transparence on symmetry, one should write the model in primordials are to in the hole aorem s tget he A fundamental u nity. I n o ther w ords, u nderstands n ature a s a w hole. [26], onethese can change their variables preserving So,model in order a better transparence on symmetry, one shouldphysics. write the in sidering that fields satisfy the Kamefuchi-O’Raifeartaigh-Salam theto get of a that better transparence on symmetry, one should write the model inmodel sidering that these fields satisfy the Kamefuchi-O’Raifeartaigh-Salam thesidering these fields satisfy the Kamefuchi-O’Raifeartaigh-Salam theterms {D , X } field basis, which is called as the constructor basis where to get a better transparence on symmetry, one should write the in µ µi terms of {D , X } field basis, which is called as the constructor basis where A non-‐abelian whole gauge symmetry appears to be understood. Considering that these fields satisfy the Kamefuchi-‐ µ µi orem [26], one can change their variables preserving physics. So, in order terms of } field basis, which is called as the constructor basis where orem [26], one can change their variables preserving physics. So, in order orem one can change their variables preserving physics. So, in order µ, X µi a O'Raifeartaigh-‐Salam theorem [26], one can change their variables preserving physics. So, in order to get a better transparence on Dµ ⌘ [26], Daµa{D tDterms is defined as Xµi } field which is calledone as the constructor basis where aµ ⌘ Dofaµ t{D as basis, a isµ ,defined to transparence on symmetry, should write the the model in X Xwhich ⌘ Dwrite isget defined as tothe get aa tbetter better transparence on symmetry, one should write model in toshould aµ tbetter transparence on symmetry, one should write the model in µ get aD , X } symmetry, one D model in is terms of {D field basis, is called as the constructor basis where µ µ i ⌘ D defined as X µ a , X } field D = A (2) µ{D terms of basis, which is called as the constructor basis where µ µI D = A (2) µ , Xµi } field basis, which µ µI as the constructor basis where X terms of}a {D is called terms of {D , Xµi field which the constructor basis where µ basis, µi µi D AµI=as (2) µ = isIcalled Dµ ≡ Dµa ta is defined as aa Dµµµ ⌘ D D A (2) t is defined as I µ µI a a µ t is ⌘ D defined as Dµµ ⌘ Dµµ taaDis defined as X µ I µ a a X a I X showing just one just field one transforming inhomogeneously and Xµi and ⌘ XX ta ⌘ areXµi ta are showing field transforming µiµi D =inhomogeneously A (2) a D AµI Dtransforming AµµµI (2)X a t (2) showing just one field transforming µI and Xµi ⌘ Xµi ta are µ =inhomogeneously µI= µ (N 1) potential fields showing just one field inhomogeneously and X ⌘ are (N 1) potential fields µi I µi a II (N 1) potential fields I X = A A (3) (N 1) potential fields a X = A A (3) µ1 µ µ 1 µ1 2inhomogeneously µ1 µ2 showing just one field transforming and X X ⌘ Xaµi ta are are aa µi ⌘ a1 X = A A (3) showing just one field transforming inhomogeneously and X t showing just one field transforming inhomogeneously and X ⌘ X t are µ1 µ µ µi a µi a 2 µi µi a µi µi =(N Aµ−1) (3) showing just one field transforming inhomogeneously and X µi ≡ X µX are pA otential fields µ2 aµ1 (N potential fields Ai tµ1 AµN (4) Xµ(N A1µ1 A (4) µ(N 1) = µN 1) = (N 1) 1) fields potentialX fields (N 1) potential Xµ(N 1) =XAX (4) µ1µ1 =A µN (3) =A (4) µ1µ1 A µ 1) X =µcovariantly. AA AA Xµ1 Aµµµ(N µ22µNNevertheless, µ1 = covariantly. 1 with i = with 2, . . . i, N and the {Dµ ,the X(3) } µ , X(3) = 2, . . .transforming , N and transforming 11 µ1 Aµ 22 µNevertheless, µi{D µi } with 2, .the . . ,isiN and covariantly. Nevertheless, the {D X µ , the µi } X = A A (4) with = 2, the . .transforming . ,basis. NX andBy transforming covariantly. Nevertheless, {D , X basisiis=not physical definition, the physical fields are that ones µ1 µN µ(N 1) basis not physical basis. By definition, the physical fields are that ones µ µi } =µN Aµ1 AµN (4) (4) µ(N µ1 1) A µN µ(N 1) 1) =XA µ1 µ(N basis is not the physical basis. By definition, the physical fields are that ones basis iis not physical basis. By definition, thefunctions. physical fields are{D that ones} which physical masses are the poles of two-points Green For the this, which physical are the poles of two-points Green functions. For, X this, = 2, ..the ..are .. ,,masses N and transforming covariantly. Nevertheless, µ, X µi} with iNphysical = 2, N and transforming covariantly. Nevertheless, the {D with i physical = 2,with . . has . ,masses and transforming covariantly. Nevertheless, the {D , X } the poles of two-points Green functions. For this, µ µi {D , X } with i = 2,…,which transforming covariantly. Nevertheless, the basis is not the physical basis. By definition, the N and µ µi µ µi which masses are the poles of two-points Green functions. For this, µ µ i one has toone diagonalize the transverse sector by introducing the matrix ⌦ [27]. to diagonalize thebasis. transverse sector bythe introducing the matrix ⌦ ones [27]. basis is not the physical By definition, physical fields are that that basis is not the physical basis. By definition, the physical fields are ones one has to diagonalize the transverse sector by introducing the matrix ⌦ [27]. basis is not the physical basis. By definition, the physical fields are that ones physical fields are that ones which physical masses are the poles of two-‐points Green functions. For this, one has to diagonalize the one has tobasis diagonalize the transverse sector by rotating introducing the matrix ⌦ this, [27]. Thus, theThus, physical {G } is obtained byof rotating as the physical basis {G is obtained by µI µI } poles which physical masses the two-points Greenasfunctions. functions. For which physical masses are poles of two-points Green For this, which physical masses are the poles ofµI btwo-points Green functions. For this, Thus, the physical basis {G }the isare rotating as {G } transverse sector by introducing the m atrix Ω [27]. Thus, pobtained hysical i s o btained b y r otating a s µIbasis Thus, the physical {Gthe }asis is by obtained by rotating as µI one has the sector by introducing the matrix matrix ⌦ ⌦ [27]. [27]. I transverse I introducing has to to diagonalize diagonalize the transverse the one has to one diagonalize the transverse by=sector introducing the Imatrix ⌦ [27]. ⌦1I GXIµµi (5) D ⌦1ID G ⌦iIXGµiby (5) µIµ = sector µ = µ = ⌦iI Gµ I Thus, the physical basis {G } is obtained by rotating as I as Dµ{G = µI ⌦basis X =rotating ⌦iIX Gµi (5) thebasis physical {G }G isIµµi obtained by rotating 1I µI D = µI ⌦1I ⌦ G (5) µ = Thus, the Thus, physical }G isµµobtained by as µI iI µ the ⌦invertible matrix invertible under theunder ⌦ matrix condition.condition. I under the Ω matrix invertible condition D G XGµi =⌦ ⌦iI GIIµ (5) II = iI G under the under ⌦ matrix condition. DIµµIµµ = =⌦ ⌦1I G (5) G XµIµµi (5) the invertible ⌦D matrix invertible condition. 1I µi µ = ⌦1I 1I µi = ⌦iI iIX µ µ µ µ 1 ⌦ ⌦KJ1 = IJ (6) ⌦IK ⌦KJ =IKIJ (6) 1 condition. under the ⌦ matrix invertible ⌦ ⌦ = (6) the invertible ⌦ matrix invertible IK KJ condition. ⌦IKIJ⌦KJ1 = IJ (6) under the under ⌦ matrix condition. a Y = Y a t , where t are the matrices which The notation to be followed is The notation to be followed is Yµ = Yµa taµ, where 11µ ata are thea matrices which ⌦ ⌦ (6) a= IK a The notation to be followed isfollowed Y = Y11µis tw ,IK where are thetafmatrices which ⌦aYhich ⌦satisfy =the (6) KJ ⌦ ⌦ = µ a, IJ The notation to be = Y are the )matrices which IJ IK IJ satisfy the Lie algebra for SU (N ). KJ µ the Lie algebra for SU (N ). KJ IK IJ µ tta t Y = Y t The notation to satisfy be followed is , w here a re t he m atrices Lwhere ie algebra or . (6) SU(N KJ µ µ a a satisfy thethe Lie algebra for SU (N ). this satisfy the algebra for SUtowork (Nstudy ). is to Thus, theLie objective of study classically on this wholewhich nona Thus, objective of this work is classically ont this whole nona ta , where The notation to be followed is = arewhole the matrices aa Y µ µ The notation to be followed is Y =isY Y , where taanon-abelian are the matrices which Thus, the objective of this work is to study classically on this whole non-‐abelian gauge theory. The paper organizes such µ where asuch Thus, the objective of this work is to study classically on this nonThe notation to be followed is Y = Y t , t are the matrices which µ tstudy µ a a Thus, the objective of this work to classically on this whole nonµ µ a a abelian gauge theory. The paper organizes new performance µ abelian gauge theory. Thealgebra paper organizes such new non-abelian performance satisfy the Lie for SU (N ). new non-‐abelian performance as theory. follows. In section 2, (N one studies the symmetry composition at constructor basis {D, Xi } . satisfy the Lie algebra for SU (N ). abelian gauge The paper organizes such new non-abelian performance satisfy the Lie algebra for SU ). abelian gauge theory. The paper organizes such new non-abelian performance follows. In2,section 2, onethe studies the symmetry composition at constructor as follows.asIn sectionthe one studies symmetry composition at constructor Thus, objective of this work is to study classically on this this whole nonThus, the objective of this work is to on whole Thus, the objective of this work isstudies to study classically this whole nonSection 3 rewrites the corresponding fields sIn trengths in the p3 hysical bsymmetry asis . corresponding Acomposition t study section classically 4, on ccomposition lassical pconstructor roperties are derived fthe rom {G } as follows. In section 2, one studies the at as follows. section 2, one the symmetry at constructor I basis {D, X }. Section rewrites the fields strengths innoni basis {D, abelian Xi }. Section 3 rewrites the corresponding fields strengths in the gauge theory. The paper organizes such new non-abelian performance abelian gauge theory. The paper organizes such new non-abelian performance basis {D, X }.theory. Section 3 section rewrites the corresponding fields strengths in the abelian The paper organizes such non-abelian performance basis }. Section 3section rewrites the corresponding fields strengths in the the physical Lagrangian L gauge is {D, left or icAt onclusion possible extension to Qnew CD. (G physical {G 4, classical properties are from derived from I }. a At physical basis }.fX 4, classical properties are composition derived the I )i . It {G Ibasis as follows. In section 2, one studies the symmetry at constructor as follows. In section 2, one studies the symmetry composition at constructor as follows. In section 2, one studies the symmetry composition at constructor physical basis {G }. At {G section classical properties are derived the physical basis At 4,left properties arefrom derived the 2 Constructor basis {D, Xconclusion }classical ILagrangian physical ). It for X conclusion a extension possible extension to i{D, Isection physical Lagrangian L(G ).I }.ItL(G is4,3left foris possible to from 2basis Constructor basis ISection ia} basis {D, X }. rewrites the corresponding fields strengths in the i {D, X }. Section 3 rewrites the corresponding fields strengths in the basis {D, X }. Section 3 rewrites the corresponding fields strengths in the i i physical L(GI ). ItL(G is left a possiblea extension to physical Lagrangian It isconclusion left for conclusion possible extension to i QCD. I ). for QCD. Lagrangian physical basis {G At section 4,a properties classical from the the II}. physical basis {G properties are derived from Eq.(1) should be rewritten by expressing connection with Yang-Mills as I physical basis {G }. At section 4, classical are derived from the QCD. QCD. IXi } {D,should Eq.(1) be rewritten by expressing a connection with Yang-Mills as 2 Constructor Basis physical Lagrangian Itthe isconclusion left conclusion a extension possible extension to II ). for physical Lagrangian extension to physical Lagrangian L(G ItL(G is left a possible to boundary condition. Thus, defines fieldfor D theDusual gauge field µ as II ).one boundary condition. Thus, one defines the field µ as the usual gauge field QCD. Eq. (1) should be rewritten by expressing a connection with Yang-‐Mills as boundary condition. Thus, one defines the field QCD.µi as a kindi of vector-matter fields transforming in the adjoint and the fields QCD. and X the fieldsi Xµ as a kind of vector-matter fields transforming in the adjoint Dµ as the usual Xµ as a kind of vector-‐matter gauge field and the fIt ields fields 3transforming in the adjoint representation. It gives 3 representation. gives representation. It gives 3 3 i 0 1 1 0 1 )U i Dµ ! DD = U!DD + µ3(@ µU µU µ µ U + (@µ U )U 1 3µ = U D g (7) g (7) i i i 1 i U Xµ U i i 1 Xµ ! XX µ = ! X = UX U

µ

µ

µ

Eq. (7) propitiates a more easy uanderstanding the symmetry properties model carries. properties Geometrically, the origin Eq.(7) propitiates more easy understanding onthat thethe symmetry Eq.(7) propitiates aon more easy understanding on the symmetry properties of the potential fields can be treated back to the vielbein, spin-‐connection and potential fields of higher-‐dimensional gravity-‐matter that the model carries. Geometrically, the originthe of the potential fields canfields be can be that the model carries. Geometrically, origin of the potential coupled theory spontaneously compactified for an internal space with torsion [17, 18, 19]. Also by relaxing supersymmetry to the vielbein, and potential fields of higherconstraints [20, treated 21, 22]. back treated back to the spin-connection vielbein, spin-connection and potential fields of higher-

dimensional gravity-matter coupled theory compactified for coupledspontaneously theory spontaneously compactified for an internalanspace with torsion [17, 18, 19]. Also by relaxing supersymmetry internal space with torsion [17, 18, 19]. Also by relaxing supersymmetry constraints[20, 21, 22]. 21, 22]. constraints[20, 4928 | P a g e Thus the most for the Lagrangian is Thus general the mostexpression general expression for the Lagrangian is dimensional Thus the most general expression for gravity-matter the Lagrangian is

J u n e 2 0 1 7 w w w . c i r w o r l d . c o m µ⌫z µ⌫ ) L = tr(Zµ⌫ Z µ⌫ ) + tr(zµ⌫

˜ µ⌫ ) µ⌫ z µ⌫ ) + ⌘ tr(Z µ⌫ + tr(Z ˜ µ⌫ L = tr(Zµ⌫ Z ) + tr(z +Z⌘ tr(Z µ⌫ z µ⌫) + tr(Zµ⌫ z )µ⌫ µ⌫ Z ) (8) 1 2 ) i µj + ⌘ tr(zµ⌫ z˜+µ⌫⌘)tr(zµ⌫mz˜ijµ⌫2)tr(X1µimXijµj tr(Xµ X ) 2 2

(8)

treated back to the vielbein, spin-connection and fields higherto vielbein, spin-connection and potential fields of highertreatedtreated back toback the vielbein, spin-connection and potential potential fields of ofcompactified highertreated back tothe thecoupled vielbein, spin-connection and potential potential fields of higherhigherdimensional gravity-matter coupled theory spontaneously for dimensional gravity-matter theory spontaneously compactified for treated back to the vielbein, spin-connection and fields of treated back to the vielbein, spin-connection and potential fields of higherdimensional gravity-matter coupled theory compactified for dimensional gravity-matter coupled theory spontaneously compactified for dimensional gravity-matter coupled theory spontaneously for dimensional gravity-matter coupled theory spontaneously compactified for dimensional gravity-matter coupled theory spontaneously compactified for dimensional gravity-matter coupled theory spontaneously compactified for dimensional gravity-matter coupled theory spontaneously compactified for an internal space with torsion [17, 18, 19]. Also by relaxing supersymmetry an internal space with torsion [17, 18, 19]. Also by relaxing supersymmetry dimensional gravity-matter coupled theory spontaneously compactified for dimensional gravity-matter coupled theory spontaneously compactified for anspace internal space with[17, torsion [17,Also 18, 19]. supersymmetry an internal with torsion 18, 19]. by relaxing an internal space with torsion [17, 18, Also by by supersymmetry relaxing supersymmetry I S S Nsupersymmetry 2 3 4 7 -‐ 3 4 8 7 an space with torsion [17, 18, by supersymmetry an internal space with torsion [17, 18, 19]. Also relaxing an internal internal space with torsion [17, 18, 19]. 19]. Also by relaxing relaxing supersymmetry an internal space with torsion [17,Also 18,19]. 19]. Also by by relaxing supersymmetry constraints[20, 21, 22]. constraints[20, 21, 22]. an internal space with torsion [17, 18, 19]. Also by relaxing supersymmetry an internal space with torsion [17, 18, Also relaxing supersymmetry constraints[20, 21, 22]. constraints[20, 21, 22]. V o l u m e 1 3 N u m b e r 6 constraints[20, 21, 22]. constraints[20, 21, 22]. constraints[20, 21, 22]. constraints[20, 21, 22].most constraints[20, 21,expression 22]. Thus the most general for the Thus the general expression for the Lagrangian the JLagrangian o for u r nthe a l Lagrangian o fis A d v a n is cise s i n P h y s i c s constraints[20, 21, 22]. constraints[20, 21, 22]. Thus the most general expression for the Lagrangian is Thus the most general expression for the Lagrangian is Thus most general expression Thus most general for isis Thus the most general expression for the is Thus the the most general expression for the the Lagrangian Lagrangian Thus the mostexpression general expression for the Lagrangian Lagrangian Thus the most general expression for the Lagrangian is ˜ µ⌫ isis ˜ µ⌫ Thus the most general expression for µ⌫ µ⌫ µ⌫the Lagrangian µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫) L = tr(Z )) + tr(z ))tr(z + tr(Z ⌘⌘µ⌫tr(Z )) µ⌫ Z µ⌫ µ⌫ µ⌫ ) µ⌫ ) ˜˜⌘⌘µ⌫ µ⌫ µ⌫)z µ⌫))z µ⌫+ LZ =µ⌫ tr(Z Zµ⌫ )z+ + tr(z zµ⌫ + tr(Z zµ⌫ +Z tr(Z ˜µ⌫ µ⌫ µ⌫z µ⌫ µ⌫ µ⌫ µ⌫ L Z + tr(z + tr(Z zz+ )) + + tr(Z Z L = tr(Z Z + tr(Z z ) tr(Z Z µ⌫= µ⌫ µ⌫ µ⌫ ˜ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫)) ˜ LL= ==tr(Z tr(Z Z ) + tr(z z ) + tr(Z + ⌘ tr(Z Z ) L tr(Z Z ) + tr(z z ) tr(Z z ) + ⌘ tr(Z Z ˜ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ tr(z µ⌫ µ⌫ µ⌫µ⌫ µ⌫ µ⌫ µ⌫ tr(Z Z ) + tr(z z ) + tr(Z z ) + ⌘ tr(Z Z ) µ⌫ µ⌫ µ⌫ µ⌫ ˜ L = tr(Z Z ) + z ) + tr(Z z ) + ⌘ tr(Z Z (8) µ⌫ µ⌫ µ⌫ µ⌫ ˜ µ⌫ µ⌫ µ⌫ (8) L = tr(Z ) +µ⌫1tr(z z ) + tr(Z z ) + ⌘ tr(Z Z ) Lµ⌫=Zµ⌫ tr(Z Z ) + tr(z z ) + tr(Z z ) + ⌘ tr(Z Z )) µ⌫ µ⌫ µ⌫ 1 (8) (8) µ⌫ µj µ⌫ 11mz˜µ⌫ µ⌫22 tr(X 2) i µj µj µ⌫ 1 µim (8) (8) µ⌫ iX 2 µj + ⌘ tr(z z ˜ ) i 1 (8) µ⌫ ij + ⌘ tr(z ) tr(X X ) (8) µ⌫ ij 1 µ⌫ 2 i µj + ⌘ tr(z z ˜ ) m tr(X X ) µ⌫ 2 i µj µ 1 + ⌘ tr(z z ˜ ) m tr(X X ) (8) µ⌫ ij (8) µ⌫ ij µ⌫ 2 i µj µ 1 2 µ⌫ 2 i µj µ ++⌘⌘tr(z z ˜ ) m tr(X X ) 1 + ⌘ tr(z z ˜ ) m tr(X X ) 2 iijX tr(zµ⌫ ) µ⌫22µ⌫z˜m m ⌘tr(z zµ⌫ ˜ijij2) tr(X tr(X µ⌫+ ij2 µj + ⌘ tr(z z˜z˜⌘µ⌫ )tr(z X )) µµiµXXµj )) + tr(X 222mµm µ⌫ µµij µ⌫2 ij) tr(X 2 2 where Z is the most general covariant field strengh an unique depenwhere is the the mostcovariant general covariant covariant fieldwith strengh with an unique depenµ⌫ µ⌫ is where Z is the most general field strengh with an unique depenwhere ZZ most general field strengh with an unique dependepenµ⌫ µ⌫ where Z is the most general covariant field strengh with an unique depenwhere Z is the most general covariant field strengh with an unique µ⌫ µ⌫ where Z is the most general covariant field strengh with an unique depenwhere Z is the most general covariant field strengh with an unique depenµ⌫ dence on fields, µ⌫ dence on fields, where Z is the most general covariant field strengh with an unique depenwhere Z µν is dence the most on general covariant field strengh with an unique dependence field on fields, µ⌫ where Z is the most general covariant strengh with an unique depenfields, µ⌫ fields, dence on dence fields, dence ononfields, 0 00 U 11 dence on on fields, dence fields, Z 0 µ⌫= ! Z U Z (9) 0µ⌫ Z Z =1U UZ Zµ⌫ U 111 (9) dence on fields, µ⌫ ! µ⌫ µ⌫ Z ! Z (9) dence on fields, Z ! Z = U Z (9) 0 0µ⌫ µ⌫ µ⌫ U 11 Z ! Z = µ⌫ µ⌫ µ⌫ 0 1Z µ⌫ µ⌫ 0U µ⌫ Z ! = U U (9) ZZµ⌫µ⌫ ! Z = U Z U (9) 0 1 µ⌫ µ⌫ µ⌫ ! Z = U Z U (9) µ⌫ ! = U Z U (9) 0 1 µ⌫ µ⌫ µ⌫ Z ! Z = U Z U (9) µ⌫ µ⌫ µ⌫ Z ! Z = U Z U (9) µ⌫ which can be decomposed as µ⌫ µ⌫ which can be decomposed as µ⌫ which can be decomposed as which can be decomposed as which can be decomposed as which can decomposed asas as which can can be decomposed decomposed as canbe decomposed which can be dwhich ecomposed awhich s be which can be decomposed as which can bebedecomposed as Z + = Z Z Z .. + (10) Z + Z(µ⌫).. (10) µ⌫ = [µ⌫] (µ⌫) µ⌫ [µ⌫] Z = Z + Z (10) Zµ⌫ = Z[µ⌫] (10) Z = Z µ⌫ (µ⌫) µ⌫ [µ⌫] (µ⌫). Z[µ⌫] = (10) ZZµ⌫µ⌫ ==ZZ[µ⌫] ++ ZZZ .. ++ZZ (10) µ⌫ [µ⌫] (µ⌫) (µ⌫) (10) Z = Z Z . (10) (µ⌫) µ⌫ [µ⌫] (µ⌫) Zµ⌫ = Z[µ⌫] + Z . (10) [µ⌫] (µ⌫) Z = Z + Z . (10) µ⌫ field[µ⌫] (µ⌫)is The granular antisymmetric strength The granular antisymmetric field strength is The granular antisymmetric field strength is Thefgranular granular antisymmetric fieldstrength strength isis The antisymmetric field strength The granular antisymmetric ield strength is The antisymmetric field The granular antisymmetric field The granular antisymmetric field strength strength is Thegranular granular antisymmetric field is strength is The granular antisymmetric field strength iiis The granular antisymmetric field strength isii Z = dD (11) Z = dD + ↵ X (11) µ⌫ + ↵i Xi[µ⌫] [µ⌫]+ ↵i X[µ⌫] µ⌫ [µ⌫] Z = dD (11) i Z = dD +↵↵iiX X[µ⌫] Z[µ⌫] =↵dD dD (11) µ⌫ i X[µ⌫] i + µ⌫ [µ⌫] µ⌫ [µ⌫]+ i Z = (11) ZZ[µ⌫] = dD (11) i µ⌫ µ⌫ i [µ⌫] [µ⌫] = dD + ↵ X (11) [µ⌫] Z = dD + ↵ X (11) µ⌫ + ↵iiX µ⌫ [µ⌫] = dD[µ⌫] i i[µ⌫] [µ⌫] Z[µ⌫] (11) µ⌫ [µ⌫] Z = dD + ↵ X (11) µ⌫ i [µ⌫] [µ⌫] with with with with with with with with with with with D⌫µ⌫ =@@⌫@D @ Dµ ,µ, D +⌫ ig[D with D = @@µµD D ]] µD ⌫+ ig[D µ , D⌫ ] µ⌫ µ D D ig[D D = @⌫⌫@µµD D @@⌫⌫⌫⌫D D + ig[D D⌫⌫]] µ⌫ = ⌫µ⌫= µ⌫⌫+ µµµ ⌫ig[D D = D D + µ⌫ µ (12) µ µ,,,D D D @ + D = @ D @ D + ig[D , D ] (12) µ⌫ ⌫ µ µ⌫ µ ⌫ µ µ ⌫ D = @ D @ D + ig[D , D ] i i i D = @ D @ D + ig[D µ⌫ = @X µDiµ⌫ ⌫ =@@ ⌫µ ⌫ig([D ii µ⌫ iiµµ⌫+ ig[D ii] ⌫ Xµµµ, + µ ,D ⌫ i] ] ii [D , X i ]) (12) D D D (12) X @ X µ ⌫ ⌫ ⌫ i X = @ X @ X + ig([D , X ] [D , X ]) µ ⌫ µ ⌫ i i i i i (12) D = @ D @ D + ig[D , D ] [µ⌫] ⌫ µ ⌫ µ µ ⌫ µ ⌫ i i i i i (12) µX ⌫ Xµiµ ,+ ⌫, iX [µ⌫] ⌫⌫i i =@@ µ ⌫⌫i i] µ[D µ (12) i µ⌫ X X X i i = @X i⌫ii⌫i+ ig([D i i]) [D⌫ , Xiµi ]) (12) µX ⌫@ ⌫X X @ ig([D , ] [µ⌫] µ µ = X @ X + µ ⌫ µ i i i [µ⌫] µ ⌫ X = X @ X + ig([D , X ] [D , X ]) µ ⌫ µ ⌫ X = @ X @ X + ig([D , X ] [D , X ]) [µ⌫] ⌫ µ ⌫ µ µ ⌫ µ ⌫ i i i i i (12) X[µ⌫] = @@µX X @⌫@⌫⌫µX X + ig([D ig([D ,X Xig([D [D X ]) [D⌫ , Xµiµ ]) ⌫⌫ =@ µ⌫iµ⌫+ ⌫⌫]] X @⌫ Xµi µµµ,+ ]µµµ]) µ[µ⌫] ⌫,,⌫i⌫X µ ,⌫X [µ⌫] = iX X [D [µ⌫] µ ⌫ [µ⌫] ⌫ µ ⌫ X[µ⌫] @sector, @⌫sector, Xgets X⌫as] the [Dcounterpart Considering the=symmetric one gets granular µ X⌫ µ ,granular ⌫ , Xµ ]) counterpart µ + ig([D Considering the symmetric as the Considering the symmetric sector, one gets as the granular counterpart Considering the swhere ymmetric sector, one the gthe ets asymmetric s the granular one csector, ounterpart Considering symmetric sector, one gets as the granular counterpart where Considering the symmetric sector, one gets as the granular Considering one gets as the granular counterpart Considering the symmetric sector, one gets as the granular counterpart i i i i i Considering the symmetric sector, one gets as the granular counterpart Considering the symmetric sector, one gets as the granular counterpart X @ X @ X + , X ] + [D , X ]). (14) Considering sector, gets as granular counterpart ithe symmetric i = i ⌫ i+ one i ↵ithe i↵i i ig([D µ ⌫ µ ⌫ (µ⌫) µ ⌫ µ Considering symmetric sector, one the counterpart X(µ⌫) = @µthe X(µ⌫) @ XiZ + ,iµ⌫ X +gets ,as X↵i ]). granular (13) (14) = X ⇢[D (13) i ig([D ↵i iiµ i g⌫µ⌫ (µ⌫) Z = X + ⇢ g X ⌫ + µ ⌫ ]↵+ µ i ↵i (µ⌫) ↵ (µ⌫) i ↵i Z(µ⌫) =⌫ ZZ X + ⇢ g X (13) ii = ↵i iZ(µ⌫) iXµ⌫ = X + ⇢⇢iiigggµ⌫ X (13) (µ⌫) ↵ ↵i⇢ i X+ ↵i iX µ⌫ X = + X (13) (µ⌫) ↵ Z = X + ⇢ g (13) i (µ⌫) i ↵i i µ⌫ (µ⌫) i i µ⌫ (µ⌫) ↵ (µ⌫) (µ⌫) ↵ Z = X + ⇢ g X (13) Z = X + ⇢ g X (13) Another typeZstrength of zµ⌫ .↵collective collective field which does (µ⌫) iX ig µ⌫ (µ⌫)field iis µ⌫a ↵i (µ⌫) (µ⌫) ↵ Iti is = Zistrength + ⇢ X (13) ↵field which i (µ⌫) where i µ⌫ (µ⌫) Another type of field is z . It is a does (µ⌫) ↵ where µ⌫ (13) (µ⌫) = i X(µ⌫)i + ⇢i gµ⌫ X↵i where where not i where where depend onX derivatives does appear in Yang-Mills theories. 4ig([D where iX i + @ X iiand i ,not iX usual i where 44ig([D Xderivatives = @ + X ] + [D , ]). (14) where not depend on and does not appear in usual Yang-Mills theories. = @ X + @ X + , X ] + [D , X ]). (14) µ ⌫ µ ⌫ µ ⌫ µ ⌫ (µ⌫) ⌫ µ ⌫ µ i i i i i (µ⌫) ⌫ µ ⌫ µ 4 ii ii ] + [D , X i iiii + @ Xii i+ iii ii ]). iii ii 4ig([Diii, X 444ig([D (14) X(µ⌫) = @@X X X @@⌫⌫⌫@X ig([D X ] + [D ,X X ]). (14) = @µ@µX X + X + ]]µµ+ + [D (14) 4@@@@⌫⌫X X + + + [D X ]). µ ⌫X µX == [D (14) ⌫ig([D (µ⌫) = µ X , ]). (14) µµ µµ µX ⌫X µµµ,,⌫,,⌫X ⌫⌫⌫,,,X ⌫i⌫⌫ + µµ ⌫⌫ig([D (µ⌫) µi]). (µ⌫) ⌫⌫i+ µµ,i+ ⌫⌫i]]µ (µ⌫) ⌫+ µ ⌫X µµ 4 (µ⌫) X = X + X + ig([D X + [D X ]). (14) µ ⌫ µ ⌫ (µ⌫)field strength ⌫zz . = µisz ⌫ collective µfield which 4µ⌫ AnotherAnother type of field is It a collective field does z + z (15) type strength of is . It is a which does µ⌫ µ⌫ [µ⌫] (µ⌫) z = z + z (15) µ⌫ [µ⌫] (µ⌫) Itisisisis is ain aµ⌫ collective field which which does Another type of field field strength is Another type field strength .usual It collective field which Another type of field strength .. It It is collective does type of strength is zzµ⌫ field does µ⌫.. It Another ofof .collective aaaa collective field which does µ⌫ collective field which does µ⌫ notAnother depend derivatives and does not appear theories. Another type offield fieldstrength strength iszzzzµ⌫ Itisis isYang-Mills collective field which does not on depend on and does not appear in usual Yang-Mills theories. ztype Another type of field strength is . It is a collective field which does not depend on derivatives and does not appear in µ⌫ µν derivatives appear in usual Yang-Mills theories. not depend depend on derivatives and does not not depend on derivatives and does not appear in usual Yang-Mills theories. not depend on derivatives and does not appear in usual not on derivatives and does not appear in usual Yang-Mills theories. not depend on derivatives and does not appear in usual Yang-Mills where usual Yang-Mills theories. usual Yang-‐Mills theories. not depend on derivatives and does not appear in usual Yang-Mills theories. where iz +j= i j z[µ⌫] z(µ⌫) z, [µ⌫] zµ(µ⌫) (15) i zµ⌫ j=[X j + z[µ⌫] a(ij) , µ⌫ X{X b[ij] {X , XX + j [ij] Xµi X⌫j (15) (16) µ[ij] ⌫ ] i+ ⌫ }i X z[µ⌫] = a(ij) [X= , X ] + b X } + (16) [ij] + z (15) z = z + z (15) z = z µzµ⌫ ⌫= z[µ⌫] µ ⌫ + µ ⌫ z = z + z + z (15) (µ⌫) µ⌫ [µ⌫] (µ⌫) µ⌫ [µ⌫] z = z z (15) µ⌫ [µ⌫] (µ⌫) (15) (µ⌫) µ⌫ [µ⌫]µ⌫ (µ⌫) zµ⌫ = z[µ⌫] + z(µ⌫) (15) [µ⌫] (µ⌫) and where where and where where where where j i i j i i j j where where aX(ij) ,i+ X{X b[ij] X + ⌫jjj [ij] XµiiiX⌫jjj (16) (16) i⌫ ] [X i µi,[ij] ↵jX where z[µ⌫] z= az(ij) [Xj= ,[X +jµbi][ij] , ↵j X + ⌫i ]jj+ ⌫] j} µ µ ⌫j[X j}{X iX i [µ⌫] j, X =a u g , X i, X i, X jµ iii= jj(ij) iii+ i} µ⌫ (µ⌫) [ij] [ij] i j i j iµi X j⌫j (16) j i j [ij] X z a [X ] b {X + (16) µ ⌫ ↵ {X , X } + X X (16) z(µ⌫)zz=a [X , X ] + u g [X , X ] = a [X , X ] + b [µ⌫] [ij] iX jµ]µ i,[ij] j} z = a [X , X + b {X , X } + X X (16) = a [X , X ] + b {X , X } + X X µ ⌫ µ ⌫ µ⌫ [ij] [ij] [µ⌫] (ij) [ij] ⌫ µ ⌫ µ ⌫ ↵ [µ⌫] (ij) [ij] [ij] z = a [X , ] + b {X X + X X (16) µ ⌫ X X (16) [µ⌫] (ij) [ij] [ij] µ ⌫ µ ⌫ µ ⌫ µ ⌫ ⌫ µ ⌫ [µ⌫] [µ⌫] (ij) (17) z[µ⌫]µ = (ij) a⌫ (ij)i[X ] + [ij] b⌫[ij] {Xµµ[ij] ,iX⌫⌫µ}↵j+⌫ [ij] (16) µ[ij] ⌫ ⌫µ [ij] Xµµ X⌫⌫i (17) µ ,j X j and and i b(ij) j {Xµ , X⌫ } + v(ij) i gµ⌫ ↵j{X↵ , X }i + j (ij) Xµ X⌫ . + and + b {X , X } + v g {X , X } + X X . and (ij) (ij) µ⌫ (ij) µ ⌫ and µ ⌫ ↵ and and and and i=a j [X i , X j ] + ui g ↵j [X i , X ↵j ] z(µ⌫) z(µ⌫) =a [X , X ] + u g [X , X ] µ⌫ [ij] [ij] i j i ↵j µ ⌫ ↵ µ⌫ [ij] [ij] µ ⌫jj [X i ↵j [X i , X ↵j] i, Xj⌫j] + ↵ iii ,=a iu It also transforms covariantly zz[X j[ij] i↵i , X↵j µ⌫ (µ⌫) ,[ij] Xgg↵j (17) =a[ij] X +[X gµ⌫ [X (17) It alsozztransforms covariantly =a ,, giX ]+ g↵j [X↵j X ]][ij] + uuµiµ[ij] ,[ij] X ]][X µ⌫ (µ⌫) =a [ij] [ij] [ij] µ,=a (µ⌫) z [X [X ,µµi X ]X u↵↵↵iu , ,XXi↵j ] ]] ↵j }i+ j X i X j+ j ⌫j [X µµ i⌫[ij] j[X iµ⌫ µ⌫ (µ⌫) [ij] µ⌫ z(µ⌫) =a X ] + u g ⌫⌫,b[ij] (17) ⌫ ↵↵ µ⌫ (µ⌫) [ij] + {X , } + v g[X ⌫ ↵{X (17) +[ij]b(µ⌫) {X X } + v g {X , X } + X X . µ⌫ (ij) (ij) (ij) µi (17) (17) i(ij)⌫ jµ⌫ i, X ↵j j. µ ↵ ⌫ (ij) (ij) (17) µ ⌫ ↵ µ ⌫ i ↵j i j (17) i j j} {X ↵j (17) ii+ jj } {X ii(ij) jj .(ij) Xiµi Xj⌫j. bb(ij) ,,X + gg↵j {X X }}iµi X + µ⌫} ↵j ivµi(ij) j⌫µ⌫ i↵ii,,1(ij) ↵j 0 vv + bb(ij) {Xµ+ ,X X + g , X } + X X i j ↵j i j + {X X } + {X X + X X . + {X , } + v g {X , X + X . (ij) ↵ ⌫ 0 1 µ⌫ (ij) (ij) (ij) µ ⌫ µ⌫ (ij) (ij) X X . b {X , X } + v g {X , X } + X X . z ! z = U z U . (18) µ ⌫ ↵ µ ⌫ ⌫⌫! ↵ µ ⌫ µ⌫ (ij) (ij) (ij) b {X , X } + v g {X , X } + X X . µ⌫ (ij) (ij) (ij) µ⌫ µ⌫ µz+ ↵ µ ⌫ µ ⌫ ↵ µ ⌫ µ⌫ µ⌫ (ij) µ(18) ⌫ zµ⌫ U (ij). ↵ ⌫ µ⌫ (ij) zµ⌫µ= U It also transforms covariantly It alsocovariantly transforms covariantly ItIt also covariantly It also transforms transforms It also also transforms transforms covariantly also transforms covariantly It covariantly ˜ Notice ItItalso covariantly also transforms covariantly At this work we going are not explore on term.that Notice Zµ⌫ ˜µ⌫ term. 0 to on 1 tµ⌫ At this work wetransforms are not to going explore Zµ⌫ that (18) 0z 1 tz ! z U 0U= 1U 1. µ⌫ µ⌫ z ! z = U z . (18) µ⌫ µ⌫ µ⌫ zit !F zas zµ⌫ Uin 1the (18) andnot zµ⌫Lie arealgebra not Liezzvalued algebra0µ⌫ it= is QCD. However, 0Uin 1.However, 11the µ⌫ is F 00 valued and zµ⌫ are QCD. in order U .U (18) in order = 0 = 1. . µ⌫ µ⌫ ! zas µ⌫ zz= ! zz0µ⌫ = U zzµ⌫ U (18) zzµ⌫ U .U (18) µ⌫U µ⌫U µ⌫ µ⌫ (18) µ⌫ z ! z z (18) µ⌫ ! zµ⌫ µ⌫ ! = U U . (18) µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ to explore the abundance of gauge scalars that such extended model offers ˜ to explore the abundance of gauge scalars that such extended model offers  At this we are not going to explore on t term. Notice that Z Z z F ˜ At this work e are nwork ot going to work eare xplore on tgoing t erm. N otice t hat a nd a re n ot L ie a lgebra v alued a s i t i s in µ⌫ Atwthis we not to explore on t term. Notice that Z µν on t µνare not going toµνexplore ˜µ⌫ term. Notice µ⌫ that µν At this work wegoing Z µ⌫ µ⌫ ˜µ⌫ one should also consider contributions from decomposing the fields strengths ˜ At this work we are not to explore on t term. Notice that Z ˜ µ⌫ µ⌫ At this work we are not going to explore on t term. Notice that Z one should also consider contributions from decomposing the fields strengths At this work we are not going to explore on t term. Notice that Z ˜ and z are not Lie algebra valued as it is F in the QCD. However, in order µ⌫ µ⌫ ˜ µ⌫ term. Zµ⌫ the QCD. However, in order to explore the abundance of gauge scalars that such extended model offers one should also consider µ⌫ At this work we are toto on ttµ⌫Notice term. Z At this work wealgebra arenot not going explore onHowever, term.that Notice that Zµ⌫ µ⌫ µ⌫ that and zµ⌫ are not Lie algebra valued as itgoing is Fµ⌫as initexplore the QCD. inNotice order µ⌫ and zµ⌫ are not Lie valued Fµ⌫ However, in order µ⌫{tin, tthe in terms of group terms tas tgauge ,is tF tµ⌫ ,scalars [tait ,isis t, }. }.µ⌫]QCD. It yields an expansion F in the QCD. However, in order and z are not Lie algebra valued it a,g,roup bais a], bas athe b t], a[t bHowever, µ⌫ are and z are not Lie algebra valued F in the QCD. However, in order in terms of group terms t , t , t t [t , t {t , t It yields an expansion z not Lie algebra valued as it in QCD. However, in order µ⌫ µ⌫ to explore the abundance of that such extended model offers µ⌫ µ⌫ a b a b b b µ⌫ µ⌫ QCD. in order t t t , t {t , t } contributions fand rom d ecomposing t he f ields s trengths i n t erms o f t erms , , , . I t y ields a n zµ⌫ valued as ininthe in and zabundance arenot notLie Lie algebrascalars valued asscalars QCD. in order order µ⌫ and µ⌫ to explore the ofalgebra gauge that such model offers aititis bisFF aµ⌫ bextended athe b QCD. a However, bHowever, µ⌫are µ⌫ to explore the abundance of gauge that extended model offers that such extended model offers where eachalso coefficient transforms covariantly the such non-irreducible sector. to eexplore explore the abundance of gauge gauge scalars to explore the abundance of gauge scalars that such extended model offers where coefficient transforms covariantly the non-irreducible sector. to the abundance of scalars that such extended model offers one should consider contributions from decomposing the fields strengths expansion where ach ceach oefficient tshould ransforms c ovariantly t he n on-‐irreducible s ector. extended model offers toone explore the abundance of gauge scalars that such extended model offers to explore the abundance of gauge scalars that such extended model offers one should also consider contributions from decomposing the fields strengths alsocontributions consider contributions from decomposing the fields strengths oneWorking shouldone also consider from decomposing fields strengths Working out the totaltacontributions antisymmetric field{t tensor, oneyields gets should also decomposing the fields strengths out the total antisymmetric one gets one also consider contributions fields strengths terms ofterms group terms tbfrom tbtdecomposing ,{t [ttensor, ,from tb }. , ,the tthe It an decomposing the fields strengths afrom bt],],decomposing ayields b }. should also consider from decomposing the fields strengths should consider contributions the fields strengths in terms of group tconsider tb , toane tbcontributions , t It an expansion Working oshould ut the in tone otal aterms ntisymmetric field inone ofalso group terms tg,,aets ,[t ta,b,,ttfield , [t , {t t }. It yields an expansion expansion at,ensor, bat], a a b a b a b , t ], {t , t }. It yields an expansion a b a b a b a b in terms of group terms t , t , t t , [t in terms of group terms t , t , t t , [t , t ], {t , t }. It yields an expansion in terms of group terms t , t , t t , [t , t ], {t , t }. It yields an expansion where each coefficient transforms covariantly the non-irreducible sector. a b a b a b a b a b a b a b a b a b a b a b a b It yields an expansion a in terms of group terms t , t , t t , [t , t ], {t , t }. It yields an expansion inwhere terms of group tba ,a=tbaA ,b tba ttacovariantly [ta a a, bnon-irreducible tb ],b {tthe yields an sector. expansion acovariantly where each coefficient transforms each coefficient transforms b}. It sector. b ,b the aa, tbnon-irreducible [µ⌫] Z[µ⌫]a terms = bAZaµ⌫ tcovariantly µ⌫ acovariantly where each eachwhere coefficient transforms the non-irreducible sector. a each coefficient transforms the non-irreducible sector. Working out antisymmetric field tensor, one gets where coefficient transforms covariantly the non-irreducible sector. non-irreducible sector. (19) where eachtotal coefficient transforms covariantly the non-irreducible sector. where each coefficient transforms covariantly the non-irreducible sector. Working out the antisymmetric field tensor, one gets Working outthe thetotal total antisymmetric field tensor, one gets (19) 0a ab 0a field ab tensor, one gets z = B 1 + A t + C t t Working Working out the total antisymmetric Working out the total antisymmetric field tensor, one gets µ⌫ a a b [µ⌫] z = B 1 + A t + C t t Working out the total antisymmetric field tensor, one gets µ⌫ µ⌫ µ⌫ a a b [µ⌫] one gets µ⌫ µ⌫ Workingout outthe thetotal totalantisymmetric antisymmetric field tensor,one one gets gets field a tensor, tat [µ⌫] Z[µ⌫] = AZaaµ⌫ ta ==AAµ⌫aµ⌫ a aZ[µ⌫] a a (19) Z[µ⌫] =A Aaµ⌫ where a ta a = A [µ⌫] = (19) 0a ab Z (19) a µ⌫ µ⌫ where where Z tt[µ⌫] [µ⌫] aaa 0ata + Cµ⌫ abta tb µ⌫ Z = A t 0a ab z = B 1 + A [µ⌫] Z = A t µ⌫ a [µ⌫] µ⌫ [µ⌫] a (19) µ⌫ [µ⌫] µ⌫ µ⌫ B 1 + A t + C t t z[µ⌫] = Bµ⌫z[µ⌫] 1 +=A0a t + C t t (19) µ⌫ a a b (19) a a b µ⌫ µ⌫ 0a ab µ⌫ µ⌫ (19) 0a t + a ab0at t (19) ab (19) ab a iaCab CA 0a ab aB a= ia µ⌫z1 a a bbX 0a ab ta tb [µ⌫] = = B 1 + A B 1 + t + µ⌫ µ⌫ 0a zz[µ⌫] + A t + C t µ⌫ a a [µ⌫] µ⌫ a [µ⌫] A = dD + ↵ µ⌫ µ⌫ µ⌫ a a b µ⌫ µ⌫ A = dD + ↵ X µ⌫ µ⌫ z = B 1 + A t + C t t i a1i +[µ⌫] a b [µ⌫] µ⌫ µ⌫ µ⌫ µ⌫ [µ⌫] z = B A t + C t t µ⌫ a a b [µ⌫] µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ µ⌫ a a b [µ⌫] µ⌫ µ⌫ where where 4929 | P a g ewhere 1 1 where i ja i ja where where J u n e 2 0 1where 7 where B = B aµ⌫ a[ij] =µa X↵ X⌫iaia Xbµ⌫ aX a + where µai X µ⌫ [ij] AbA = dD ⌫ia aµ⌫ dD N w w w . c i r w o r l d . c o m Aaaµ⌫ = dD ↵ X N (20) µ⌫+= µ⌫ + ↵i X[µ⌫] [µ⌫] (20) i a a ia µ⌫ [µ⌫] a = dDaaa a+ ↵ Xia iaa ia Aa0a ia iiX a+ a i+ ↵ji Xia ia 0aµ⌫ jabc A = dD dD +i== ↵ X A dD µ⌫ = µ⌫ [µ⌫] abc a a 1 A ↵ µ⌫ µ⌫ [µ⌫] i µ⌫ [µ⌫] A = dD + ↵ X 1 µ⌫ µ⌫ [µ⌫] i ja A C X X i A = C X X i +µcµa ↵i iX X[µ⌫] µ⌫ µ⌫ [µ⌫] µ⌫X [µ⌫] 1B ja µ⌫µ⌫= (ij) ⌫b µ⌫ (ij) µci dD ⌫b bja µ⌫ µ⌫ [ij] bµ⌫ µ⌫X= [ij] X µa X⌫⌫ Bµ⌫ X 11 bbBµ⌫ 11⌫ja ii N ja µ⌫ = 1 [ij] µa (20) ia jb ja i ja N ia jb i (20) B = X X i 1 ja µ⌫ = [ij] i⌫jXja ja Cbb(ij) =X X Xi µa B(ij) X X B = µa ⌫ (20) C =N X B X [ij] µ⌫ [ij] µ⌫ [ij] [ij] 0a abc µi X µa⌫X ⌫b µ⌫ [ij] µ B = b X jX⌫⌫ µa ⌫b 0a abc iX µ⌫ [ij] N µ⌫ [ij] µa ⌫ B = X X (20) µa A = C X j N N µ⌫ [ij] 0a abc i N Aµ⌫ = C(ij) Xµc µaX⌫b ⌫ (20) (20) (20)

Working out the total antisymmetric field tensor, one gets Za[µ⌫] = Aaµ⌫ ta Z[µ⌫] = AZ (19) µ⌫ ta = Aa t 0a ab [µ⌫] I S S N 2(19) 3 4 7 -‐ 3 4 8 7 z[µ⌫] = B0aµ⌫µ⌫1 a+ Aab µ⌫ ta + Cµ⌫Vtaotbl u m e 1 3 N u m b e r(19) 6 z[µ⌫] = Bµ⌫ 1 +=AB CAµ⌫0atta tb+ C ab t t µ⌫ ta1+ z[µ⌫] µ⌫ J+ o u rµ⌫n aa l o f µ⌫ A ad bv a n c e s i n P h y s i c s where where where a ia Aaaµ⌫ = dDµ⌫ + ↵i X[µ⌫] ia a = dD aa + ia a ia Aaµ⌫ ↵ X ii [µ⌫] Aµ⌫ µ⌫ µ⌫ [µ⌫] µ⌫ = dD µ⌫ + ↵i X[µ⌫] 1 i 1Bµ⌫ = ii 1bja X⌫jaja [ij] Xµa ja i Bµ⌫ b X X [ij] N µa ⌫ µ⌫ = [ij] (20) N B0aµ⌫ =µaNabcb⌫ [ij] iXµajX⌫ (20) (20) jj Xµc X⌫b abc ii C(ij) 0a = CA abc µ⌫ j 0a abc i A0a X= XC⌫b µ⌫ (ij) µc A = X X µ⌫ (ij) µc ⌫b (ij) iaµc jb⌫b µ⌫µ⌫ µ⌫ =X jb ia µ⌫ = C(ij) [ij] XµiaX⌫jb ia jb µ⌫ C(ij) X [ij] C µµ = ⌫⌫ [ij] X X [ij] (ij)

with

where

and

a

(ij)

µ

⌫

with with with one expands the symmetric abc Similarly, field strength abc cba Cia =f abc ia f abc + b[ij]strength dcba (21) Similarly, one expands the symmetric (ij) abc abc cbafield abc C(ij) =symmetric + b[ij] dabc (21) one expands the symmetric field strength (ij) C(ij) = ia f + b dcba (21) Similarly, Similarly, one expands the field strength (ij) [ij] (ij) [ij] (ij) (ij) Similarly,Similarly, one expands the symmetric field strength one the symmetric field strength Similarly, one expands the symmetric field strength iaexpands ia↵ ia ia↵ Similarly, one the strength Z xpands +tzhe =expands ( zX + ⇢symmetric gµ⌫ )t field (µ⌫) Z + X iaX↵+ (µ⌫) (µ⌫) ia s= ia↵ (µ⌫) Similarly, one Ze(µ⌫) symmetric f(µ⌫) ield trength 5 ↵↵ia↵)t Z + zX =+((⇢iiX X + ⇢⇢)tiiggaaµ⌫ )taa µ⌫ X (µ⌫) (µ⌫) = ( g X (µ⌫) 5 ia ia↵ µ⌫ (µ⌫) + z(µ⌫) ia ia↵ (µ⌫) ↵ 5 ia ia↵ 5 ia X ia↵)t ib↵ i↵ ia+ jb i↵ Z + z = ( X ⇢ g )t ia jb i µ⌫ a (µ⌫) Z + z = ( X + ⇢ g X Z(µ⌫) + z = ( X + ⇢ g X )t (µ⌫) ↵ Zz(µ⌫) = X ⇢gii⌫jb ga)t )t ++ (az[ij] [X ,(X u+ ,X ] ↵i↵, X ib↵ µ⌫ i (µ⌫) µ⌫ (µ⌫) ia ia↵ ↵ µ⌫ a (µ⌫) (µ⌫) [X ,, X ]][X +X↵u ggaµ⌫ (µ⌫) ↵[ij] µ⌫ ib↵] ↵[ij] ⌫+ ⌫ ⌫iaX ia (a jb i↵ ib↵[X Z(µ⌫) +(µ⌫) = ((µ⌫) X +[ij] ⇢]](µ⌫) g+ + (a X [X ] i[X µ⌫ a+ u (µ⌫) (µ⌫) ↵ (22) µ⌫ [ij] [ij] + (a [X , X + u g [X , X ] ⌫ ⌫ ↵j , Xib↵ (22) ia jb i↵ ib↵ µ⌫ [ij] [ij] ia jb i↵ ⌫ ⌫ ↵ ia jb i↵ ib↵ i j i ia jb i↵ ib↵ iu jg i]t] } j + b(a [X ,(a X ]][X + [X ,,i↵ X ]] X (22) µ⌫ [ij] [ij] [X , X ] + u g [X , X (a [X+ , X + u g [X X + , X , t ] + v g {X , }{t , ⌫⌫+ ⌫⌫})[t ↵ + (a , X ] + u g [X , X (22) µ⌫ [ij] [ij] b {X , X })[t , t ] + v g {X , X }{t , t } ia jb ib↵ µ⌫ [ij]{X [ij] a b µ⌫ a b (ij) (ij) ⌫ ⌫ ↵ µ⌫ [ij] [ij] i j i j ↵ µ ⌫ µ ⌫ a b µ⌫ a b (ij) (ij) ⌫ ⌫ ↵ ⌫ })[t µ , X⌫ }{t ,(22) i+ b ,jX{X + (a{X ]aµµ+ u] [ij] gvµ⌫ [X , Xvii(ij) ]gjjµ⌫}{t , X , t ] + {X t } (22) [ij] [X (22) a b a b (ij) ⌫ ⌫ ↵ (22) + b , X })[t , t + g {X , X , t } ⌫ µ ⌫ i j b µ⌫ a b (ij) (ij) µ ⌫j })[t iii, t j] jj+ v µ i, X i , X ⌫j }{t i,i t } jj + bb(ij) {X {X a bbj]t⌫+ a bbX })[t ]] + vv(ij) ggµ⌫ {X }} , b⌫ijX ,, tX v(ij) gtbbµ⌫ {X , iX }{t , ,t,X }⌫⌫}{t µ µ ⌫⌫ + bj(ij) {X X })[t + }{taa,,ttbb(22) + Xµii+ X t(ij) aa,, tg + t jXµ j{X a µ⌫ aµµ (ij) {X (ij) µ⌫ µµX (ij) a⌫⌫t})[t b{X i µ µ a b (ij) ⌫ ⌫ b(ij) X })[tX, t ]t + v(ij) gµ⌫ {Xµ , X⌫ }{ta , tb } +µj ,t(ij) X ++(ij) X{X ii X attb⌫ iiµ ajj⌫ ba b µ + X X a + + (ij) Xi ⌫⌫⌫j t(ij) t(ij) aj tbbXµµX⌫⌫taatbb (ij) Xµ µ + Xµ X We should split(ij) the Lagrangian in antisymmetric and symmetric Wenow should now split in antisymmetric and symmetric ⌫ ta tbthe Lagrangian and symmetric Wenow should now split the Lagrangian in antisymmetric We should split the Lagrangian in antisymmetric antisymmetric and symmetric symmetric parts. It gives, parts. It gives, We should now split the Lagrangian in and should the Lagrangian antisymmetric and WeLagrangian should split the Lagrangian in antisymmetric and symmetric symmetric We now split thesplit Lagrangian in antisymmetric and symmetric We should now should tWe he in now anow ntisymmetric and symmetric It in gives, parts. It gives, parts. Itsplit gives, We should now split the Lagrangian inparts. antisymmetric and symmetric parts. It gives, parts. It gives, parts. It gives, 1 1 parts. It gives, 2 i µj (23) L(D, Xi ) L(D, = LA X +i )L= mijL22SXµii , 1Xmµj (23) ij 2Xµi , X µj S LA1+ µj L(D, X ) = L + L m X , X (23) 1 2 2 i A S ij 1 1 µ 1 L(D, X ) = L + L m (23) 22 Xµ ii , X µj i A S ij 2 i µj 2 i µj µj L(D, X = L + m ,,2iX (23) 1L ii) A S ij L(D, X = L m ,,X (23) L(D, X = L + L m X X (23) µ L(D, X ) = L + L m X X (23) ii))L A SSX ij A2+ ijµjX 2 A S ij µ µ µ where where L(D, Xi ) = LA + LS 22 mij X22µ , X (23) where [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] 2 +z Z where =[µ⌫]1 Z (24) +[µ⌫]2Z z[µ⌫] (24) where AZ 2 [µ⌫] [µ⌫]z+ 3 z [µ⌫] z+[µ⌫]3 Z[µ⌫] z [µ⌫] [µ⌫] where LA = 1 ZL where where [µ⌫] [µ⌫] L = Z Z + z z + Z z (24) A 1 2 3 [µ⌫] [µ⌫] [µ⌫] where [µ⌫] [µ⌫] [µ⌫] L = Z Z + z z + Z z (24) [µ⌫] [µ⌫] [µ⌫] [µ⌫] z [µ⌫] + 3Z[µ⌫] [µ⌫] [µ⌫] A = 1Z [µ⌫]Z [µ⌫] + 2z[µ⌫] [µ⌫]z [µ⌫] [µ⌫] L (24) A 1 2 3 [µ⌫] [µ⌫] [µ⌫] L = Z Z + z z + Z z (24) A 1 2 3 [µ⌫] [µ⌫] [µ⌫] L = Z Z + z z + Z z (24) L = Z Z + z z + Z z (24) A 1 2 3 [µ⌫] [µ⌫] [µ⌫] A 1 2 3 [µ⌫] [µ⌫] [µ⌫] [µ⌫] and A [µ⌫] [µ⌫] and LA = 1 1[µ⌫] Z[µ⌫] Z [µ⌫] + 2 (µ⌫) + 3 3(µ⌫) Z[µ⌫] z [µ⌫] (24) 2 z[µ⌫] z and (µ⌫) (µ⌫) (µ⌫) (µ⌫) and LS Z = ⇠1 Z+ ⇠2 + z(µ⌫) z (µ⌫) z+ ⇠3 Z(µ⌫) z (µ⌫) (25) (25) ⇠2Zz(µ⌫) z+(µ⌫) ⇠3 Z (µ⌫) and and LS = ⇠1 Z(µ⌫) and and (µ⌫) (µ⌫) L = ⇠ Z + ⇠ z z + ⇠ (25) SZ 1 Z+ 2+ 3 Z(µ⌫) z(µ⌫) (µ⌫) (µ⌫) and (µ⌫) (µ⌫) (µ⌫) (µ⌫) (µ⌫) L = ⇠ Z ⇠ z z ⇠ Z z (25) (µ⌫) (µ⌫) (µ⌫) S = ⇠ 1ZL 2 z(µ⌫) 3z (µ⌫) (µ⌫) (25) (µ⌫)= (µ⌫)+ (µ⌫)+ (µ⌫) (µ⌫) (µ⌫) ⇠ Z Z ⇠ z ⇠ Z z (25) L Z + ⇠ z + ⇠ Z z + ⇠ Z z (25) S 1 2 3 (µ⌫) (µ⌫) (µ⌫) S 1 2 3 (µ⌫) (µ⌫) (µ⌫) L = ⇠ Z Z + ⇠ z z + ⇠ Z z (25) S 1 2 3 (µ⌫) (µ⌫) (µ⌫) L = ⇠ Z Z + ⇠ z z + ⇠ Z z (25) S 1 2 3 (µ⌫) (µ⌫) (µ⌫) It yields the following equations of motion: (µ⌫) (µ⌫) (µ⌫) S 1 (µ⌫) 2 motion: It yields the following equations of LS =the ⇠1 Zfollowing + ⇠2(µ⌫) z(µ⌫) z of motion: +3⇠3(µ⌫) Z(µ⌫) z (25) (µ⌫) Z It yields equations It yields the following equations of motion: ⇣following ⌘ It yields the following equations of motion: motion: It yields equations of ⇣following ⌘[µ⌫] It yields It yields the following ethe quations of the motion: It It yields following equations ofofmotion: motion: g of g equations ⇣4d ⌘ [µ⌫] b [µ⌫] ib yieldsthe the following equations motion: [µ⌫] b4i ib ⇣ ⌘ @ Z t + (d D + ↵ X )Z [t , t ] ⌫4i [µ⌫] a D⌫ +g↵i X⌫ ⌫ i[ta⌫,ibtb ] ⌘[µ⌫] a b ⌘ @⌫ Z 1[µ⌫] ta + (d )Z ⇣ 4d b 1⇣ g ⌘ ⇣ ⌘ ⇣4d ⌘ b ib [µ⌫] N @⌫⇣4i Z[µ⌫] ta D +b4i+gg ↵(dXD + ↵i[tX⌘ibib [ta , tb ] gg (d 1 t + N ⌫, t)Z b⌫)Z [µ⌫] Z[µ⌫] ⇣ @@⌫⌫⇣Z ⌘[t [µ⌫] [µ⌫] [µ⌫] aib b ] [µ⌫] 1 4d iX ⌘⌫⌫ib ibb⌫b)Z [µ⌫] ⌫b +N↵ 4d Z t(d + 4i (d D +[µ⌫] ↵[µ⌫] X )Z ttaaa@@+ 4i D [t ,, )Z tt)Z aib iiX [taaa,,,tttbbb]]] g(d bb]] 11 4d ii⌫) ⌫ 4d Z t 4i (d D + ↵ [t ⌫)a a+ ⌫ (µ⌫) µib 4d4d @⌫@⇣Z111[µ⌫] + 4i D + ↵ X )Z [t ⇣ ⌘ ⌫ ⌫ N ⌫⌫⌫(µ⌫) a i [µ⌫] b ib ⌫ ⌫ a ib µib ⌫ ⌫ N N 4⇠1 iZ X⇢⌫ibi(d Z ⌘X[tµib tag⇣ D⌫N , at,b ]tb ] i+ iiX 1 1 i g ⇣⌫ Z 4⇠ X(µ⌫) Z+(⌫ ⇢⇢↵ ,Z ⇣+ 4i ⌘[ta[t N ⌫ a)Z b ] ⌫)⌘ ⌫)i ⌘ ⇣ ⇣i X ⌘ µib ib (µ⌫) i X µib + 4⇠⌫ib iZg(µ⌫) Z X Ztt(⌫ [ta , tb ] N ⌫) 1⇣ ⌫) ib (µ⌫) ⌫ ⌘(26) ⌫) (⌫ µib (26) ib (µ⌫) µib 4⇠ i g X + ⇢ X Z [t , ] ⌘ ⌫) ib (µ⌫) µibaZ 1⇣ 4⇠ X Z +(⌫ ⇢g⌫) X [t (µ⌫)iiX µibZ ⌫ib (⌫⇢ 4⇠ ii gg iii4⇠ X +⌫⌫[µ⌫] ,,btt(⌫ [taaaib,,,tttbb⌘ ] 11 iiiX a bbb]] ii⇣Z ggib2d + Z [t b]][µ⌫] gZ ⌫)[t ⌘ ⌫⇢ 4⇠4⇠ X Z ⇢ X Z [t (26) 11⌫1 i+ iiiX (µ⌫) µib (⌫ a (⌫ [µ⌫] b ib [µ⌫] ⌫ (⌫ + @ z t + 2i (d D + ↵ X )z [t , t ] ⇣ ⌘ g (26) +g ⇢(diaXD⌫ + Z(⌫↵i X⌫[t⌫)z i a ,⌫ib ⇣ ⌫taZ+⌫2i[µ⌫] ⌘⌘(26) 1 i g@⌫ zi3X + 3⇣ 2d tb)z ]⌘ (26) ba , tb ] [t [µ⌫] a b ⌘ (26) ⇣ (26) N g ⇣ ⌘ + 2d @ z t + 2i (d D + ↵ X [t , t ] (26) [µ⌫] b ib [µ⌫] gg (da Db +g ↵ X ibbbb⌫)z[µ⌫]⌘[t 3 t + i ,ib a b [µ⌫] [µ⌫] N ⌫ ⇣⌫⌫2i [µ⌫] ibt)z [µ⌫] ⇣⇣ ⌘[t + 33 2d 2d @⌫⌫ zz3[µ⌫] 2i (26) [µ⌫] ib a@ iX a⌫ b]] [µ⌫] + 2d z t + 2i (d D + ↵ X , t ] ⌘ [µ⌫]2d [µ⌫] ⌫b +N↵ ⌫ib⌫)z + @ t + (d D [t , t a i a b X )z [t , t ] g a i a b ⌫) 3 ⌫ a i a b + @ z t + 2i (d D + ↵ X )z [t , t ] ⌫ ⌫ ib (µ⌫) µib ⌫ ⌫ ⇣⌫ + ⌘i[t , t ] ++3 2d2d ta(µ⌫) 2i DD + ↵ XX 3 ib[µ⌫] a b N (d bN ib [µ⌫] ⌫) i↵ µib ⌫ ⌫)z za(µ⌫) ⇢+ N ⇣@⌫@iz2⇠ ⌘X t+ 2i (d )z=⌘ iX i⌘ N ⇣ ⌘ 0[ta[t⌫aa, ,btbt]b ]= 0 ⌫ z3⌫izg ⇣ a + ia , z 2⇠3 i3 g ⇣ X ⇢⌫ibiN X z⌫+ [tµib t⌫(⌫ N b ] ⌫) (⌫⌫ ⌫) ⇣ ⌘ 2⇠ i g X z + ⇢ X z [t , t ] = 0 ib (µ⌫) µib ⌫) 3 i i a b ib (µ⌫) µib ⌫) (⌫]⌫) µib z (µ⌫) µib ⌘ 2⇠33ii gg ⇣2⇠ X33⌫iib + ⇢ib⌫iizX X(µ⌫) [t = 00[t µib X⇢ +(⌫⇢⇢⌫) ,t ] = ⌫)i X[t a,,ztt(⌫ ibg (µ⌫) ii+ µibz 2⇠ zz(µ⌫) = 000 ⌫ib iiX bb]] = X [t ⌫ izX µib ⌫) [ta 2⇠2⇠ z+(⌫ ,zt(⌫(⌫ 0[taaa,,ttbbb]] = 3⌫i⌫ gzib (µ⌫) i+ iiX ⌫⇢ 3 i gi g 2⇠ iX a[t bt = (⌫ X z + ⇢ X z , 3 i ⌫ i a b] = 0 (⌫

for Dµ -‐field. And,

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for X µ -‐fields. for

[µ⌫] b [µ⌫] b [µ⌫] [µ⌫] 4 [µ⌫] 4 1 ↵i @⌫ Z +⌫ Z i g[µ⌫] Dbt⌫aZ+ [taD, ⌫tb bZ] [µ⌫] [ta , tb ] 1 ↵ti a @ [µ⌫]i g 4 1 ↵i @⌫ Z4 1 ↵tia + , t⌫bZ ] @⌫iZg D⌫jb tZ gaD [ta , tbjb] [µ⌫] jb [µ⌫] a + i[t jb µ⌫ + Xjb 2a [t , t ] + (2b + 3 2a(ij) [taX , t⌫jbb ]Z+µ⌫ (2b + )X {ta⌫jb , tZb }[µ⌫] {ta , tb } 3⌫ Z a b [ij]jb [ij] )X [ij] [ij] µ⌫(ij) ⌫+Z[µ⌫] + 3 2a(ij) [taX , t⌫b ] Z +µ⌫(2b )X Z [ij]!{t +X⌫3 Z2a(ij) [ta[ij] , tb+ ] + [ij] (2b + )X a ,⌫ tbZ} !{ta , tb } ⌫ [ij] (µ⌫) µ t ⌫)⇢ @ µ Z ⌫) !ta ! ⇢(µ⌫) a ⌫) tai µ (⌫ ⌫) i @⌫ Z(µ⌫) tai @⌫ Z i @µ Z (µ⌫) ⇣ ⇣ tai @⌫ Z ⇢i @ Zta(⌫(⌫⌫) ⌘ t⇢ai @ Z(⌫⌫) ⌘ta + 4⇠⇣1 i @⌫bZ (µ⌫) + 4⇠1 ⌘ µb ⇣ D⇢b iZD(µ⌫) + 4⇠1 ⇢ D[tµb +i g 1 i D+i , t(⌫b ]⌫) ⌘ [ta , tb ] + 4⇠ aZ ⌫b (µ⌫) b⌫ Zg(µ⌫) i+ µb Z+ (⌫⌫) i [t µb +i g D Z + ⇢ D Z , t] i i a +i g D Z + ⇢ D Z [t⌘a1, tb ] ⌫ (⌫ i i ⌘ b(⌫ 1 0 ⇣ 0 ⇣ ⌫ ⌫)⌘ µjb ⌫) ⌘1 0 ⇣a 0 jb ⇣(µ⌫) jb (µ⌫)µjb 1 X Z + u X Z [t , t ] a X Z + u X Z [t , t ] ⌫) b ⌫) [ij] jb a b [ij] ⌫jb [ij](µ⌫) [ij] µjba (⌫ ⌫ (µ⌫) µjb (⌫ a X Z + u X Z [t , t ] @ ⇣ ⌘ A @ ⇣ ⌘ a b [ij] [ij] + u(⌫[ij]⌫)X⌘ Z(⌫ ⌫) A[ta , tb ] A + 2⇠ +⇣2⇠3 ⌫ a[ij] X⌫ Z jb ⇣(µ⌫) µjb jb (µ⌫) µjb , t } ⌘ + 2⇠33@ + @ + 2⇠ b(ij) X Z + v X b(ij) X⌫jb(ij) Z (µ⌫)µjb+Zv(⌫ {t , t } A ⌫) X {taZ 3 (ij) ⌫+ (µ⌫) + b(ij) X⌫jb Z + v X Z {t , t(⌫bb}⌫) {taa , tbb } + b(ij) X⌫ (ij) Z + v(⌫(ij) X µjbaZ (27) jb [µ⌫] jb (⌫ [µ⌫] jb [µ⌫] jb [µ⌫] (27) + 2 2 2a+ [tX ] + (2b + )X z {t 2a [t , t ] + (2b + )X a ,⌫tb z a⌫, tbz} {ta , tb } (ij)2Xjb [ij] [ij] [µ⌫] jb [µ⌫] 2⌫ z a b (ij) [ij] [ij] ⌫ [µ⌫] + 2 2 2a(ij) z [tX, tjbb ]z+ (2b z {ta ,jbtbz}[µ⌫] {ta , tb } + 2X⌫[µ⌫] [ta[ij] , tb+ ] + [ij] (2b)X 2 2a(ij) a [ij]⌫ + [ij] )X ⌫ b [µ⌫] ⌫ + 3 2↵i+ (@⌫ z3[µ⌫]2↵tai (@ +⌫ izg[µ⌫] Dbt⌫az[µ⌫] [t , t⌫bbz])[µ⌫] [ta , tb ]) + i g aD [µ⌫] bb ])[µ⌫]⌘ + 3 2↵ z t + i g D z [t , t 0 i (@ ⇣ 1 ⌫ a a 0 ⇣ ⌘ 1 2↵i (@⌫ z ⌫ta + i g D⌫ z ⌫)⌘[ta , tb ])⌫) 1 0 +⇣ a3 0 jb ⇣(µ⌫) µjb jb (µ⌫) µjb ⌘ 1 X z + u X z [t , t ] a[ij] X⌫jb[ij] z (µ⌫)µjb+ u(⌫[ij] ⌫) X a z(⌫ b ⌫) [ta , tb ] [ij] jb (µ⌫) µjb X⌫⌫ z⇣ + u X z [t t @ +⇣a ⌘ A a, z b] [ij]@ [ij] ⌘ A + 4⇠2@ (⌫ a X z + u X [t , t ] 4⇠ [ij] ⌫ ⇣ b 2 @X jb z⇣(µ⌫) Aa b A µjb [ij]⌫)⌘ µjb (⌫ ⌫) ⌘ + 4⇠2 jb (µ⌫) + + v X z {t , t } + 4⇠ ⌫) + b X z + v X z {t , t } a b (ij) (ij) 2 ⌫ (µ⌫) a b (ij) v ⌫ µjb + b(ij) X⌫jb X µjb+zv(⌫(⌫(ij) X{t a ,zt(⌫ b }⌫) {t , t } +z b(ij)+ X⌫jb(ij) z (µ⌫) a !b (ij) ! (⌫ ⌫) (µ⌫) µ ⌫) (µ⌫) µ !t t ⌫) i @⌫ z(µ⌫) ta i @⌫ ⇢ i @µ z a z t ⇢ @ z ⇣ i @⌫ z ⇣ ta @ ⇢zi(µ⌫) @ zt(⌫a(⌫⌫) ⌘ ta⇢i @ µ z(⌫ ⌫)⌘ta= 0 ! = 0 + 2⇠3 + 2⇠ ⇣ ⌘ i ⌫ a 3 b (µ⌫) µb (⌫⌫) a 0 (µ⌫) + 2⇠3 ⇣ + +i g 3 i D+i ⇢⌫bi D [ti µb ⌫)⇢i D a ,ztb ] ⌘ = g D z + b⌫ z(µ⌫) µb z(⌫ + 2⇠ = 0 i a , tb ] (⌫ +i g i D+i +D ⇢ibD z(⌫+ ⇢ D [taµb, ztb ] ⌫) [t (µ⌫) ⌫z g z [t , t ] i i a b ⌫ (⌫

Xµia -fields. for Xµia -fields. for Xµia -fields. ia a X -fields. Takingforthe trace in above equations, one gets for Dgets µ Taking the trace in D above equations, oneD for Dµaa aµ a Taking trace in above equations, one gets for Taking the trace in athe bove e quations, o ne g ets f or µ µ Taking the trace in above equations, one gets for Dµ [µ⌫]a [µ⌫]c ib [µ⌫]a [µ⌫]c 2 d @ Z 1 d+@⌫gZfabc (d+Dbg⌫b fabc↵(d Xib⌫ibb)Z[µ⌫]c ⌫b⌘ ↵i X⌫ib )Z [µ⌫]c 2 11 d @⌫⌫Z⇣2[µ⌫]a + g⇣ fabc (d D⌫ ↵iiXD )Z [µ⌫]a ⌫ ⌘ Z + g fµib abc (d D ⌫)c⌫⌘ ↵i X⌫)c ⇣2 1Xdib@Z⌫(µ⌫)c ⌫ )Z ib⇢i X (µ⌫)c µib + 2⇠1 g fabc + Z ⇣ ⌘ ⌫)c i ib (µ⌫)c µib ⌫ + 2⇠ g f X Z + ⇢ X Z (⌫ abc i +⌫⇢i X + 2⇠1 g fabc i1X⌫ Z (µ⌫)cZ(⌫ i µib (⌫ ⌫)c + [µ⌫]a 2⇠1 g fabc i X⌫ib Z + ⇢ X Z i [µ⌫]c (⌫ib [µ⌫]c + d @ z[µ⌫]a + g zf[µ⌫]a Db ↵(d Xib⌫ibb)z[µ⌫]c abc (d+ ↵i X⌫ib )z [µ⌫]c ⌫ + 33 d @⌫⌫+ z⇣ 3 d+@g⌫ f⇣ (d D⌫bg⌫ fabc ↵iiXD )z ⌘ [µ⌫]a b abc ⌫ ⌘ + z + g fµib ↵i X⌫)c ⌫)c⌫⌘ 3 d abc (d D ⇣ ib@⌫(µ⌫)c ⌫ )z ib⇢i X (µ⌫)c µib + ⇠3 g fabc+ ⇠3i X z + z = 0 ⇣ ⌫)c ib (µ⌫)c µib ⌫ X⌫⇢ zX z+(⌫⇢i X µib z0(⌫ ⌫)c ⌘ = 0 i+ + ⇠3 g fabc i Xg⌫fabc z (⌫ ⇢ X = z + ⇠3 g fabc i X⌫ib iz (µ⌫)c + =0 i (⌫

(28) (28)

ia

and for X µ

7 7

7 7


(27) (27)

(28) (28)


ia and ia for X µia and for X and for X ia µ µ and for Xµ

Zf[µ⌫]a g [µ⌫]c fabc D⌫bbZ [µ⌫]c [µ⌫]a [µ⌫]aD b Z [µ⌫]c 1 ↵i @⌫ g 2 1 ↵i @⌫22Z[µ⌫]a ↵ @ Z g abc b [µ⌫]c ⌫ ⌫ fabc D⌫ Z ◆ 2 1 ↵i ✓@⌫ Z 1 i✓ ✓ g fabc D⌫ Z ◆ jb [µ⌫]c 1◆ ✓ + ◆ 1 1 i a f + b d + jbdabc [µ⌫]cX⌫jbZ [µ⌫]c 3 abc abc [ij] [ij] [ij] 1 d[ij] + i a+ Zabc a b[ij]fd abc++b[ij] abc2 X [ij] f3abc i+ jb [µ⌫]c X⌫ Z abc abcd+ [ij] ⌫d + 33 i a[ij] fabc + b[ij] [ij] dabc + 2 [ij] dabc 2 X⌫ Z ! 2(µ⌫)a ⌫)a µ ⌫)a ! ! (µ⌫)ai @⌫ Z (µ⌫)a µ + ⇢i @ µZ(⌫ ⌫)a ! + ⇢Zi @µ Z(⌫+ @ ⌘ ⌫)a⇢i @ Z(⌫ i @⌫ Z(µ⌫)a ⇣ i ⌫ 2⇠ ⌘ ⇣ ⇢i @b Z(µ⌫)c @⌫ Z + ⌫)c ⌘ 2⇠ (⌫ 2⇠11 i⇣⇣ ⌘ µb ⌫)c +gDfbabc (µ⌫)ci D⌫bZ (µ⌫)c µb + ⇢ iD µbZ(⌫ ⌫)c 2⇠11 +g fabc Z(µ⌫)c + ⇢ D Z +g D Z + ⇢ D Z ⌫)c i fb⌫abc i i ⌫ (⌫ i (⌫ +g fabc + ⇢i Dµb Z(⌫ i D⌫ Z ! ⌫)c ! ! µjb ! Z(⌫ ⌫)c ⌫)c µjb (i+u[ij] fabc +)X ⌫(ij) µjbdabc )X (i+u⇠[ij] f ⌫ d Z (i u f + ⌫ d )X Z ⌫)c abc abc (ij)abc (ij) µjb abc (⌫ (µ⌫)c (i+u ⇠33 fabc + ⌫a[ij] )X Z (⌫abc +⇠ (ij) d +(i fabc +)X b(ij) )X⌫jb jb d(⌫ (µ⌫)c jbZ (µ⌫)c abc + ⇠33 +(i[ij] a[ij] fabc b[ij] d Z +(i+a[ij] f + b d )X Z abc (ij) abc abc (ij) jb (µ⌫)c ⌫ ⌫ +(i a[ij] fabc + b(ij) dabc )X⌫ Z (29) jb [µ⌫]c (29) (29) + 2ia f + 2b d + X jb d[µ⌫]c jbz [µ⌫]c 2 abc abc abc (ij) [ij] [ij] ⌫ (29) + 2ia+ + (ij) 2b[ij] dabc fabc ++ 2b[ij] ddabc abc+ X[ij] (ij) f2abc2ia [ij] jb abc X⌫ z ⌫ zd[µ⌫]c + 22 2ia(ij) fabc + 2b d + d X z [ij] abc [µ⌫]a[ij] abc ⌫ b [µ⌫]c @⌫ z [µ⌫]aD+b gf [µ⌫]a [µ⌫]c 3 ↵i + abc D⌫b z [µ⌫]c + 3 ↵i @+ + gf ⌫ z[µ⌫]a b⌫ z [µ⌫]c 3 ↵i @gf ⌫ z abc + abc D⌫ z ! + 3 ↵i @⌫ z + gfabc D⌫ z ! jb!(µ⌫)c (ia f + b d )X z ) ! jb (µ⌫)c jb (µ⌫)c abc [ij] (ij)z abc ) ⌫ (ia f + b d )X (ia f + b d )X z ) abc abc [ij] (ij) abc abc [ij] (ij) jb (µ⌫)c ⌫ ⌫ + 2⇠ ⌫)c 2 fabc + b(ij) dabc )X z (ia2⇠[ij] ))X µjb + 2⇠ + ⌫)c +(iu ffabc v⌫v(ij) 2 µjbdabc⌫)c + 2⇠22 +(iu[ij] fabc + [ij] v[ij] dabc+ zabc +(iu +)X )X µjbzz(⌫ ⌫)c (ij) abc (ij) µjb d (⌫ (⌫ +(iu[ij] fabc + v(ij) dabc )X z(⌫ ! ⌫)a (µ⌫)a ⌫)a µ ! ! ⌫)a (µ⌫)a i @⌫ z (µ⌫)a µ + ⇢i @ µz(⌫ ! @ z + ⇢ @ z @ z + ⇢ @ z ⇣ ⌘ ⌫)a i ⌫ i i ⌫ i (µ⌫)a µ (⌫ (⌫ ⇠⇠3 i⇣@⌫ z ⌘ µb ⌫)c ⌘ ⇣ ⇢i @b z(µ⌫)c + ⇠ (⌫ 3 ⇣ +gfbabc(µ⌫)ci D ⇢⇢i⌘ D µb + ⌫)c bz (µ⌫)c µbz(⌫ ⌫)c ⌫⇢ ⇠33 +gfabc D z + D z +gf D z + (µ⌫)c i ⌫ i µb (⌫⌫)ci D z(⌫ +gfabc iiD⌫b⌫zabc + ⇢i D z(⌫ = = 0. = 0. 0. = 0. Multiplying the equations of motion by t and taking the correk Multiplying the equations of taking motion t andbytaking again theagain correMultiplying thetk equations ofby motion tk and taking again thefollowing correMultiplying the equations of motion by and again trace, we finally have the Multiplying the equations of motion bythe tkkcorresponding and taking again the corresponding trace, we finally have the following equations of motion sponding finally the following trace, wetrace, finallywe have the have following equationsequations of motionof motion equations of msponding otion sponding trace, we finally have the following equations of motion ◆ ✓ [µ⌫]a ✓ ◆ ✓ ◆ if[µ⌫]a aek aek )@ ✓ d(d )@⌫⌫Z Z [µ⌫]a+ + ◆ [µ⌫]e d(daekb d(d if )@ aek aek+ aek ⌫ Zif [µ⌫]a 1 ib d(daekib⌫b +if )@ fabc dcek )Z [µ⌫]e aek ⌫ Z abc f+ 1 ib)(if [µ⌫]e iX g(dD + ↵↵abc fcek g(dD1b⌫b +g(dD ↵i Xib⌫ ⌫)(if f⌫cek dcek iX abc abc dcek )Z ! 1 abc cek )Zf[µ⌫]e ⌫ )(iff g(dD⌫ + ↵i X⌫ )(ifabc fcek fabc dcek )Z µib ! ⌫)e ! ⌫)e !Z g⇢ if )X µib µibdcek ⌫)e i (f abc d cek abc g⇢ (f d if d )X Z(⌫ g⇢ (f d if d )X Z + ⇠⇠1 abcd cek i if abcabccekcek abc cek µib (⌫ (⌫⌫)e ib (µ⌫)e +ii(f d )X Z + ⇠1 g⇢ 1abc +g cek i (fabc abcdcek cek if ib d (µ⌫)e cek )X ⌫ibZ (µ⌫)e + ⇠1 dcek ifabc d(µ⌫)e +g i (f✓ dcek i (fabc ifabc dcek )X Z(⌫ cek )X⌫ Z abc+g ib ⌫abc ◆ +g (f d if d )X Z i abc abc cek [µ⌫]a ⌫ ✓ cek ◆ ✓ ◆ (30) d(d if 3 [µ⌫]a aek aek )@ ⌫ zz [µ⌫]a ◆ ✓ (30) d(d if )@ (30) d(d if )@ z 3 3 + aek aek ⌫ aekb aek ⌫ [µ⌫]a ib z [µ⌫]e (30) + 2 b +g(dD d(daekib⌫b +if )@ + 3 ↵ X f f d )z aek ⌫ ib)(if [µ⌫]e [µ⌫]e i abc cek abc cek ⌫ +g(dD + ↵abci X fdcek fabc dcek + 2 +g(dD ↵ X )(if fcek 2 + abc cek )z[µ⌫]e ⌫ )(iff ! )z +g(dD⌫b⌫ + ↵iiX⌫ib⌫ ⌫)(ifabc fcek fabc 2 !)z ⌫)e ! abc dcek µib ⇠⇠3 g⇢ ddcek if )X µibz(⌫ ⌫)e µibdcek⌫)e ⇠3 g⇢ii(f (fabc ifabc )X! z g⇢ (f3 abc dcek ifabc dcek )X z(⌫ ⌫)e + abc cek abc d cek µib ib (⌫ (µ⌫)e +ii(f g⇢ d if d )X z + ⇠3 abc +g cek i (fabc abcdcek cek 2 if d )X (⌫ ib ib d (µ⌫)e cek )X⌫ zz (µ⌫)e + 2 dcek ifabc +g 2i (fabc+g dcek i (fabc ifabc dcek )X z(µ⌫)e abc cek ⌫ ib ⌫ 2 +g i (fabc dcek ifabc dcek )X⌫ z = = 00 =0 =0 a for Dµ . And,

8 8

88


Dµa . for aDµafor . And, for Dµa. And, for✓D Dµaµa.. for for Dµ . And,


And, ✓ ◆ And, And, [µ⌫]e◆ [µ⌫]e ↵ (d if )@ Z i aek aek ⌫ ✓ ◆ ifaek )@⌫[µ⌫]e Z ✓↵ (d ◆ ✓1 ◆b Z [µ⌫]e ◆ 1↵✓(di aek if [µ⌫]e )@ [µ⌫]e i +g↵ aek i (if aek ⌫ Zif bdcek [µ⌫]e [µ⌫]e ↵ (d if )@ Z f f )D ii (d aek aek ⌫ abc cek abc ↵ )@ Z ⌫ +g↵ (if f f d )D Z ↵ (d if )@ Z aekabc⌫ cekaek ⌫ ii ✓ abc cek aek aek 1 b ⌫[µ⌫]e ◆ b [µ⌫]e [µ⌫]e +g↵ fceki(if ✓i11(if ◆ 1 abc abc cek )D +g↵ (iffabc fdcek f⌫dabc d[µ⌫]e )D bZd abc cek abc cek)D +g↵ f f +g↵ (if f f d )D Z a (if f)abc⌫b⌫ Z fZcek ) i cek 3 i abc cek abc cek abcf⌫abc cekf+ (ij) ✓3 + ◆ a (if d + ✓ ◆ X jb Z [µ⌫]e abc cek cek (ij) jb ◆ ◆) jb ⌫ +3 ✓ X Z [µ⌫]e a✓ (if1abc d+ +[ij] fabc fcek ) cek cek1(if abc cek+ (ij)+(b a (if d + f f ⌫ )[µ⌫]e 3 )(d d id f abc cek (ij) 2 jb [µ⌫]e abc abc cek [ij] a d f f ) )(d fcekcek ) X⌫jbZ [µ⌫]eX⌫jb 3a d(ij) + abc fdabc fcekid ) abcabc + 32 + cekcek [ij] [ij] abc cek21abc (ij)+ ++(b Z [µ⌫]e 1 1(if 2 0 X Z + 2 0+(b X Z + +(b )(d d id f ) 1 1 ⌫ abc cek abc cek [ij] [ij] +(b + )(d d id f ) 1 ⌫ 2 abc cek abcfcek cek ) [ij] [ij])(dabc 2 +abc dcek ⌫)e )(d dcek id(µ⌫)e fcek )⇢idi (if abc [ij] 2 0 +(b2(if (µ⌫)e cek abc [ij]0+ 2i (if [ij] 1µ Z(⌫ ⌫)e 1 )@⌫+ Z + )@ 22d 1 aek aek)@ µd aek d[ij] Z ⇢ (if d Z aek cek )@⌫(µ⌫)e i aek aek ⌫)e 0 (ifi 0 1 ⌫)e C µ (⌫ (µ⌫)e µ B di (if ⌫)e )@ Z + ⇢ (if d )@ Z ⌫)e b (µ⌫)e (µ⌫)e µ B+ C d )@ Z + ⇢ (if d )@ Z i ⇠ aek cek ⌫ i aek aek (µ⌫)e µ aek aek aek )@ Z(⌫ bdaek (⌫ (if )DZ (if d cek )@ Zffiabc +fabc ⇢iid(if daek @ A Z + (if )@ icek abc cekdaek id ⌫⌫⇢ aek +B ⇠1 @ i (if1aek (if Z (µ⌫)e ⌫Z cekaek ⌫+g aek C A(⌫ i)@ abc fcek aek )D B+g+g C b ⌫(µ⌫)e b (⌫ (µ⌫)e B C ⌫)e + ⇠1 @ (if f f d )D Z B + C A b (µ⌫)e µb + ⇠ +g (if f f d )D Z i abc cek abc aek ⌫)e b (µ⌫)e @ A ⌫ i(if abc cek abc aek)D µb ⌫ ⇠11 @+g+g⇢ +g (if f f d )D Z +g⇢ f f d Z A + ⇠1 @ (if f f d )D Z abc cek abc aek cek abc fiicek abc fabc dcek )D Z ⌫ (⌫ ⌫)eA i i (if abcabccek abc aek ⌫ cek ⌫)e µb (⌫ µb ⌫)e fceki(if )Dffabc +g⇢ (iffabc fdcek d(⌫cek⌫)e )Dµb Z Z⌫)k 0+g⇢ 1 i (ifabc abc µbZd 0 (⌫1 +g⇢ )D 1ja 1µja +g⇢ f(µ⌫)k f(µ⌫)k dcek )D µja i ja abc abcZ cek ⌫)k i (if abc cek abc cek 1 1fcek (⌫ (⌫ b X Z + v X Z 0 1 (ij) (ij) 0 1 b X Z + v X Z ⌫ N ⌫)k 10 jaN µja 1 (⌫ µja ◆ (⌫ ⌫)k N (ij) 1⌫ (µ⌫)k + 1 ◆ ⌫)k ja 1 v (µ⌫)k 0 ✓bN(ij)(ij) X⌫✓ Z X 1 1Z ⌫)k C b X Z + v X ja N (µ⌫)k µja Z(⌫1 (ij) 1 jaN (µ⌫)k µja (ij) (ij) B ✓N1B C (⌫ ⌫ ba(ij) Z +fN vf(⌫cekX)jb ◆(µ⌫)e Z(⌫CjbC(µ⌫)e C (if dvcek + ◆ X + Z ✓ abc abc [ij] dX 1 ✓ ⌫+ (ij)fX NZabc cek cek )N ⌫(if Nfabc B B +✓1NB ◆X Bba(ij) C +a[ij] Z ◆ (ij) X Z daacek + f f ) ⌫jbC (if d + f f ) 21 abc abc cek [ij] (if B ⌫ B C 1 ⇠23 B C abc cek abc cek + [ij] jb (µ⌫)e B C (µ⌫)e C +b (d d id f ) +B ⇠B (if + f f ) abc cek abc cek (ij) +b (d d id f ) 3 a (if d + f f ) + X Z abc cek abc cek [ij] 1 B C + X Z abc cek abc cek jb (µ⌫)e (ij) abc cek abc cek [ij] B C ◆ XC B ++21 ⇠✓ C ⌫jb (µ⌫)e B C ⌫ Z ◆ 2✓ + ⇠3 B + X Z +b (d d id f ) @ A +b (d d id f ) B C 3 ⌫ abc cek abc cek (ij) B C @ A 2 C abc cek abc cek (ij) ⌫ C 2 ⇠3 B u (if(d dabc + f)◆ fcek )µjb)◆ + ⇠3 @ +b dcek idabc fcek ✓(d ◆ ⌫)e ⌫)e C abc cek abc [ij] u (if d + f f +b d id abc (ij) µjb B abc cek cek [ij] abc cek (ij) B +✓ C A ✓ @ A ✓ @ ◆ fcek X AZ(⌫ ⌫)e A X Z u[ij]+(ifabc+v duucek + f f ) (if d + f ) ⌫)e abc cek abc cek abc [ij](if @ ++ µjb (⌫ µjb (d id f ) d + f f ) abc cek abc cek (ij) ⌫)e (d d id f ) u+v (if d + f f ) X Z abc cek abc cek [ij] + X Z ⌫)e µjb cek abcabc cekcek abc abc (⌫ +(ij) X Z(⌫ 0[ij] 1 (dabc dcek fcek + +v Xfµjb +v (did dcek idabc )(⌫ 0 1 cek ) id (ij) abcabc cekZ (ij)(d (⌫ [µ⌫]k 1d ja +v ) 1 ja f[µ⌫]k +v dcek id f ) abc cek abc cek (ij) abc cek 0 1 (ij) (dabc (2b + )X z 0 1 (2b + )X z [ij] [ij] ⌫ [ij] [ij] N 1 ⌫ ✓ ◆ ja [µ⌫]k 1 1 ja [µ⌫]k N ✓0 ◆ 0 1 (31) (2b + )X z (2b + )X z 1 ja [µ⌫]k [ij] [ij] [ij] [ij] 1 @ A ja [µ⌫]k ⌫ ⌫ N (2b N ◆z) aabc (if f)abc ✓a(ij) ◆ jbA [µ⌫]e (2b + +[ij] )Xf⌫cek + 2 @+✓ (if[ij] d+cek + fdabc abc)X cek (ij) (31) A [ij] N [ij] jb ◆ [µ⌫]e ⌫ fzcek N A⌫jbz [µ⌫]e @ +a✓ ✓ 22 @ ◆) jb X + X z + 2@ (if d + f f ) (31)A + a d + f f 1(if abc cek abc cek (ij) abc cek abc cek (ij) 1 ⌫ [µ⌫]e @ A +(b )(d d)cek ) X⌫jb z [µ⌫]e ad+ dcek fabc fid )Xf⌫jbcek )(d id fcek )abc [ij] + 2 ++ 2 +(b +[ij] fdabc fabc z [µ⌫]e abc cek (ij) +a[ij] abc cek abc [ij] abc cek21(if cek+ (ij)+ 1(if 2 + X z )(d+ fcek ) idabc +(b[ij] +abc12dcek )(did dcek ✓ [ij] + +(b ◆ff⌫cek + +(b X z) ⌫ ✓ abcabc [ij])(d 21 [ij] [µ⌫]e [µ⌫]e abc cek + )(d dz[ij] id d f◆ [ij] ✓↵[ij] ◆ cek ) ✓3 ◆ abc cek cek )idabc [ij] 2 if ↵i (d 3+(b (d if )@ 2 aek aek )@⌫ zabc i aek aek ⌫ [µ⌫]e [µ⌫]e ✓ ◆ ◆ b [µ⌫]e ↵ifi(d (d if )@ z[µ⌫]e +3 ✓ + 3↵i (daek ↵ )@⌫ z if aek aek)@ aek bd ⌫[µ⌫]e [µ⌫]e (if faek 23 ↵ii(if ++g↵ f)@ )D + 32 + i abc aek ⌫ z )D⌫b z [µ⌫]e (d+g↵ if zcek cek abc ficek abc aek aek ⌫d ⌫ zcek babc[µ⌫]e +g↵ (iffabc f⌫abc fceki(if + 2 0+g↵2i (if cek cek )Db⌫ z[µ⌫]e abc cek )D 0 abc 1 bz [µ⌫]e 1 +g↵ ffdcek ddcek )D⌫ z⌫)kµja 1 ⌫)k 2µja i2ja abc fcek f(µ⌫)k dja )D z+ (µ⌫)k 2 0 +g↵2i (if 2 abc 2fabc abc cek 0 1 ⌫ b X z v(ij) X µjaz(⌫ ⌫)k b X z + v X z (ij) ⌫ja2 (µ⌫)k ⌫)k (ij) (ij) 2 2 2 ja (µ⌫)k µja ⌫ N N 0 1 (⌫ N N 0 ◆z(⌫1 ⌫)k b(ij) X +µja X µja X✓ z + X 2 (µ⌫)k 2 zv◆ jaN2zv (µ⌫)k (ij) ⌫)k (ij) ⌫ ⌫ Bbb(ij) C ja N N (⌫ N2 B ✓✓ C X+⌫+ zdvabc +◆ z(⌫CjbC(µ⌫)e C ✓(if ◆ (ij) X✓ X zvf(ij) X ab[ij] (if )jb ◆ a[ij] dcek f+ (ij) (ij) abc cekf)N ⌫ Nzabc BB + (µ⌫)e Nf ✓ NB ◆ f(⌫cek + XC z (µ⌫)e C acek dfcek + fabc )⌫(µ⌫)e a (if d + f ) X z B abc cek abc cek [ij] (if abc abc cek [ij] ⌫jbC B B jb B + ⇠ a (if d + f f ) + ⇠B (dabc idabc ) ◆ XC +(if z (d+b dcek idcek )abc ++ ⇠2✓Ba+b Xfcek z (µ⌫)e 2B d[ij] + fabc fcek abc cek (ij) abc abc (ij) abc cek cekf)cek [ij] ⌫jbC C ⌫ C jb ✓ ◆ + ⇠2 B + X z (µ⌫)e C +b dcek id dcek fcek ) id Xff⌫cek z )) ◆ C B (ij) (d C abcabc abc (ij) (did ⌫ A abc @ A B@+ C +✓⇠22+b ✓ ◆ +b (d d + ⇠2 B u (if d + f f ) abc cek abccek cek (if dcek + (ij) +b (d d[ij] idfdabc ffcek ⌫)e C B abc cek ⌫)e abc abc cek))abc [ij] @ u(ij) A abc cek µjb @ A B + C µjb ◆ X ✓ ✓ ◆ u (if + f f ) + z u (if d + f f ) X z ⌫)e abc cek [ij] @[ij]+ abc+v A cek abc cek+ f abcf cek µjb (⌫ µjb) ) ⌫)e @+ (⌫XA u (if d z ⌫)e (d id f X z abc cek abc cek [ij] (d d id f ) u+v (if d + f f ) µjb abc cek abc cek (ij) abc abc cek abc cekcek abc [ij] (⌫ (⌫ ⌫)e +(ij) X z(⌫ +v dcek idabc (dabc dcek fcek + +v Xfµjb abcabc cekz) (ij) (did 0(ij) cek ) id 0 1 ⌫)e 1 +v dcek ⌫)e abc abc fcek )(⌫ (ij) (d(µ⌫)e 0 1 +v (d d id f ) (µ⌫)e µ 0 1 µ abc cek abc cek (ij) (if d )@ z + ⇢ (if d )@ (if d )@ z + ⇢ (if d )@ z ⌫)e µ z(⌫ ⌫)e 1 i 0aek i (ifaek aek (µ⌫)e ⌫daek )@⌫ z (µ⌫)e i aek⇢i (ifaekµ daek 1 (⌫ )@ + z ⌫)e (if d )@ z + ⇢ (if d )@ z i aek aek ⌫ i aek aek ⇠ ⇠0 (µ⌫)e µ ⌫)e i aek aek ⌫ i aek aek C (⌫ C 3 B ⇠3 B µ b(⌫d(µ⌫)e b (µ⌫)e (if dfaek )@ +)D ⇢id(if z(⌫ id aek aek )@ (if3 @ )@ z (µ⌫)e + ⌫⇢zfi (if )@ z⌫b(⌫z (µ⌫)e ⇠3 B B +g C (if )D + (if+g zcek CA aekaek ⌫ abc aek aek A @+ cekd cekf (µ⌫)e ⌫d +i ⇠23aek +g (iffabc fabc f⌫babc d )D z B C @+g i (ifi abc+g A + ⇠3 @ fcekiicek d )D z abc cek abc cek A b (µ⌫)e C 2B + ⌫ abc cek ⌫)e ⌫)e b (µ⌫)e µb µb (if f f d )D z A 2 @+g+g⇢ i(ifabc abc cek abc cek +2 @ fcek fabc dcek )D zdcek ⌫ z ⌫)e A +g⇢ f )D f f d )D z i (if abc ⌫)e µb ⌫ i cek abc i (if abc cek abc cek µb (⌫ (⌫ 2 +g⇢ fcek fabc dcek 2 +g⇢i (ifabc+g⇢ fceki (iffabc abc cek )D ⌫)e)Dµb z(⌫ ⌫)e µbzd fdcek fabc +g⇢i (ifabc fceki (ifabc fabc dcek )D z(⌫(⌫cek )D z(⌫ =0 = 0 =0 =0 = 0 ia = 0ia ia for X for . X ia. for X µ . for Xµia for Xµiaia.µfor µ . X BianchiBianchi identities are for XµThe . corresponding The identities are µ . corresponding The corresponding Bianchi identities are The corresponding Bianchi identities are The corresponding Bianchi are The identities corresponding Bianchi identities are The corresponding Bianchi identities are D Z D +µµZ +⌫⌫ Z (32) + ⌫ Z+ ⇢ Z[µ⌫] [⇢µ]D DD ZD[⌫⇢] + DD ZD[⇢µ] +D D=⇢ Z Z0[µ⌫] = = 00 [⌫⇢] [⇢µ] Dµ Zµ[⌫⇢][⌫⇢] +D (32) ⌫Z ⇢Z [⇢µ]++D [µ⌫]+=D0⇢Z [µ⌫] = 0 Z Z µ ⌫Z ⌫ ⇢Z ⇢ [µ⌫] [⌫⇢] [⇢µ] Dµ Z[⌫⇢] + D (32) [⇢µ] + D [µ⌫] = 0 where where where where where i D Z D @[⌫⇢] i [⌫⇢] g[⌫⇢][d+ Diiµgg+[d ↵D , Z↵ ]µii ,, Z (33) where where = @@i+µµgZ µ Z[⌫⇢] iiX [⌫⇢] µ iiX [⌫⇢] ] µ+ D= ZZ = Z + [d D + ↵ X µ [⌫⇢] Dµ Zµ[⌫⇢][⌫⇢] =D @µµµZ + [d D + ↵ X , Z ] (33) i Z[⌫⇢] ] µ i [⌫⇢] [⌫⇢] µi + ↵ X µ Z = @ Z + i g [d D , Z ] [⌫⇢] Dµ Z[⌫⇢] = @µ µ Z[⌫⇢] Dµ + ↵i Xµµ, Z[⌫⇢]i] µ [⌫⇢] (33) [⌫⇢] + iµg [d 9 9 9 and 9 and and 9 and and Dµ z[⌫⇢] +D Dµ⌫zz[⌫⇢] +D D⌫⇢zz[⇢µ] =D0⇢ z[µ⌫] = 0 (34) [⇢µ]+ [µ⌫]+ Dµ z[⌫⇢] +D Dµ⌫zz[⌫⇢] +D D⌫⇢zz[⇢µ] =D0⇢ z[µ⌫] = 0 (34) [⇢µ]+ [µ⌫] + where where where where where D z =D@ zz = i i i gµ z[d D+ (35) i g↵[d , z[⌫⇢] ] µ [⌫⇢] µ+ i XD [⌫⇢] + @ [⌫⇢]↵]i Xµ µµ [⌫⇢] [⌫⇢] µ i ,µz+ i Dµ z[⌫⇢] =D@µµzz[⌫⇢] + i@gµ z[d[⌫⇢] D+ (35) i g↵[d µ+ i XD [⌫⇢] = [⌫⇢]↵]i Xµ , z[⌫⇢] ] µ ,µz+ Notice that theythat are they Bianchi with sources. Notice are identities Bianchi identities with sources. Notice that they are Bianchi identities with sources. Notice that they are Bianchi identities with sources. Notice that they are B ianchi i dentities w ith s ources. The associated energy-momentum tensor is tensor is The associated energy-momentum The associated energy-momentum tensor is tensor is The associated energy-momentum The associated energy-‐momentum tensor is 1 ⇢ 1 ⇢ ✓µ⌫ = Tµ⌫✓µ⌫ + 1=@T (S + 1+@S S⌫⇢µ ) (36) Sµ⌫⇢ S⌫⇢µ ) µ⌫⇢ ⇢µ⌫ + ⇢ µ⌫ ⇢µ⌫ ⇢ (S ✓µ⌫ = Tµ⌫✓µ⌫ + 2=@T (S S⌫⇢µ ) (36) + 2+@S (S Sµ⌫⇢ S⌫⇢µ ) µ⌫⇢⇢µ⌫ + µ⌫ ⇢µ⌫ 4933 | P a g e 2 2 J u n e 2 0 1where 7 where www.cirw o r l d . c where o m where @L @L @L @L i i Tµ⌫ = T@L @⌫ X ⌘@µ⌫ L↵a @L @L = @⌫ D ⌘µ⌫ L ↵a + @⌫ D@L ↵a µ⌫ ↵a + ⌫X i i i µ µ i µ µX @(@ X ) Tµµ⌫D= @ D + @ X ⌘µ⌫ L Tµ⌫ = @(@ @⌫ D + @ X ⌘ L D ) @(@ ) ↵a )@(@ ⌫ ↵a ⌫ ↵a↵a ⌫ µ ↵a↵a µ⌫ ↵a ↵a µ i µ i (37) @(@@L D↵a )@(@ D↵a ) @(@ X@L ↵a )@(@ X↵a ) @L @L

(31) (31) (31)

(32) (32) (32) (33) (33) (33)

(34) (34) (35) (35)

(36) (36)

(37)

µi g [d [⌫⇢] Dµ [⌫⇢] µ i iD [⌫⇢][⌫⇢] = + + ↵]iX Xµi ,, zz[⌫⇢]]] (35) where [⌫⇢] [⌫⇢] + D+ (35) ii gg↵[d Dµµii µ,µµµz+ ↵ where Dµµ zµ[⌫⇢] D@µµzzzµ[⌫⇢] = @i@ @µgµµzzz[d + [d (35) µ+ iX [⌫⇢] =D [⌫⇢] = [⌫⇢] where ii µ [⌫⇢] [⌫⇢] [⌫⇢] ii where where Dµ z[⌫⇢] =D@µµµzz[⌫⇢] + i g [d D + ↵ X , z ] (35) µ i [⌫⇢] [⌫⇢] µ i = @ z + i g [d D + ↵ X , z ] (35) µ µ i [⌫⇢] [⌫⇢] [⌫⇢] Notice Notice that they are Bianchi identities with Notice they are Bianchi with D zBianchi = i@gµidentities zwith +sources. isources. g↵with [d D + (35) D =are @@µidentities + [d D + zsources. ]] i Xiµµ , z[⌫⇢] ] (35) [⌫⇢] [⌫⇢] that they are µ iX [⌫⇢] [⌫⇢]↵ µi ,,µsources. Notice that that theythat are Dµµzthey zBianchi zBianchi + i µgidentities [d D+ + ↵with (35) Notice Bianchi Notice that they are identities with µidentities µsources. iX [⌫⇢] = [⌫⇢] = [⌫⇢] D z @ z i g [d D ↵ X , z ] (35) I S S N 2 3 4 7 -‐ 3 4 8 7 µµ z+ µ i [⌫⇢] [⌫⇢] [⌫⇢] µ Notice that they are Bianchi identities with sources. The associated energy-momentum tensor is The associated energy-momentum tensor is The associated energy-momentum tensor is V o l u m e 1 3 N u m b e r 6 Notice that they Bianchi identities with Thethat associated energy-momentum is tensor The associated energy-momentum is Notice that they are are Bianchi tensor identities with sources. sources. The associated energy-momentum tensor Notice they are Bianchi identities with sources. Notice they are Bianchi identities with Jsources. Thethat associated tensor is Notice they areenergy-momentum Bianchi identities with energy-momentum o utensor r n a lsources. is The associated Thethat associated energy-momentum tensor iso f A d v a n c e s i n P h y s i c s The associated energy-momentum tensor is 1 1 1 ⇢ ⇢ The associated energy-momentum tensor is 11+ + The associated energy-momentum = T✓µ⌫ +1=@ T (S Stensor S (36) (36) ✓✓µ⌫✓µ⌫= Tµ⌫ + (S S S⌫⇢µ )) ) S⌫⇢µ ) (36) ⇢@ ⇢⇢ ⇢µ⌫ µ⌫⇢ ⌫⇢µ µ⌫⇢ ⇢(S + (S + Sis µ⌫ 2 µ⌫⇢µ⌫ ⇢µ⌫ + µ⌫⇢ + (36) 1=@2TT + S S⌫⇢µ ) + 112+@@ @S (S (36) µ⌫ = Tµ⌫✓✓µ⌫ µ⌫⇢ µ⌫ ⇢µ⌫ µ⌫⇢ ⇢(S µ⌫ = µ⌫ ⇢µ⌫ ⇢µ⌫ S⌫⇢µ µ⌫⇢ ⇢ ✓µ⌫ = Tµ⌫✓µ⌫ + 112=@T S⌫⇢µ ) (36) 22+@ S ⇢(S ⇢µ⌫ µ⌫⇢ + + S S ) (36) ⇢T(S µ⌫ ⇢µ⌫ µ⌫⇢ ⌫⇢µ ✓ = + @ (S + S S ) (36) 1 µ⌫ 2 @ ⇢(S µ⌫⇢µ⌫ 2 ⌫⇢µ ✓✓µ⌫ = Tµ⌫ + + S )) (36) µ⌫⇢⇢µ⌫ S ⌫⇢µµ⌫⇢ ⇢ + @ (S + S S (36) 2 µ⌫ = Tµ⌫✓µ⌫ ⇢µ⌫ µ⌫⇢ ⌫⇢µ where where where = T + @ (S + S S ) (36) 2 µ⌫ ⇢µ⌫ µ⌫⇢ ⌫⇢µ where where 2 where 2 where where where @L @L@L @L where where i i @L @L @L @L where whereTµ⌫Tµ⌫ @L + @⌫@X = T@L D↵a + X ⌘X @L @@⌫@D + L iiiL i ↵a ⌘ ⌫ ↵a ⌫ ↵a µ⌫ = @ D @µ⌫ L µ µ i µ µ i µ⌫ ⌫ ↵a ⌫X µ⌫ L T = @ D + @ ⌘⌘µ⌫ Tµ⌫ = = @(@ D + @ X ⌘ L @L @L ↵a ⌫⌫@(@ ↵a ⌫ ⌫ ↵a ⌫ µ⌫ @(@ D↵a ) X ) Tµ⌫ = @ D + µ µ i D ) @(@ X ) ↵a ↵a @L @L i µ⌫ ↵a ⌫ µ⌫ L ↵a µ µ i µ µ i ↵a ↵a ↵a )@(@ @(@ D )) @@(@ @(@ X @L @L ii µD µX i )) ↵a @(@ ) D ) X Tµ⌫ = @(@ @ D + @ X ⌘ L @L @L ↵a ↵a (37) (37) ↵a ⌫ ↵a ⌫ µ⌫ (37) @(@ D @(@ X ) ↵a ↵a ↵a T = D + @ X ⌘ L i ↵a µ µ i µ⌫ ⌫⌫ D ↵a ⌫⌫ X @L @L ↵a = @@(@ + )@(@ @µ⌫ ⌘µ⌫ @L @L µ µ ii )⌘ µ⌫ ↵a µ⌫ L(37) i ↵a @(@T@L D X@L T @@⌫ D + @@⌫ X L @L ↵a )@(@ (37) µ⌫ = ↵a↵↵a µ↵D µ↵a ↵a@L ) X ii↵a ↵ ⌘ i L ↵ )↵@ iX @L T = D + X ⌘ L µ µ i ↵a @(@ D ) @(@ X µ⌫ ⌫ ↵a ⌫ µ⌫ @L @L @L @L T = @ D + ↵a ↵a ↵ (37) @(@ D ) @(@ X ) S⇢µ⌫ =@(@ ⌃µ@L ⌃i@L ↵a µ⌫ ⌫a + ↵a S = + µ D= µ i ) + µ⌃µ⌫ ↵ ii ↵ DD ↵ X⌫X i a ↵a ↵a )⌃µ⌫ (37) a⌃ ↵a µ⌫ µ⌫ a ↵ ↵ ↵ Xµ⌫ S ⌃ D @(@ X ⇢ ⇢ i (37) ⇢ ⇢ i ⇢µ⌫ a S = D + ⌃ ↵a S⇢µ⌫ = ⌃ D + ⌃ X @L @L @(@ D ) @(@ X ) µ⌫ µ⌫ a ↵a (37) ⇢µ⌫ a ⇢µ⌫ D )µ⌫@L @(@ X )µ⌫↵a S@(@ =↵a D⇢X X i aa (37) ))@(@ ))@(@ @L ↵D↵a )a⌃µ⌫ ↵ iii i a⌃µ⌫ ⇢µ⌫ a ↵a ↵a ⇢⇢ ⇢⇢ ⇢D i+↵a µ⌫@(@ µ⌫ ↵ @(@ @L @L ↵ ⇢ ⇢ D ) X ) ↵a @(@ D @(@ X S⇢µ⌫ = @(@ ⌃ D + ⌃ X (37) @L @L ↵a ↵ ↵ i ↵a ↵a @(@ a⌃ D ) @(@ X ) ↵a ↵a µ⌫ µ⌫ a S = D + ⌃ X ↵ ↵ i ↵a ⇢ ⇢ i ⇢µ⌫ a @L @L ↵a S⇢µ⌫D= DX + )⌃µ⌫@L µ⌫ @L a ↵a ⇢⇢↵DD )a⌃+ ii i)a⌃µ⌫ S = @(@ ⌃ µ⌫↵@(@ µ⌫↵ Xi aa ↵a )@(@ ↵a S⇢µ⌫ ⌃µ⌫ ⌃µ⌫⇢⇢↵X X ⇢ D= i )@(@ @(@ XX ↵a ⇢µ⌫ = @(@ S⇢µ⌫ ⌃+ D⇢⇢X ↵a )a µ⌫ DD @(@ ↵a )a⌃µ⌫ X a a + ⇢ D↵a ) i )@(@ with with with µ⌫ ↵a ⇢ ⇢ i @(@ ) @(@ X with ↵a @(@ D↵a ) ↵a @(@ X↵a ) with with with with with with with ↵ ↵ ↵ with ↵ ⌃↵µ⌫ i(µ↵↵↵↵↵µ↵⌫ = (38) (38) ⌃ i( (38) ↵ ↵ == with ⌫ i( ⌫↵ ⌫↵ ↵ µ ))µ ) ↵ ↵ µ) ⌃ = i( ⌃ ⌃µ⌫ = i( (38) µ⌫ µ ⌫⌫ µ ⌃µ⌫ (38) µ ⌫↵ µ↵ µ ⌫⌫ ↵ ↵↵ ⌫ = i( µ⌫ µ ⌫ ⌫ µ) ↵ ⌃µ⌫ = i( ) (38) ↵ ↵ ↵ ⌃ i( )) (38) µ⌫↵ µ ⌫ = ⌫ µµ ⌫ ↵ ↵ ⌃µ⌫ = i( (38) ⌫⌫ µ ↵ ↵ ↵ µ⌫ µ ⌫ µ ⌃ = i( ) (38) ↵ ⌫ in ⌫ ↵µantisymmetric ↵ µ⌫ µtensor ⌃ = i( ) (38) Splitting the energy-momentum in and symmetric Splitting the energy-momentum tensor antisymmetric and symmetric ⌃ = i( ) (38) µ⌫ µ ⌫ ⌫ µ Splitting the energy-momentum tensor Splitting the energy-momentum tensor in antisymmetric and symmetric µ⌫ µ ⌫ ⌫ µ Splitting the energy-momentum tensor in antisymmetric and symmetric Splitting the energy-‐momentum tensor antisymmetric and symmetric sectors, Splitting thein energy-momentum tensor in antisymmetric and symmetric sectors, Splitting the energy-momentum tensor in tensor antisymmetric and symmetric sectors, Splitting the in antisymmetric and sectors, sectors, Splitting the energy-momentum energy-momentum tensor in antisymmetric and symmetric symmetric sectors, Splitting the energy-momentum tensor in antisymmetric and symmetric sectors, A A S Santisymmetric and symmetric Splitting the energy-momentum tensor in ✓ = ✓ + ✓ (39) A S sectors, A S ✓ = ✓ + ✓ (39) Splitting the energy-momentum tensor in antisymmetric and symmetric A S µ⌫ µ⌫ sectors, µ⌫ µ⌫ µ⌫✓ µ⌫ A S = ✓ + ✓ sectors, ✓ = ✓ + (39) ✓ = ✓ + ✓ (39) µ⌫ µ⌫ µ⌫ µ⌫ + ✓µ⌫ µ⌫ sectors, µ⌫ µ⌫ (39) A ✓µ⌫ S ✓ µ⌫ ✓= A µ⌫ µ⌫ sectors, sectors, ✓µ⌫ = ✓A + (39) A + ✓S S ✓ = ✓ (39) µ⌫ µ⌫ S µ⌫ ✓ = ✓ + ✓ (39) µ⌫ µ⌫ it yields for the antisymmetric sector the following expression: µ⌫ yields for the antisymmetric sector the following expression: µ⌫ µ⌫ A S it yields for the itantisymmetric s ector t he f ollowing e xpression: ✓✓sector ✓ thesector + ✓✓µ⌫ Athe (39) µ⌫ = S yields for the antisymmetric following expression: it yields for the antisymmetric sector the it yields it for the antisymmetric following expression: (39) µ⌫ = ✓µ⌫ = ✓ + ✓ (39) it yields for the antisymmetric sector the following expression: µ⌫✓+ µ⌫ µ⌫ µ⌫ µ⌫ it yields for the antisymmetric sector thesector following expression: it yields for ⇢]athe antisymmetric ⇢]a ⇢]a following ⇢]a ⇢]a ⇢]a ⇢]athe ⇢]aexpression: A A A A it for the antisymmetric sector the following expression: ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a AA = 4Z Z + 4z z + 2z Z + 2Z z ⌘µ⌫ L (40) ⇢]a ⇢]a ⇢]a ⇢]a = 4Z Z + 4z z + 2z Z + 2Z z ⌘⌘µ⌫ L it✓✓yields yields for the antisymmetric sector the following expression: A✓µ⌫ A [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a µ⌫ [⌫ [⌫ [⌫ [⌫ it yields for the antisymmetric sector the following expression: [⌫ [⌫ [⌫ [⌫ ✓ = 4Z Z + 4z z + 2z Z + 2Z z ⌘(40) LA A A (40) ✓ = 4Z Z + 4z z + 2z = 4Z Z + 4z z + 2z Z + 2Z z L [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a Z[⌫ [µ⇢]a z[µ⇢]a [µ⇢]a Z[⌫ µ⌫ [µ⇢]a [µ⇢]a [µ⇢]a [⌫ ⇢]a[µ⇢]a µ⌫ [⌫ [⌫ [⌫ ✓µ⌫ =[⌫4Z + 4z[µ⇢]a +[⌫ ⇢]a 2z[µ⇢]a +[⌫ ⇢]a 2Z[µ⇢]a z[⌫ (40) A A (40) ⌘µ⌫ [⌫ ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a µ⌫ LA A µ⌫ ⇢]a ⇢]a ⇢]a ⇢]a [⌫ [⌫ [⌫ [⌫ A A (40) ✓Aµ⌫ = 4Z Z + 4z z + 2z Z + 2Z z ⌘ L (40) µ⌫ [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a ✓ = 4Z Z + 4z z + 2z Z + 2Z z ⌘ L µ⌫ A [⌫ [⌫ [⌫ [⌫ µ⌫ [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a µ⌫ [⌫ [⌫ [⌫ [⌫ µ⌫ [µ⇢]a [µ⇢]a [µ⇢]a [µ⇢]a ⇢]a ⇢]a ⇢]a ⇢]a µ⌫ A A [⌫ [⌫ [⌫ [⌫ ✓ = 4Z Z + 4z z + 2z Z + 2Z z ⌘ L (40) with ⇢]a [⌫ ⇢]a [⌫ ⇢]a [⌫ ⇢]a A Z[⌫ µ⌫ = 4Z[µ⇢]a with ✓with + 4z + 2zz[µ⇢]a + 2Z ⌘µ⌫ µ⌫ [µ⇢]a z+ [µ⇢]a Z [µ⇢]a z ✓[µ⇢]a Z [µ⇢]a +[⌫2z[µ⇢]a Z [µ⇢]a +[⌫ 2Z z L (40) ⌘µ⌫ LA (40) µ⌫ with [⌫ 4z[µ⇢]a with with µ⌫ = [⌫4Z[µ⇢]a [⌫] [↵ [↵ ] [⌫ [↵ ][µ⇢]a [⌫ with A A [⌫ [↵ ] [↵ ] [↵ ] A(Z [↵ ]][↵ ⌘L L= = ⌘L Z(Z z[↵ + Z[↵ z ])Z ) z [↵ (41) with ⌘µ⌫ (Z zz][↵ Z [↵ [↵ [↵ ]] (41) ] ++ [↵ µ⌫A µ⌫ [↵ ][↵ ]]]]z+] + [↵ µ⌫ with ⌘⌘µ⌫ ]Z ]z ]z = ⌘⌘⌘µ⌫ Z A[↵ [↵ [↵ [↵ ] [↵ ]) L = Z + z+ zz[↵ (41) (ZAA Z][↵zz[↵ (41) µ⌫ µ⌫ [↵ [↵ [↵ µ⌫ LA = ⌘⌘ [↵ ]Z ] z [↵ ] z]][↵+ with [↵ (Z ]+ ])Z[↵ ⌘ = (Z Z + (41) µ⌫ L µ⌫ [↵ z]][↵ [↵Z]][↵ [↵ ]] z [↵ [↵ ]] +] + ]]) (41) with A [↵ [↵ ] [↵ ⌘ L = ⌘ (Z Z + z z ) with µ⌫ A µ⌫L [↵=] ⌘µ⌫ [↵ ] [↵+ [↵ ] [↵+] Z[↵ ] z A ⌘ (Z Z z z ) (41) A [↵ ] ] µ⌫ [↵ ] [↵ ] µ⌫ µ⌫ [↵ ] [↵ ] [↵ ] A where L is defined at eq.(24). where L is defined at eq.(24). A [↵ ] [↵ ] [↵ ] A A ⌘⌘LL L = ⌘⌘µ⌫ (Z Z + zz[↵ [↵] zz ] + Z )) (41) µ⌫A [↵ ]eq.(24). [↵[↵] z where is defined Aat ] [↵ ] A where L is defined at eq.(24). where is defined at eq.(24). L = (Z Z + + Z z (41) µ⌫ µ⌫ [↵ ] [↵ ] [↵ ] ⌘ L = ⌘ (Z Z + z z + Z z ) (41) LAA issymmetric defined at eq.(24). Awhere µ⌫ µ⌫ [↵ ] [↵ ] [↵ ] where L is dwhere efined at L eq. (24). issymmetric defined eq.(24). For Lthe isatdefined atsector, eq.(24). Awhere ForLthe sector, where is defined at eq.(24). A For the symmetric sector, For Lthe symmetric sector, where defined eq.(24). where LA is at defined at eq.(24).⇢)ia For the symmetric sis ector, ⇢) S ⇢)ia ⇢) ✓ = 4 Z X +X 4⇢i Z(⇢ia⇢)a X(µ⌫) ia S i (µ⇢)a µ⌫ ⇢)ia ⇢) (⌫i Z ✓Sµ⌫ = 4 i✓ZS(µ⇢)a X + 4⇢ ⇢)ia ia (⌫ (⇢ + a 4⇢(µ⌫) =X 4 (⌫ X ia i Z(µ⇢)a iZ ✓µ⌫ = 4 i Zµ⌫(µ⇢)a +X 4⇢(⌫i⇢)ia Z(⇢ a X(µ⌫)(⇢ ⇢) a (µ⌫) ia ⇢)+ 2⇢i z ia X(µ⌫) +X 2 i⇢)ia z⇢)ia X (µ⇢)a (⌫i ⇢)ia (⇢ ia + 2 i z(µ⇢)a + 2⇢ z X ⇢)+ ia⇢)a X (µ⌫) (⌫ (⇢ a + 2 z X 2⇢ z i + 2 i z(µ⇢)a X(⌫i (µ⇢)a + 2⇢(⌫i z(⇢i a X (µ⌫) ⇢j(µ⌫) a (⇢ a ⇢) i j a 4z[X ]a u+ 4z i a[ij] ⇢j[X a ⌫ i , X ⇢j⇢) ia u[ij]j[X a µ i , X⌫ j ] a (µ⇢)a (⇢ ⇢) + 4z(µ⇢)a a+ , X ] + 4z [X , X ] ⇢) i ⇢j a i j a [ij] [ij] ⌫ a ⌫ [X , X ] [X⌫ ,4z X(⇢ ] au+ 4z u[ij] (µ⇢)a + 4z(µ⇢)a a+[ij]4z [X [X µ ,aX ⌫ , X[ij] ] + ⌫ ] µ i ⌫ ja i (⇢⇢jaa [ij] (⇢µ ⇢) ⇢) + 4z {X , X } + 4z v {X i b[ij]⇢ja i ja (µ⇢)a [ij] ⌫ ja } (⇢ ⇢)ia, X ja ⇢ja v[ij] {X + 4z(µ⇢)abb+[ij]4z {X X } ⌫+ +i ,4z 4z } µ i ,, X ⇢ja ⌫ i ,,bX µ ,v ⌫ {X {X X X (⇢⇢)} av+ 4z (µ⇢)a [ij] [ij] + 4z(µ⇢)a } {X X } ⌫ µ ⌫ } (⇢ a [ij] {X [ij] ⌫ µ⇢) ⌫ (⇢ i ⇢j a i j a ⇢) + 2Z(µ⇢)a a [X , X ] + 2Z u [X , X i [ij] ⇢j10 a 10 ia [ij]j a µ i ⌫ i 2Z ⇢j⇢) ⌫ j ]a (⇢ ⇢) a u[ij] [X 10 + 2Z a [X , X ] + , X ] i ⇢j a i j a (µ⇢)a [ij] 10⌫ ,2Z ⌫ a 2Z [X X (⇢ ]10 u ⌫⌫ [X a + 2Z(⇢µ ,a X (µ⇢)a + 2Z(µ⇢)a+ a[ij] [X ] µ ,iX⌫ ]ja 10 ⌫ , X[ij] ]10+ µ ⇢) [ij] (⇢ a u[ij] [X i ⇢ja 10 + 2Z(µ⇢)a {X , X } + 2Z v {Xµ i , X⌫ ja } ⇢) i b[ij]⇢ja i ja [ij] 10 (⇢ ⇢)ia, X ja ⇢ja⇢) v[ij] {X + 2Z 2Z(µ⇢)a+ b[ij] {X X } ⌫+ +i ,2Z 2Z } ,X } ⇢ja 10 ⌫ i ,,bX µ a, v ⌫ {X 2Z {X X } 2Z (⇢10 a+ (µ⇢)a [ij] [ij] + } v {X X ⌫ ⌫ (⇢ (µ⇢)a b[ij] {X [ij] ⌫⇢ µ ⌫ ia} µ ia (⇢ a ⇢ (42) 4 i @ ⇢ (Z ⇢ ia (µ⇢)a X⌫ ia⇢) + 4 i @ ⇢ (Z ia (µ⌫)a X⇢ ia ) (42) (42) 4 @ (Z X ) + 4 @ (Z X ) ⇢ ia ⇢ ) +(µ⌫)a ia i i (µ⇢)a (42) 4 @ (Z X 4 @ (Z X ) ⌫ ⇢ 4 i @ (Z(µ⇢)aiX⌫ )(µ⇢)a + ⇢) 4 i⌫@ia (Z(µ⌫)aiX⇢⇢ )(µ⌫)a ⇢ ia 4⇢ X ⇢) 4 i @ ⇢ (Z ⇢) @µ (Z ia ia (⌫⇢)a Xµ ia ) ⇢) i @ 4⇢i@@µ(Z (Z(⇢4⇢ Xµ⌫(Z X ia )(⇢ ⇢)a4Xi⌫@ia ⇢)(Z(⌫⇢)a ia ) 4 @ µ(Z ai X i 4⇢ ) 4 @ (Z X )(⌫⇢)a Xµia ) ⌫ (⇢ a i µ i (⌫⇢)a (⇢ a ⌫ ⇢) ia ⇢µ 4⇢ @ (Z X ) + 2 @ (z ⇢) ⌫ ia (⇢ ⇢)a µ ia⇢ i ⇢ ia (µ⌫)a X⇢ ia ) ⇢) i @ 4⇢i@@⌫(Z (Z(⇢4⇢ X⌫µ(Z X ia ) + 2Xi @⇢ )(z+ (µ⌫)a 2 @ ⇢ia (z) X ) a i 4⇢ X ) + 2 @ (z X µ i ⌫ (µ⌫)a i ⇢⇢ )(µ⌫)a ⇢ia (⇢ a ⇢µ (⇢ a i ia 2 @ (z X ) 2 @ (z ⇢ i ⇢ ia (µ⇢)a ⌫ ia⇢ i ⇢ ia (⌫⇢)a Xµ ia ) (z(µ⇢)a 2Xi @⇢)(z(⌫⇢)a ia ) 2 X @ ⌫(z 2 X (z)(⌫⇢)a X ) i @µia 22 ii@@⇢ (z (µ⇢)aiX⌫ )(µ⇢)a ⇢) 2 i⌫@ia (z(⌫⇢)a X µ ⇢ ) ↵) µ ia ⌫ 2⇢ X ) ⇢) i @µ (z ↵) @ ⇢ (Z ia ⇢) X⌫ ia⇢) + 4⇢ ia ↵) ↵) i @ 2⇢i@@µ(z (z(⇢⇢) X X⇢(Z ) ⌫µ⌫ aa X⇢⇢ ia ) µµ⌫ ia )(⇢+a4⇢ ia (↵ 2⇢aiX @µ⌫(z Xii⌫@@⇢ (Z )(Z+(↵4⇢ ai X 2⇢ ) + 4⇢ ) (⇢ a (↵ i µ (⇢ ⌫ ⇢ µ a (↵ a ⇢ ⇢) ↵) ia ia ⌫ 2⇢ 2⇢ ⇢) i @⌫ (z ↵) i @ ⇢ (z ia (⇢ ⇢)a Xµ ia⇢) + ↵) ia (↵ ↵) ⌫ a X⇢ ia ) µ ⇢) ia ⇢ ia ⌫ 2⇢ @ (z X ) + 2⇢ @ (z X ) 2⇢ @ (z X ) + 2⇢ @ (z X ) µ⌫ i ⌫ i µ i ⌫µ (⇢ aX (↵ ai ⇢ 2⇢i @⌫ (z(⇢ a µS )(⇢+ a2⇢iµ@ (z(↵ a X⇢ )(↵µ a ⇢ ⌘ LS LSS ⌘µ⌫ µ⌫L µ⌫ L ⌘⌘µ⌫ S Swhere LS is defined at eq.(25). Eq.(39) provides the symmetry relationship S where Leq.(25). is defined defined at eq.(25). Eq.(39) provides the symmetry symmetry relationship Swhere L pisrovides defined at Eq.(39) eq.(25). Eq.(39) the provides the symmetry relationship where L is dwhere efined at L Eq.(39) the symmetry relationship is at eq.(25). provides relationship µ⌫ ⌫µ (43) µ⌫ ⌫µ✓ µ⌫ = ✓ ⌫µ = ✓✓⌫µ (43) ✓ = ✓ (43) ✓✓µ⌫ = (43)

and the conservation law and

the conservation law and the the conservation conservation law and the conservation law µ⌫ and law 4934 | P a g e µ⌫ @ µ ✓ µ⌫ = 0 µ⌫ @ ✓ = = 00@µ ✓ = 0 (44) (44) J u n e 2 0 1 7 @ µµ ✓

(44) (44)

w w w . c i r w o r l d . c o m

33

Ia 3 Physical Basis Ia {GIa Ia Physical Basis Basis {Gµµ }} {Gµµ } 3 Physical } Physical Basis {G

We should now to study such non-abelian whole gauge symmetry in terms

where L is defined at defined eq.(25).atEq.(39) symmetry relationship where LS is eq.(25).provides Eq.(39)the provides the symmetry relationship S S at eq.(25). Eq.(39) where L where is defined provides the symmetry relationship L is defined at eq.(25). Eq.(39) provides the symmetry(43) relationship ✓µ⌫ = ✓⌫µ I S S N 2 3 4 7 -‐ 3 4 (43) 8 7 ✓µ⌫ = ✓⌫µ V o l u m e 1 3 N u m b e r 6 µ⌫ ⌫µ µ⌫ ⌫µ ✓ = ✓ ✓ J o=u r✓ n a l o f A d v a n c e s i(43) n P h y s i(43) c s and the conservation law and the conservation law µ⌫ @µ ✓ = 0 @ ✓µ⌫ = 0 (44) (44) and the conservation law µ and the conservation law @µ ✓µ⌫ = 0@ µ ✓µ⌫ = 0 (44) (44) Ia 3 Physical Basis {G µ } {GIa 3 Physical Basis µ } Ia Ia Ia 3 should Physical Basis {G 3{G Basis } gauge symmetry in terms }to study 3 Physical µ } {Gµwhole µ Physical WeBasis now non-abelian We should now such to study such non-abelian whole gauge symmetry in terms of physical fields. Changing basis, one gets We sWe hould should now to sof tudy such wChanging hole non-abelian gauge symmetry ione n terms of physical fields. Changing asis, one gets physical fields. gets now tonon-‐abelian study such whole gauge symmetry in bterms We should now to study suchbasis, non-abelian whole gauge symmetry in terms 0 1 0 1 0 1 of physical fields. Changing basis, one gets of physical fields. Changing one gets 0 basis,1G µ1 Gµ1physical where ⌦ matrix isinDµ defined inConsequently, eq.(5). Consequently, the fields physical fields suffer Dµ where ⌦ matrix is defined eq.(5). the suffer 0 10 B C . 1 0 1 B C the @ A where ⌦ matrix is defined in eq.(5). Consequently, physical fields suffer . . 0 1 = ⌦ (45) @ A @ A where ⌦ matrix is defined in eq.(5). Consequently, the physical fields suffer . where ⌦ matrix is defined in eq.(5). Consequently, the physical fields suffer . G the following gauge transformation =Consequently, ⌦ @the (45) where ⌦ matrix matrix is defined inDµ eq.(5). Consequently, the physical fields suffer A thefields µ1 Gµ1 . physical the following gauge transformation where ⌦ matrix is defined in eq.(5). physical fields suffer Dµ where ⌦ is defined in eq.(5). Consequently, suffer the following gauge transformation where ⌦ matrix is defined in eq.(5). Consequently, the physical fields suffer where ⌦ matrix is defined in eq.(5). Consequently, the physical fields suffer where ⌦ matrix is defined in eq.(5). Consequently, the physical fields suffer B C X . where ⌦ matrix is defined eq.(5). Consequently, the physical fields suffer the following following gauge transformation µi A in X B C G the following gauge transformation . @ µN . the gauge transformation ⌦µi@A = (45) @=Consequently, G..µN the gauge following gauge transformation . ⌦A@ the (45) the following gauge transformation where ⌦ matrix is defined in eq.(5). fields suffer A the following gauge transformation the following following gauge transformation iphysical the transformation the following gauge transformation iConsequently, 0 eq.(5). 1 1 where ⌦ matrix is defined in the physical fields suffer 0G 1 1 X ! G = U G U + @ U · U (46) i µi µI µI µ G µI Gtransformation =! UG Uµi=1U+ @ U11 + ·GU (46) the following gauge 0X 1 µN µI ! tG µI µN 0µI iifIollowing where Ω matrix is defined the in eq.(5). Consequently, he ptransformation hysical fields tG he gauge G GµI Uµµ U U ·· U (46) giiI111@@µtransformation 0µIUsuffer 1 µI µI 1U following i g GµI !G G =! UG G + @ · U (46) 0 1 1 G G = G U + U U (46) µI gauge 000µI 1 i i G ! = U U + @ U · U (46) µI µI µ i gigI11@µ U · U 11 µI µI 0µI 1 ·+ G= ! G =11 U U GggµI UµµU (46) µI G ! G U G U + @ U (46) 0µI µI µI µI 0 1 I G ! G = G U + @ U · U (46) GµI ! G = U G U + @ U · U (46) I G ! G = U G U + @ U · U (46) µI! µI µI µI µI µ + gI@µµU · U I µI µI µ G = U G U (46) i µI g µI µI a 11 II11 0 and the µI 1 1coupling constants, g g = g , gI Iconstants, aGµI g g = G t presence of different i I G ! G = U G U + @ U · U (46) a I g g1 where GµIwhere = G t and the presence of different coupling g = , µI a µI µI µ 0presence of different 1 1 I µI µIGaµI = Ga tG where the coupling ⌦ggI11 ,gI = ⌦ I1 g , Gof = U GgµI Udifferent + constants, @µ U · U constants, (46) a and µI µI ! where G GµI =G GaaaµI tµI and the presence of different coupling constants, g = I coupling g 11 , µI 1 , gI = ⌦I1 a µIwhere a I g G = G t and the presence of coupling constants, where = t and the presence different g = a 11 1 ,yields µI ⌦ a I g g where G = G t and the presence of different coupling constants, g = g a µI means the possibility for coupling with different currents. Eq.(45) the a where G = G t and the presence of different coupling constants, g = g ⌦ 11 µI a I I1 a I 1 , g = ⌦gI11 ,, µI II = µI meansG possibility for with different currents. Eq.(45)constants, yields the athe µI = G tpresence and the different coupling constants, where Gthe = GG taaaµIand and ofpresence different coupling constants, gIEq.(45) = ⌦⌦⌦I1 I11 where = G tthe the of different constants, gyields aand 1,yields µIwhere µI I111, µIG means possibility for coupling with different currents. the µI where = G tcoupling the presence ofofcoupling different coupling g µI µI apresence II = ⌦⌦ g µI ⌦ means the possibility for coupling with different currents. Eq.(45) the a I1 I1 means the possibility for coupling with different currents. Eq.(45) yields the I1 , fields strengths redefinition means the possibility for coupling with different currents. Eq.(45) yields yields the g I1 where G = GµIstrengths tthe the presence of different coupling constants, gIEq.(45) = ⌦the following fields redefinition means possibility forwith coupling with different currents. 1 yields the µIfollowing a and means the possibility for different currents. Eq.(45) g a coupling means the possibility for coupling with different currents. Eq.(45) yields the following fields strengths redefinition means the possibility for coupling with different currents. Eq.(45) yields the GµaI ta and where Gµ I =means the presence of different coupling constants, , means the possibility for coupling with g = I1 the possibility for coupling with different currents. Eq.(45) yields the where G = G t and the presence of different coupling constants, g = I 1, following fields strengths redefinition means the for coupling with different currents. Eq.(45) yields µI possibility a I µIstrengths following fields redefinition ⌦the following fields strengths redefinition following fields strengths redefinition I1 Ω−1 following fields strengths redefinition means the possibility for coupling with different currents. Eq.(45) yields the I1 following fields strengths redefinition I I I J following fields strengths redefinition followingfollowing fields strengths I with I strengths redefinition D =II⌫ ⌦ @⌫ G [GIµ , GJ⌫ ])Eq.(45) (47) the possibility coupling different currents. the µ⌫µfor 1I@(@ µG I[GIµ ,ig⌦ Dfields =fields ⌦redefinition G GJJ⌫ 1J ]) (47)yields(47) µ µ⌫ 1ID ⌫G 1J different currents. Eq.(45) ymeans ields the strengths following s(@ trengths redefinition Iµ ⌫II⌫ ig⌦ = ⌦ (@ G @ G ig⌦ [G , G ]) following fields redefinition µ⌫ 1I µ ⌫ 1J I I J I I I J µ µ ⌫ D = ⌦ (@ G @ G ig⌦ [G , G ]) (47) I I I J µ⌫ = ⌦ 1ID µG ⌫G 1J I I I J = ⌦ (@ G @ G ig⌦ [G , G ]) (47) ⌫ µ µ ⌫ Dfields (@ @ ig⌦ [G , G ]) (47) followingD strengths redefinition µ⌫ 1I µ ⌫ 1J µ⌫ 1I µ ⌫ 1J I I I J D ⌦1I@@(@ (@G @⌫1JG (47) µµ Iµ IG I ig⌦ = ⌦ (@ G=I⌫⌫I ⌦ ig⌦ [G G ]) (47) µ⌫ µIµµ 1J [G µ ,, G µ⌫ 1I µµG I ⌫⌫⌫I ig⌦ Iµ J⌫⌫ ]) µµ,,,ig⌦ G GI[G [G ]) I (47) (47) Dµ⌫ ⌦I1I (@µ⌫ G ig⌦ [G GJ⌫⌫⌫J⌫1J ]) (47) µµ 1J[G i= ⌦ iµ I1I J µ⌫ 1ID 1J D (@ @⌫⌫⌫µG G ]) µ µ i i = I 1I Iµµ, ⌦ J⌫⌫]) 1J D =G= ⌦ (@ G @@⌫ ⌫G GG (47) ⌫I⌫ µ⌫Ii@ X = ⌦ (@ G @ ) ig(⌦ ⌦ + ⌦ )[G , G ] (48) IµG I ig⌦ J )[G ⌫ µ µ ⌦1I i I I I J µ ⌫ iJ 1J iI X[µ⌫] = ⌦ (@ G G ) ig(⌦ ⌦ + ⌦ , G ] (48) [µ⌫] ⌫ µ ⌫ µ ⌫ 1I iJ 1J iI ⌫1I (@ µ µ ⌫ D = ⌦ G @ G ig⌦ [G , G ]) (47) ii iiIXi[µ⌫] I I I J = ⌦ (@ G @ G ) ig(⌦ ⌦ + ⌦ ⌦ )[G , G ] (48) µ⌫ µ ⌫ 1J ⌫ µ µ ⌫ µ ⌫ 1I iJ 1J iI iIi@⌫ GII )I⌫I ig(⌦ IµI ⌦iJ +I ⌦1J ⌦iI )[GII I, GJJ J] I 1I I µII (48) ⌫J Xi[µ⌫] =⌦ ⌦iX (@iiµµG G= J] I= IX ⌫iI (@ G @@G ⌦ + ⌦ )[G G (48) G [G ,G⌦ G (47) X = (@ G ig(⌦ ⌦@iJ +µ⌦ ⌦1J )[G ,1JG (48) µµI⌦ ⌫⌫G 1I iJ 1J i@ I ,,(48) J] µ⌫I@ µ ⌫ ig(⌦ 1J ⌫= 1I iJ 1J iI)[G ⌦ G G )⌦ ig(⌦ ⌦ + ⌦ ⌦ )[G G (48) i I⌫D⌦ Iµ Iµ,µ J [µ⌫] ⌫I⌫ (@ig(⌦ µIµ)) ⌫Iantisymmetric ⌫] ]) X = ⌦ (@ G G )))1I + ]iI 1Iig⌦ iJ i[µ⌫] iIIiX I= Iµµ J [µ⌫] I (@ µ µµfollowing ⌫(@ 1I iI i(@ i the I) Iµ J⌫⌫⌫ ] [µ⌫] ⌫⌫⌫ ⌦ µµ ⌫⌫⌫iI G @ G ig(⌦ ⌦ + ⌦ ⌦ )[G , G (48) Thus we obtain following granular and collectives fields X[µ⌫] = ⌦ G @ G ig(⌦ ⌦ + ⌦ )[G , G ] (48) µ ⌫ 1I iJ 1J Thus we obtain the antisymmetric granular and collectives fields ⌫ 1I iJ 1J iI X = ⌦ (@ G @ G ig(⌦ ⌦ + ⌦ ⌦ )[G , G ] (48) [µ⌫] I ⌫ µ µ [µ⌫] I µ µ µ =⌫I ⌦I (@ ⌫µ G 1I iJig(⌦1I 1J⌦iJiIgranular X @ ) + ⌦ )[G , G ] (48) Ii [µ⌫] µ µ ⌫ Gµ 1J ⌦⌫JiIand Thus we obtain the following antisymmetric collectives fields i I I ⌫ µ ⌫ Thus we we obtain the following antisymmetric granular and collectives fields X = ⌦ (@ G @ G ) ig(⌦ ⌦ + ⌦ ⌦ )[G , G ] (48) µ ⌫ 1I iJ 1J iI Thus we obtain the following antisymmetric granular and collectives fields Thus obtain the following antisymmetric granular and collectives fields strengths, [µ⌫] I ⌫ µ µ ⌫ ThusX we following i obtain i theantisymmetric I Iantisymmetric I fields strengths, Thus we obtain the following granular and collectives Thus we obtain following granular and collectives fields strengths, Thus we obtain the following antisymmetric granular and collectives = ⌦the (@ G ig(⌦ + ⌦ )[G ,fields GJ⌫ ] (48) Thus strengths, we oThus btain the following antisymmetric ranular µ ⌫IG 1I ⌦ iJand 1JI⌦iI we obtain the following granular fields I antisymmetric I strengths, J collectives [µ⌫] Igthe ⌫and c@ollectives µ ) fields µ Thus we obtain following antisymmetric granular and fields Iantisymmetric Iiga J collectives strengths, strengths, Z = a (@ G @ G ) [G , G ] (49) I I I J strengths, Z = a (@ G @ G ) iga [G , G ] (49) I µ ⌫ [µ⌫] (IJ) strengths, ⌫ µ µ ⌫ I µ ⌫ [µ⌫] (IJ) Thus we obtain the following antisymmetric granular and collectives fields ⌫ µ µ ⌫ I I I J strengths, Z = a (@ G @ G ) iga [G , G ] (49) strengths, I@ G µ I )I⌫I ⌫ µ [µ⌫] strengths, µ ⌫J I J] II [GIIµ , G(IJ) Zwe = aaIIZ (@ GI= iga (49) J collectives strengths,Z µG [µ⌫]obtain (IJ) J ⌫= a µ)antisymmetric ⌫ ] [GIII , G following granular and fields = (@ G iga [G ,G (49) @@ (49) I )[G ZIthe aI@@ (@ @J⌫⌫G µ ⌫⌫IG [µ⌫] (IJ) IG Iiga µµIIG [µ⌫] (IJ) µ ⌫J Z = a (@ G ) , ] µµ µ,G I@(@ [µ⌫] (IJ) J⌫I⌫I iga I Iµµ JG I⌫J]] J (49) I⌫J⌫I= J strengths, Thus I µ ⌫ [µ⌫] (IJ) I J µ ⌫ Z a (@ G G ) iga [G (49) I I I J Z = a (@ G G ) iga [G , G ] (49) I µ ⌫ [µ⌫] (IJ) I(@µ µ= ⌫µ [µ⌫] (IJ) z b [G , G c {G , G } + G (50) ⌫] + µ µ, G ⌫G Z = a G @ G ) iga [G , G ] (49) I J I J I J ⌫ µ µ ⌫ z = b [G , ] + c {G , G } + G (50) IJ IJ [µ⌫] [IJ] I ⌫ [µ⌫] (IJ) Z = a (@ G @ G ) iga [G ] (49) ⌫ µ ⌫ µ ⌫ IJ IJ [µ⌫] [IJ] ⌫ µ µ ⌫ ⌫cJ ⌫IJ µ ⌫ [IJ] [µ⌫] Iµ = J J Iµ J IbIJI[Gµµ, G I ⌫⌫ IIµ J +J strengths, z[G ] ,+G {G ,IG, G }(IJ) (50) µ Gµ ⌫I G⌫J [µ⌫] µ ⌫ I J I I J I J z = b , G ] + c {G } + G (50) Z = a (@ @ G ) iga [G ] (49) I J I J IJ IJ [µ⌫] [IJ] I IIµ,µ= IIµ J⌫ I GJI G J (50) µ IJIµ ⌫G = bbIJ GJJ⌫⌫J⌫bbb]]IJ +[G {G GcJJ⌫cc⌫J⌫IJ G G G {G ,,µiga G + I+ µ z[G = [G G + (50) IJz IJ I]µ] ,+ [µ⌫][µ⌫] [IJ] [µ⌫] [IJ] µ,, G zz[µ⌫] = + cc⌫IJ G }}}(IJ) + G (50) IJ IJ [µ⌫] µIµ,,,{G ⌫⌫J} Iµ I(IJ) [IJ] z[G = G {G GJG }µµ+ +J⌫⌫J⌫[G G (50) µ Z a+ (@ G @c⌫⌫⌫IJ G ]GJ⌫⌫for (49) J⌫⌫⌫⌫Iµµ],,,+ Iµ = [G Gb= ]reproduces +the {G G +) I µIµ,[IJ] G G (50) Notice that eq.(49) the usual Yang-Mills field strenght for(50) the IJ [µ⌫] [IJ] I[G [µ⌫] IJ IJ [µ⌫] = [IJ] Notice that reproduces usual Yang-Mills strenght the µµG ⌫field zzeq.(49) bbIJ [G ,G ]reproduces cIµcIJ {G G } + G (50) µ µ ,G ⌫µG µ= [µ⌫] z [G , ] + {G G } µ ⌫J⌫IJ µ µ ⌫ Notice that eq.(49) the usual Yang-Mills field strenght for the IJ [µ⌫] [IJ] I I J I J ⌫ µ ⌫ field strenght µ ⌫for the Notice that eq.(49) reproduces the usual Yang-Mills z = b [G , G ] + c {G , G } + G G (50) IJ IJ [µ⌫] [IJ] Notice that eq.(49) reproduces the usual Yang-Mills field strenght for the µ ⌫ µ ⌫ µ ⌫ Notice that eq.(49) reproduces the usual Yang-Mills field strenght for the sector, Notice that eq.(49) reproduces the Notice that eq.(49) symmetric reproduces the usual Yang-‐Mills fthe ield fYang-Mills or the symmetric sector, symmetric sector, Notice that eq.(49) reproduces usual field strenght for the I strenght I Yang-Mills I J Notice that eq.(49) reproduces the usual field strenght for(50) the symmetric sector, Notice that eq.(49) reproduces the usual Yang-Mills field strenght for the for zeq.(49) [G ,usual GJ⌫ ] + cIJusual {G , Yang-Mills GJ⌫field } +↵)I G⌫for Notice that eq.(49) reproduces the Yang-Mills strenght the IJ [µ⌫] = breproduces [IJ] G symmetric sector, ↵)I I µ µ µ Notice that the field strenght the I ↵)I symmetric sector, symmetric sector, I G+↵)I Z = G ⇢ g G (51) symmetric sector, Z = G + ⇢ g (51) symmetric sector, I I µ⌫ (µ⌫) I I µ⌫ (µ⌫) Notice that eq.(49) reproduces the usual Yang-Mills field strenght for the (µ⌫) (↵ (µ⌫) I symmetric sector, Z = G + ⇢ g G (51) (↵ symmetric sector, sector, ↵)II µ⌫ (↵ ↵)I symmetric sector, Zeq.(49) = reproduces G(µ⌫) + ⇢⇢IIIG G (51) for(51) I(µ⌫) I G ↵)I symmetric IZ µ⌫ (µ⌫) = III(µ⌫)= ↵)I (↵ Z G + ggµ⌫ (51) Notice thatZ the usual field strenght the ↵)I I(µ⌫) + ⇢⇢IIgg Yang-Mills = G IZ µ⌫ (µ⌫) = III G µ⌫ G + ⇢ g G (51) (µ⌫) (µ⌫) = µ⌫G (µ⌫) ↵)I (↵ (µ⌫) symmetric sector, I I (µ⌫) (↵ ↵)I Z + G (51) (µ⌫) Z = G + ⇢ g G (51) I (↵ I I µ⌫ (µ⌫) where µ⌫ (µ⌫) = Z (µ⌫) where Z(µ⌫) + ⇢I G G+ (51) (↵ (µ⌫) (↵↵)I IIG IIg(µ⌫) µ⌫ = ⇢ g G (51) where (µ⌫) (↵ I µ⌫ (µ⌫) I symmetric sector, (↵ where I I = IG I I µ⌫ I I J J I (µ⌫) J +@⌫⇢([G where Ig (µ⌫) where I += I ig III , G I,G J ]↵)I where where (↵ G ig [GII⌫ ,, G GJJµ ]) ]) (51) (52) GII(µ⌫) = @ZµG G @⌫@@G ] ([G + [G ,G ]) [G (52) where µG (µ⌫) µ µ ⌫ ⌫J µ+ I⌫ = Iµ ⌫⌫ + Iµ G J I⌫G J where GG G + @J⌫([G G ig ([G , G ] + (52) where where µ(µ⌫) J] ⇢ Z = G + g (51) (µ⌫) µ µ ⌫ ⌫ µ where I I I I J I J G = @ + @ G ig , G + [G , G ]) (52) I I µ⌫ I I I I I I I I J I J (µ⌫) ⌫ J II(µ⌫) = @ µ I I I J I J (↵ ⌫ µ µ ⌫ ⌫ µ G G + @ G ig ([G , G ] + [G , G ]) (52) I I I I J I J G = G + GI µµIµµI,,G (52) = G @J@@J⌫([G G ig ([G [G IG J+ G @@µµµG G @@⌫⌫⌫@@@G ig [G ,G G ]) (52) µIµ ⌫([G (µ⌫) = µ ⌫J]J]J+ µ µµ (µ⌫) µ,I⌫,⌫IG where (µ⌫) ⌫⌫⌫ + µ ⌫⌫ ]J ⌫⌫,, G µµ]) I(µ⌫) I⌫⌫I + J G = + G ig ([G ] + [G G ]) (52) ⌫ µ G = G + G ig G + [G , G ]) (52) I I I J I J ⌫ J J (µ⌫) µ µ and and (µ⌫) = @G ⌫, G]µµ G⌫I⌫ +=@@⌫µGGµIµ⌫ +ig@J⌫ ([G + [G ]) ⌫ , Gµ ]) (52) µ (µ⌫) Gµ µIµ, G ig⌫J⌫J]([G , ⌫IG (52) and G(µ⌫) I J+ [G µ ⌫ where and G = @ G + @ G ig ([G , G ] + [G , G ]) (52) µ ⌫ J and ⌫ µI ⌫ and (µ⌫) and and Iµ I ⌫ J µ I and and and GII ,I(µ⌫) iguIJ,([G G + [G ,{G GvJµI ]) JJ =I @µ G II ⌫ GJ ↵J ↵J I J J⌫ +@ ↵J ↵J and and µ µ ,][G J ↵J ↵J z = b [G G ] + c {G , G } + g G ]⌫+ + {G G(52) z = b [G , G ] + c {G , G } + u g [G G +⌫I↵Iv],,(IJ) gµ⌫ } II↵,, G µ⌫ µ⌫ {G (µ⌫) (IJ) [IJ] (IJ) µ⌫ (µ⌫) [IJ] (IJ) [IJ] µ ⌫ µ ⌫ ⌫[IJ] µ ⌫ ↵ Iµ= bJ I J I ↵J I↵ , G ↵J z [G , G ] + c {G , G } + u g [G G ] v(IJ) gg↵J }} and µ⌫ µ⌫ (µ⌫) [IJ] (IJ) [IJ] µ ⌫ µ ⌫ ↵ ↵ I J I J I ↵J I z = b [G , G ] + c {G , G } + u g [G , G ] + v g {G , G } I J I J ↵J I J I J I ↵J I µ⌫ µ⌫ (µ⌫) = b [IJ] (IJ) [IJ] (IJ) IIµ,= JJ IIµ+ JJ Iu ↵J II↵, Gg↵J ⌫ ⌫ ↵ I J I J I ↵J I ↵J z [G G ] + c {G , G } + u g [G , G ] + v g {G } z b [G , G ] c {G , G } + g [G , G ] + v {G , G } I ↵J ↵J µ⌫ µ⌫ (µ⌫) = b[IJ] [IJ] (IJ) [IJ] (IJ) zz(µ⌫) cc(IJ) {G ,, G }} [G + (53) (53) µ⌫ z(µ⌫) [G ,=G ]] + cIc(IJ) {G ,G }} + uuµµµ[IJ] G ]] + {G ,,G }}{G (µ⌫) [IJ] (IJ) [IJ] µ+ ↵,[IJ] µ,, G and µ⌫ (IJ) Iµµ= J [G G {G GJggg⌫⌫⌫µ⌫ + I↵u gµ⌫ [G ,(IJ) G↵Jgggµ⌫ ] {G + vI↵↵(IJ) g↵J µ ⌫⌫⌫]]Iµ+ (53) ⌫⌫⌫} = bb[IJ] [G= GbbJ⌫⌫⌫[IJ] +[G {G G +Iu [G Gg↵J +Ivv↵↵v,(IJ) {G Gg(53) µ⌫ µ⌫} (µ⌫) [IJ] (IJ) (IJ) J{G ↵J } I↵↵ , G µG µ⌫ µ⌫ (µ⌫) = [IJ] (IJ) [IJ] µ,,G µ,,cG ↵,,[IJ] ↵ zz(µ⌫) [G ] + c + [G G ] + , G µ⌫ µ⌫ (IJ) [IJ] z b [G , ] + {G , G } + u [G G ] + v {G , G } µ ⌫ µ ⌫ ↵ ↵ J following I (IJ) J I[IJ] ↵J I basis ↵J µ⌫to ↵ µ⌫ (µ⌫) [IJ] (53) ⌫{G µu with ⌫g to theb following relationships with respect constructor basis with relationships respect (53) zwith = [GwIµith ,the G + tµco (IJ) +constructor v(IJ) gµ⌫ {G(IJ) , G(53) } ↵ (53) (53) with the following relationships with respect constructor µ⌫ [G (µ⌫) [IJ] (53) ⌫ ]relationships µ, G ⌫ } +respect ↵ , G ]to ↵basis with the following relationships respect constructor basis with the[IJ] following with to constructor basis I J I J I ↵J I ↵J (53) with the following relationships with respect to constructor basis with the following relationships with z = b [G , G ] + c {G , G } + u g [G , G ] + v g {G , G } with the following relationships with respect to constructor basis with the following relationships with respect to constructor basis µ⌫ µ⌫ (µ⌫) [IJ] (IJ) [IJ] (IJ) (53) µ ⌫ relationships µ ⌫ torespect ↵ with the relationships following with to ↵constructor basisi with the thewith following relationships withiirespect respect constructor basisi irelationships with following with to constructor basis i the following with respect to constructor basis a = d⌦ + ↵ ⌦ a = a ⌦ + ↵ ⌦ ⌦ a = d⌦ + ↵ ⌦ a = a ⌦ + ↵ ⌦ ⌦ I 1I i I 1J i 1I (IJ) I 1I i I 1J i 1I (IJ) a = d⌦ + ↵ ⌦ a = a ⌦ + ↵ ⌦ ⌦ I J (53) J I ↵ ⌦ii 1I with i IIirespect 1J⌦ii i 1I J with the following aI = = d⌦ d⌦relationships + a(IJ)to= =constructor aI ⌦ ⌦(IJ) +j ↵ ↵iI⌦ ⌦basis 1I a 1J 1Ii⌦ jiI jj ii iiJjj iii i ii⌦ i + i↵ + ↵ ⌦ aa aa + = d⌦ + ⌦ i⌦ i⌦ I = 1I Ib 1J i⌦ 1I1J (IJ) = I⌦= 1I i⌦ I⌦ 1J 1I ⌦J (IJ) aaabI(IJ) d⌦ + ↵ a a ⌦ ↵ a d⌦ + ↵ a = a + = a ⌦ b b ⌦ ⌦ Iia JJ ↵iii⌦ I J b = ⌦ = b ⌦ ⌦ 1I i I 1J i 1I (IJ) I 1I i I 1I (IJ) (ij) [IJ] [ij] j j i i (IJ) (ij) [IJ] [ij] I J with the following relationships with respect to constructor basis I b = a ⌦ ⌦ b = b ⌦ ⌦ = d⌦ + ↵ ⌦ = ⌦ + ↵ ⌦ ⌦ I I J J I 1I i I 1J 1I (IJ) i i I I J I J i i (IJ) (ij) [IJ] [ij] I = d⌦ 1I i I 1J i 1I (IJ) I I I J J J abbI(IJ) + ↵ ⌦ a a ⌦ + ↵ ⌦ ⌦ j j j j = a ⌦ ⌦ b = b ⌦ ⌦ i i 1I i jjJIi1I + ↵ Ia(IJ) 1J iI ⌦1I (IJ) a(ij) = d⌦ + jjJj a Ji jjjj↵i ⌦1I ⌦J (ij) [IJ] = b [ij] jj Ijj iiI i⌦= Ib(IJ) 1Jiiiii⌦ iiiI⌦ ii iiii⌦ = a ⌦ ⌦ b ⌦ j = a ⌦ ⌦ b = b ⌦ (IJ) [IJ] [ij] (ij) [IJ] [ij] babc(IJ) = a ⌦ b = b ⌦ ⌦ b = a ⌦ ⌦ b = b ⌦ ⌦ = d⌦ + ↵ ⌦ a = a ⌦ + ↵ ⌦ ⌦ b ⌦ ⌦ c c ⌦ ⌦ I I I I J J J J c = b ⌦ ⌦ c = c ⌦ ⌦ jJ jJ [ij] (ij) [IJ] [ij] (IJ) (ij) [IJ] I[IJ] = a1I i jJjJ I [ij] I 1J i 1I (IJ) [ij] (ij) iII ⌦ i I [IJ] (IJ) (ij) ⌦ b = b ⌦ ⌦ I I J J J I I (IJ) (ij) [IJ] [ij] j j i i I I J J I I [IJ] [ij] (IJ) (ij) i i J J (IJ) (ij) [IJ] [ij] Ii⌦jJj i Ii Jjj J Jjjbj bcc(IJ) = bab[ij] ⌦j= bcc[IJ] ⌦⌦ jJ (54) ⌦⌦= ⌦ =bcc[ij] ⌦IiIi⌦ ⌦ (54) iI= (ij) i ba(IJ) = ⌦ bac[IJ] ⌦1J ⌦ [IJ] = (IJ)= (ij) Jj= iiIIi⌦ [ij] iiii+ jj↵i ⌦1I ⌦ i⌦ Ii↵ I⌦ JjJjjI J iIii⌦ jJjab1I jJ a ⌦ = b(ij) ⌦ ⌦ cu(ij) ⌦ ⌦ d⌦ + ⌦ = ⌦ i⌦ [IJ] [ij] (IJ) (ij) [IJ] [ij] (IJ) cbcu[IJ] cbcu(IJ) = ccu ⌦ i⌦ I(ij) (IJ) (54) cuIc⌦ = ⌦ c c ⌦ = bbau[ij] = b ⌦ ⌦ ⌦ = ⌦ ⌦ I⌦ II⌦ J JjJ J u ⌦ u = ⌦ jJ= jJJ (ij) (IJ) (ij) [IJ] [ij] [IJ] [ij] (IJ) [IJ] = [ij] (IJ) (ij) j j iIIiI⌦ i I [ij] (ij) = ⌦ ⌦ = ⌦ ⌦ I I I I I [IJ] [ij] (IJ) (ij) J J J J J j j i i I J I I [IJ] [ij] (ij) (54) J J (54) [IJ] [ij] (IJ) (ij) i i Ii j Jj (54) cu ⌦⌦ cu ⌦⌦ j (54) u[IJ] =bu u[ij] ⌦Iiii⌦ ⌦ u(IJ) =cu u(ij) ⌦IiIii⌦ ⌦= iiiIii J⌦jjjjJ jjjJ c[ij] ba[ij] ⌦⌦ ⌦ c(ij) = c[ij] ⌦ [IJ]= [ij] (IJ)= (ij) JjJ= JjJjjjJ (54) I= I= j [IJ] (IJ) (ij) 1(IJ) ii ⌦ 1⌦i⌦ I⌦ I⌦ J jJj = ⌦ = ⌦ u u u u ⌦ b = b b ⌦ iI⌦ 1 i i (54) [IJ] (IJ) [IJ] [ij] (ij) u u ⌦ ⌦ u u ⌦ ⌦ (IJ) (ij) [IJ] cu[IJ] = b ⌦ ⌦ c = c ⌦ ⌦ = g = g ⌦ u u ⌦ u = u ⌦ I I I J J J I I = ⌦ ⌦ g = g ⌦ j j J J [IJ] [ij] (IJ) (ij) (54) = ⌦ [ij] (IJ) (ij) I [IJ] (ij) [IJ] [ij] (IJ) (ij) j I I iiI ⌦ iI ⌦ JJ [ij] JJ (ij) I = u ⌦ u = u ⌦ (ij) I I [ij] (IJ) I J J I I 1 j j J J I I J I I I [IJ] [ij] (ij) J J [IJ] [ij] (IJ) (ij) i i I I J u[IJ] ⌦ ugg(IJ) u(ij) ⌦Ii ⌦ jju (54) = u[ij] == i j jjj jJ iI i⌦ u = ⌦⌦ u =gJj(ij) ⌦jJ I = [IJ] = [ij] (ij) iIiiiI ⌦ 111 i⌦ I = [ij] (IJ) (ij) ii⌦ i⌦11 iI⌦ 1⌦iI⌦ jJb[ij] JJ ⌦ = ⌦ ⌦ ggg⇢(ij) ciiI[IJ] = c⌦ cu(ij) ⌦ [IJ] [ij] [IJ] [ij] (ij) ⌦ ⌦ gug⇢II(IJ) g= ⌦[ij] [IJ] u[IJ] ⌦ ⌦= u⌦ ⌦ ⇢gg(IJ) = ⇢g(ij) ⌦ Ii ⌦ j= I == iu I= i⌦ =I= J [IJ] [ij] I[IJ] iJ(ij) IIIIII⌦ [IJ] iIIII= 1 JJ iI(ij) = ⌦ ⌦ = ⌦ [ij] J ii⌦[ij] IIi ⌦ Ii⌦1III I J I I j J I J I [IJ] [ij] (ij) (54) J = ⌦ ⌦ g = g ⌦ = ⌦ ⇢ = ⇢ ⌦ j [IJ] [ij] I = i⌦ i⌦iig = ⌦ iiiI[IJ]Ii= Iu⇢ iiI⌦ iIi I ⌦i ⌦j II1I= ⌦ =g= ⇢(ij) ⌦ ⇢⇢IIIII = = ⇢⇢(ij) Ii⌦ J I= i[ij] i⌦ uI [IJ] =JjJ u ⌦ ⌦ u I = i⌦ ⇢ ⇢ Ii⌦ ⌦ ⇢ = ⇢ = ⌦ ⌦ g g [ij] (IJ) (ij) I[IJ] iii⌦ i I I I I J J I i I i i i = ⌦ = ⌦ [ij] (ij) I i I i I I I I I I I I i Physics does not depend on the fields basis. The symmetry results obtained in the constructor basis can be transposed to I = ⇢ ⌦i ⇢I = ⇢i ⌦Ii⇢ I = i ⌦IiII = i ⌦iI i j I = gi I⌦1 = strengths =the ia⌦bove ⇢I = ⇢it⌦ I the physical basis [28]. Notice tIhat fields are he Ig covariance [ij] ⌦I ⌦ (ij)property I J also preserving I [IJ] i ⇢I = ⇢i ⌦iI I = i ⌦I

12 12 12 12 12 12 12 12 4935 | P a g e 12 12 12 J u n e 2 0 1 7 12 w w w . c i r w o r l d . c o m

Zµ⌫Z(G Zµ⌫ZI(G = U ZUIµ⌫Z (G )UIZ)U )! 1 (GI )U 1 0=(G I) Z I )0 IZ IU )Z ! (G )Zµ⌫= µ⌫ µ⌫ (G I! µ⌫ µ⌫ Zµ⌫ (G Z(G ) = U(G Zµ⌫I))(G 0= (G(G (G U Zµ⌫ I) ! µ⌫ I )U I )I! µ⌫ µ⌫ (GI )U 1 Zµ⌫ 0 0Zµ⌫ (GI0 ) = U Z 1 µ⌫1 (GI )U1 µ⌫ (GI ) ! zµ⌫z(G ) ! z (G ) = U z (G )U (55) z (G ) ! z (G ) = U z 0 1 0 1 I µ⌫ I µ⌫ I (G ) ! z (G ) = U z (G )U (55) (55) µ⌫ Iµ⌫ ) )! µ⌫U (G I ) I )U µ⌫ ) I! I=z µ⌫0 =)U I µ⌫ (G zµ⌫ (G zµ⌫(G (G z(G zzµ⌫ (55) (55) 1 I )I ! µ⌫ µ⌫ (G I )U I depend µ⌫I(G I U Physics does not fields = UzzThe 2 3 4 7obtained -‐ 3 4 (55) 8 7 µ⌫ (Gon I thezµ⌫ I )basis. µ⌫ (Gsymmetry I )U I S S Nresults Physics does not depend on the fields basis. The symmetry results obtained o l u m e basis 1 3 N[28]. u m bNotice e r 6 in the constructor basis can be transposed to theV physical basis can be transposed toJ othe u r physical n a l o f basis A d v a[28]. n c e sNotice i n P h y s i c s in the constructor AA further consistency onon how such non-abelian whole symmetry implethat the above fields strengths are alsonon-abelian preserving the covariance property AA further consistency on how such whole symmetry implefurther consistency how such non-abelian whole symmetry implefurther consistency on how such non-abelian whole symmetry impleA further consistency on how such non-abelian whole symmetry imple that the mentation above fields strengths are also preserving the covariance property A further consistency on how such non-abelian whole symmetry implementation at SU (N ) gauge group was studied in a previous work [25]. Anat SU (N ) gauge group was studied in a previous work [25]. Anmentation at SU (N ) gauge group was studied in a previous work [25]. An0 1 SU (N group wasIin previous Anmentationmentation at SU (N )at gauge group studied a= previous [25].work An- [25]. Z))µ⌫gauge (Gwas Zµ⌫ (G )studied U Z1µ⌫in (GaaIwork )U I) ! mentation at SU (N gauge group was studied in previous work [25]. An0 alyzing through the constructor basis one derives eight aspects attached to alyzing through the constructor basis one derives eight aspects attached to alyzing through the constructor basis one derives eight aspects attached to Zconstructor ! constructor Zµ⌫ (GI )one = basis U Zµ⌫ (G )U µ⌫ (GI ) I alyzingthe through the one derives eight aspects attached to alyzing through basis derives eight aspects attached to 0 1 alyzing through the constructor basis one derives eight aspects attached to z (G ) ! z (G ) = U z (G )U (55) µ⌫ I µ⌫ I µ⌫ I group generators and gauge parameters. They are from groups generators: 0 1 are group generators and gauge parameters. They from groups generators: group generators and gauge parameters. They are from groups generators: group generators gauge They are from groups generators: zµ⌫gauge (GI ) and ! zµ⌫ (GI ) parameters. = They U zµ⌫ (G (55) group generators and parameters. are from groups generators: I )U group generators and gauge parameters. They are from groups generators: algebra closure through Jacobi identities and Bianchi identities. From gauge algebra closure Jacobi and Bianchi identities. From algebra closure through Jacobi identities and Bianchi identities. From gauge algebra closurethrough through Jacobiidentities identities andidentities. Bianchi identities. From gauge gauge algebra closure through Jacobi identities and Bianchi From gauge algebra closure through Jacobi identities and Bianchi identities. From gauge parameters: Noether Theorem, gauge fixing, BRST symmetry, global transA further consistency o n h ow s uch n on-‐abelian w hole s ymmetry i mplementation a t g auge g roup w as s tudied in SU(N ) parameters: Noether Theorem, gauge fixing, BRST symmetry, global transparameters: Noether Theorem, gauge fixing, BRST symmetry, global transparameters: Noether Theorem, gauge fixing, BRST symmetry, global impletransparameters: Noether Theorem, gauge fixing, BRST symmetry, global transA further consistency on how such non-abelian whole symmetry parameters: Noether Theorem, gauge fixing, BRST symmetry, global transa previous work [25]. Analyzing through the constructor basis one derives eight aspects attached to group generators and gauge formations (BRST, ghost scale, gauge global); charges algebra, covariant A further consistency on how such non-abelian wholealgebra, symmetry imple- covariant formations (BRST, scale, gauge global); charges algebra, formations (BRST, ghost scale, gauge global); charges covariant formations (BRST, scale, gauge global); charges algebra, formations (BRST, ghost scale, gauge global); charges covariant mentation at SUalgebra (Nghost )ghost gauge group was studied aalgebra, previous work From [25]. gauge Anparameters. They are from formations groups generators: closure through Jacobi identities inand Bianchi algebra, identities. (BRST, ghost scale, gauge global); charges equations of motion plus PoincarÃľ lemma. mentation at SU (N ) gauge group was studied in a previous work [25]. An- covariant equations of motion plus PoincarÃľ lemma. equations of motion plus PoincarÃľ lemma. equations of motion plus PoincarÃľ lemma. equations of motion plus PoincarÃľ lemma. parameters: Noether Theorem, gauge through fixing, BRST symmetry, global transformations (BRST, ghost scale, gauge global); charges alyzing the constructor basis one derives eight aspects attached to equations of motion plus PoincarÃľ lemma. As example to express the consistency for introducing gauge families, let alyzing through the basis one derives eight aspects attached to As example express the consistency for introducing gauge families, let example to express the consistency for introducing gauge families, let example to express the consistency for introducing gauge families, algebra, covariant eAs quations of As mAs otion pto lus constructor Poincaré lemma. example to express the consistency for introducing gauge families, let group generators and gauge They are from groups generators: As example tounification. express theparameters. consistency for introducing gauge transformafamilies, let us us take the Jacobi identity Taking the infinitesimal transformagroup generators and gauge parameters. They are from groups generators: take the Jacobi identity unification. Taking the infinitesimal transformaus take Jacobi unification. Taking the infinitesimal us take the Jacobi identity unification. Taking the infinitesimal transformaus take Jacobi identity unification. Taking the infinitesimal transformaAs example to ethe xpress the cthe onsistency for iidentity ntroducing gauge fidentities amilies, let us take the Jacobi iidentities. dentity unification. Taking the algebra closure through Jacobi and Bianchi From gauge useq.(46) take the Jacobi identity unification. Taking the infinitesimal transformation from eq.(46) algebra closure through Jacobi identities and Bianchi identities. From gauge tion from tion from eq.(46) infinitesimal transformation from efrom q.(46) tion eq.(46) tion from eq.(46) parameters: Noether Theorem, gauge fixing, BRST symmetry, global transtion from eq.(46) parameters: Noether Theorem, gauge global transa a a a a fixing, aa BRST b b c symmetry, a c a b c a a a b c GaµI == !G + gf GbcbµI !µIgf (x) formations (BRST, scale, algebra, covariant a (x) agauge cglobal); µG GµI @ (x) + gf G ! (x) =+ (x) gf GµIµI !charges @@µgf !!bc (x) ++ (x) bc µ µ G = @@ghost (x) (x) µI µI! bcbc a= aG aG b! c(x) formations (BRST, ghost gauge charges algebra, covariant µ !G µI scale, µI ! (56) = a@global); + gf G ! (x) (56) (56) µa! (x) µI bc µI (56) equations of motion plus PoincarÃľ lemma. aa = [D !(x)] a[DµI!(x)] (56) µI = !(x)] = [D !(x)] equations of motion plus PoincarÃľ lemma. = [D µI µI [DµI !(x)] a = [D As example to=express the consistency for introducing gauge families, let µI !(x)] a a gauge As example toJacobi expressidentity the for introducing families, let a consistency one hashas toone verify acting onacting thethe field Gthe a hasto to verify Jacobi identity acting onG field GaaµI to verify Jacobi acting on field Gfield µI one has Jacobi identity on the Gidentity one has to verify Jone acobi identity atake cting othe n verify the field us Jacobi unification. Taking the G infinitesimal transformaone has to verify identity acting on the field atransformaµ Iidentity µI µI us take the Jacobi identity unification. Taking theoninfinitesimal one hasJacobi to verify Jacobi identity acting the field GµI µI a a tion {[ from eq.(46) a + [2 ,32[,, [3[3]]3,1]][,,++ ]]} G =11]]} 0 G [ 212,]]} ,[ [aµI = 0 (57) 1 , [ ,2[, 23,{[ 2,]][ + tion from eq.(46) = 0GaµI (57) (57) 1[, [+ 3,,µI ,]] ,3]][, [[+122,1[,, [[222]],33]][,,++ 311,[]]} µI {[ {[ ]]]]13+ G = 0]]} 3G a = 0 (57) 1 , [1 2 , {[ 3{[ 3 1 1+ 2[]][32+ µI , [ , ]] + [ , [ , ]]} G (57) 1 , [ 2 , 3 ]] 3 1 2 2 3 1 a a a b c µI = 0 G a a µI = @µ !a (x) b +cgfbc GµI ! (x) Showing that Showing that G = @ ! (x) + gf G ! (x) that thatµI Showing tShowing hat Showing µ Showing bc µI that (56) a Showing that (56) = [D !(x)] a µI a a abc b c b c a b c a abcµIb!(x)] b cc b ca b cc c (D c ↵ =↵abc [D a abc bµI ]G2 ]G = gf2gf [(D )b[(D ↵[(D ↵µI1c1)↵ )b (D ↵)(D ]= = gf Dbc2cc]µI (↵ ↵abcaac2b1D )DcµI (58) aµI abc c2b ↵ a↵↵ b(↵ ]G =µI ↵2 )2(D )↵ ↵ )bbc ]D = gf ↵(58) (58) ↵ )1c1↵µIµI gf ↵ ) (↵2b(58) µI 2↵ µI 2b↵ 1)) [ [= ,1 ,= gfgf = (58) 2↵ µIgf [[ 11[,, 122,]G gf ↵µI22)abc (D gf (↵ 1↵ 2 ]↵= 2 ]G 11) bc a=[(D b↵ bD µI µI µI[(D 1 1 1↵ 2 µI bc1 )D1µI (↵ µI 1[µI 2 ]µI 2a , ]G = gf [(D ↵ ) ↵ (D ↵ ) ↵ ] = gf (↵ ↵11c ) (58) 1 to2 verify µI 2 acting µI the 1 field µI µI 1 2 bc 2 4 Classical Equations one has Jacobi identity on G 4 Classical Equations a µI one has to verify Jacobi identity acting on the field G one verifies eq.(56) similarly to the YM case. Consequently, eq.(46) is saying µI one verifies eq.(56) similarly to the YM case. Consequently, eq.(46) is saying one verifies eq.(56) similarly to the YM similarly case. Consequently, eq.(46) is saying that it is possible to eq.(46) derive is a eq.(46) Lagrangian the verifies eq.(56) similarly to YM case. Consequently, is saying 4 one Classical Equations one verifies eq.(56) similarly to the YM case. Consequently, iswhere saying one verifies eq.(56) tosimilarly thethe YM case. Consequently, eq.(46) saying 4 Classical Equations a theory. one verifies to the YM case. Consequently, eq.(46) is saying number of potential fit ields ipossible s not necessarily eeq.(56) qual to t,he n2Lagrangian umber of Lagrangian g[ roup g1enerators a[s number r2uled b,y of Y1number ang-‐Mills Tfields he introduction that is to derive a Lagrangian where the number potential fields {[ [ , ]] + , [ , ]] + , [ ]]} G = 0 (57) a that it is possible to derive a where the of potential fields 1 3 3 2 3 that it is possible to derive a where the of potential µI of The is + to where nature works as objective this work where nature works that itobjective is of to Lagrangian the= potential fields that itClassical is possible to derive athis Lagrangian where ofnumber fields {[it , [possible ]] + [derive ,work [ to [ promote , where ]]}a Gmodel 0potential (57) as 1possible 2 , 3of 3is 1 ,apromote 2 , [a the 3model 1number µI 4 The that is to derive a2 ]]of Lagrangian the number ofYangpotential fields of Yang-‐Mills families in necessarily a same gEquations roup becomes a rto ealistic wnumber hole p hysics to where be where understood. SU(N ) equal 4 Classical Equations is not necessarily equal the of group generators as ruled by Yangis not equal to the number group generators as ruled by The objective of this work is to promote a model nature works as is not necessarily to the number of group generators as ruled by Yanggrouping. At this way from physical fields set {G } sharing a common gauge grouping. At this way from physical fields set {G } sharing a common gauge isThe notShowing necessarily to theis number of group generators asnature ruled by Yangnot necessarily equal toofequal the number of group generators by YangIas ruled objective this work to promote a model where works as that 4 isMills Classical Equations is not necessarily equal to the number of}Igroup generators as ruled by YangShowing that Mills theory. The introduction ofthe Yang-Mills families in aSU same SU (N )theory. group theory. The introduction of of Yang-Mills families in aissame same SU (N ) group group grouping. At this way from physical fields set {G sharing a common gauge The Mills Yang-‐Mills extension becomes realistic. Eq.(46) underly basis for deriving a whole non-‐abelian gauge It I step Mills theory. The introduction Yang-Mills families in a same (N ) group transformation as eq.(46) says, a next to build up the wholeness transformation as eq.(46) says, a next step is to build up the wholeness Mills theory. The introduction of Yang-Mills families in a same SU (N ) group theory. The introduction of Yang-Mills families in a SU (N ) The objective of this is abc tofrom promote acpromote model where works as works grouping. Atwork this way physical fields set {G }nature sharing anature common gauge The objective ofphysics this work is understood. to aorentz model where as Mills theory. introduction of families in same aThe b Yang-Mills b cIup the aa wholeness bSUc 1(N 1 ) group becomes a realistic whole physics to be understood. becomes a realistic whole to be transformation as eq.(46) says, a next step is to build ( , ) contains scalar becomes and v ector s ectors t o b e u nderstood. T his f act i s a lready p redicted f rom L G roup r epresentation . A n ext a abc b c b c a b c [ , ]G = gf [(D ↵ ) ↵ (D ↵ ) ↵ ] = gf D (↵ ↵ ) (58) becomes a realistic whole physics to be understood. principle through (2N +↵7) interconnected classical equations. From principle through 7) interconnected equations. From eq.(55), 1agf 2(2N µI 2 a↵ µI 1 is µI gauge 2eq.(55), whole physics to be a2transformation realistic whole physics to be µI+as 11 classical 2µI bc 2 1 2wholeness says, amodel step to build the grouping. At this way from physical fields {G }= sharing a(↵ common [Classical ]G = [(D ↵eq.(46) (Dunderstood. ↵ ]I understood. gf Dnature ) up (58) The objective of this work is to where as 1 ,becomes µI 2 ) promote µI set µI 1 physics bc 2 ↵1works grouping. At this way from physical fields set {G }Eq.(46) sharing a common gauge becomes arealistic realistic whole to)next be2understood. Equations 4 Classical Equations The Yang-Mills extension becomes realistic. underly the basis Iunderly The Yang-Mills extension becomes realistic. Eq.(46) the basis principle through (2N + 7) interconnected classical equations. From eq.(55), step is to open 4 it tone hrough t he c orresponding d ynamics. The Yang-Mills extension becomes realistic. Eq.(46) underly the basis one gets a systemic Lagrangian gets a systemic Lagrangian The Yang-Mills extension becomes realistic. Eq.(46) underly the basis The Yang-Mills extension becomes realistic. Eq.(46) underly the basis through (2N + 7) interconnected classical equations. From eq.(55), transformation as eq.(46) says, a next step is to build up the wholeness grouping. Atprinciple this way from physical fields set {G } sharing a common gauge I The Yang-Mills extension becomes realistic. Eq.(46) underly the basis one verifies eq.(56) similarly to the YM case. Consequently, eq.(46) is saying transformation as eq.(46) says, a next step is to build up the wholeness for anon-abelian whole non-abelian gauge theory. It scalar contains scalar and vector forfor deriving aderiving whole non-abelian gauge theory. Ittheory. contains and vector one verifies eq.(56) similarly to the YM case. Consequently, eq.(46) is saying gets aone systemic Lagrangian deriving a whole gauge theory. It contains scalar and vector transformation for deriving a whole non-abelian gauge theory. It contains scalar and vector for deriving a whole non-abelian gauge It contains scalar and vector gets a systemic Lagrangian principle through (2N + 7) interconnected classical equations. From eq.(55), as eq.(46) says, a next step is to build up the wholeness The objective of this work is to promote a model where nature works as The objective of this work is to promote a model where nature works as principle (2N +L7)= equations. From for deriving aunderstood. whole non-abelian gauge Itofcontains scalar andeq.(55), vector thatunderstood. it is possible derive ainterconnected Lagrangian where the number of potential fields sectors tothrough be This fact is already predicted from Lorentz Group sectors to be This fact is already predicted from Lorentz Group that it isto possible derive ato Lagrangian where the number potential fields L +theory. Lclassical (59) L += LisL (59) A S predicted A Spredicted sectors to be understood. This fact is already predicted from Lorentz Group sectors be understood. This fact is already from Lorentz Group sectors to be understood. This fact already from Lorentz Group one gets a systemic Lagrangian principle through (2N + 7) interconnected classical equations. From eq.(55), 1way 1 Lagrangian 4 Classical Equations grouping. Atnot this way from physical fields set {G }group sharing asharing common gauge 1 be grouping. At from physical set {G }through a Yangcommon gauge sectors to understood. This fact is already predicted from Lorentz one gets systemic is to the number of as ruled YangIthrough LA =next L + L (59) by Group Ithe 1a 1this representation ,1equal ).step step is to open itgenerators the corresponding A S fields 1,). representation ( 112necessarily ,(equal A next is to open it corresponding is not necessarily to1(A ofstep group generators as ruled by 2the 21 number representation next step is to open it through the corresponding L = Lit +open L (59) 2 ).2 ). representation ( , A next step is to open corresponding A through S it the representation ( , ). A next is to through the corresponding 1 2 one gets a systemic Lagrangian 2 2 transformation as eq.(46) says, a next step is to build up the wholeness 2( ,a 2eq.(46) as says, next step isit tothrough up the wholeness representation ). A is toas open the corresponding Mills theory. The ofaYang-Mills families in a(N same SU (N )fields group The Mills objective of transformation this work is introduction to promote model where nature works grouping. At build this way from physical set dynamics. dynamics. 2 introduction 2 of theory. The Yang-Mills families in a same SU ) group dynamics. L = LA +LLclassical (59) dynamics. S LA + LS dynamics. = (59) principle through (2N + 7) interconnected equations. From eq.(55), principle through (2N + 7) interconnected classical equations. From eq.(55), dynamics. A S a common gauge transformation as eq.(46) says, a next step is to build up the wholeness principle through {GI } sharing a realistic whole physics to be understood. becomes a becomes realistic be [µ⌫] physics [µ⌫] [µ⌫] (µ⌫) (µ⌫) [µ⌫] whole [µ⌫] [µ⌫] L[µ⌫] =z3to L +zunderstood. L (59) A S [µ⌫] L =gets Z + z + Z z13 +⇠ Z(µ⌫) +⇠ z3 Z +⇠ L gets = 1one Zc[µ⌫] Z + z + Z +⇠ Z Z1(µ⌫) +⇠Z2Eq.(46) z(µ⌫) z (µ⌫) +⇠ z (µ⌫) one a systemic Lagrangian a systemic 1Z 2eLagrangian 3 2 zunderly 3 Z(µ⌫) z 2 z[µ⌫] 1 [µ⌫] (µ⌫) [µ⌫] (µ⌫) (µ⌫) (µ⌫) The Yang-Mills extension becomes realistic. the lassical e quations. F rom q.(55), o ne g ets a s ystemic Lagrangian (2N + 7) interconnected [µ⌫] [µ⌫] becomes [µ⌫] (µ⌫) (µ⌫)the (µ⌫) basis The Yang-Mills extension realistic. Eq.(46) underly basis 13 L = 1 Z[µ⌫] Z + z + +z 3 Zz[µ⌫] z+13+⇠ Z+⇠ Z+⇠2 zZ z +⇠ +⇠ z+⇠ Z(60)z (µ⌫) [µ⌫] [µ⌫] (µ⌫) 13 2 z[µ⌫] 1 Z13 3 Z(µ⌫) (µ⌫) (µ⌫) (60) L Za[µ⌫] z (µ⌫) for= whole2 non-abelian theory. Itscalar contains and vector 1 Z[µ⌫]non-abelian 3 Zgauge 1 (µ⌫) 2 zscalar 3 (µ⌫) [µ⌫]gauge [µ⌫] z13 (µ⌫) for deriving a deriving whole theory. It contains and vector (60) L = L + L (59) L = L + L [µ⌫] [µ⌫] [µ⌫] (µ⌫) (µ⌫) (µ⌫) (59) (µ⌫) A SZ A S Eq.(59) provides the following whole non-abelian Lagrangian Eq.(59) provides the following whole non-abelian Lagrangian [µ⌫] [µ⌫] [µ⌫] (µ⌫) (µ⌫) (60)z (µ⌫) L = Z Z + z z + Z z +⇠ Z +⇠ z z +⇠ Z z [µ⌫] [µ⌫] [µ⌫] (µ⌫) (µ⌫) +⇠Group sectors to2 be understood. fact1 ispredicted already from 1 to[µ⌫] 3 z 2 Z 3 Lorentz [µ⌫] [µ⌫] (µ⌫) (µ⌫) sectors be This fact isThis already from Lorentz Group L =understood. Z + + non-abelian +⇠11Lagrangian Zpredicted +⇠ z(µ⌫) 1 2 3 2 3 (µ⌫) 1 Z[µ⌫] 21z[µ⌫] 3 Z[µ⌫] 2 z(µ⌫) 3 Z(µ⌫) [µ⌫]the [µ⌫] [µ⌫] z (µ⌫) (µ⌫) (µ⌫) (µ⌫) Eq.(59) provides following whole [µ⌫] [µ⌫] [µ⌫] (µ⌫) (µ⌫) 1 Eq.(59) provides the non-abelian Lagrangian (60) , 23). Afollowing isitZtothrough open through corresponding L representation = 1 Z[µ⌫] Zrepresentation +( 1 2, z1[µ⌫] zA (next + Z znext +⇠step +⇠2 zit(µ⌫) z corresponding +⇠the z 1 Zwhole 3 Z(µ⌫) [µ⌫] (µ⌫) ). step is to open the (60) 2 2 2 L = LKL+=LLI K++Lm (61) (61) LI + Lm Eq.(59)dynamics. provides the following whole non-abelian Lagrangian (60) dynamics. Eq.(59) provides whole non-abelian Lagrangian L the =Lagrangian Lfollowing L (61) K +LL= I + m Eq.(59) provides the following whole non-‐abelian LI Lagrangian + Lm (µ⌫) (µ⌫) [µ⌫] [µ⌫] [µ⌫] [µ⌫] LK + [µ⌫] (µ⌫) (µ⌫) (61) (µ⌫) (µ⌫) provides the following =Z [µ⌫] z[µ⌫] +non-abelian Z[µ⌫] z(µ⌫) +⇠ Z+⇠ Z(µ⌫) zthe z(µ⌫) LEq.(59) = 1 ZL +that z[µ⌫] z+ 2+ Zzshould +⇠ Z z+⇠ z+⇠unique 1Z 3 1considered 3 Z(µ⌫) z [µ⌫] (µ⌫) (µ⌫) 2Z 3whole 1Z 2z 3Z [µ⌫] [µ⌫] z Eq.(60) shows Yang-Mills not be more as2+⇠ unique Eq.(60) shows that Yang-Mills should not be more considered as the L= LK +L L= Lmore (61) I + m L + L + L (61) Eq.(60) shows that Yang-Mills should not be considered as the unique K I m (60) (60) 13 L = L + L + L (61) I C, m C,considered 13At(N Lagrangian performed from Yang-Mills SU (N ).SU Apendice one studies Lagrangian performed from ).KAt not Apendice didactically, Eq.(60) shows that should bedidactically, more as one the studies unique L = L + L + L (61) K I m Lagrangian performed from SU (N ). At Apendice C,isdidactically, one Eq.(59) provides the following whole non-abelian Lagrangian Eq.(59) provides the following whole non-abelian Lagrangian each building block for eq.(60). The kinetic term givenis by each building block for eq.(60). The kinetic term given bystudies Eq.(60) shows that Yang-‐Mills should not be more considered as the unique Lagrangian performed from SU(N ) . At Apendice C, Lagrangian performed from SU (N ). more At Apendice C, didactically, one studies Eq.(60) shows that Yang-Mills should not be considered as the unique Eq.(60) shows that Yang-Mills should not be more considered as Eq.(60) shows that Yang-Mills should not be more considered as the the unique unique each building block for eq.(60). The kinetic term is given by didactically, Eq.(60) one s tudies e ach b uilding b lock f or e q.(60). T he k inetic t erm i s g iven b y each building block for eq.(60). The kinetic term is given by Lagrangian performed from SU (N ). At Apendice C, didactically, one studies shows that Yang-Mills should not be more considered as the unique I µ ⌫J Lagrangian performed from SU (N ). At Apendice C, didactically, one studies I µ ⌫J L = L + L + L (61) L = L + L + L (61) LagrangianLK performed SU I(N ).)(@ At Apendice C, one studies IJ )(@ µm = 2( a2( ) didactically, LK1 a=Ifrom ⇠1m G )(@ G ) JK+ I + JK µIIG 1 a⇠I1aJ ⌫ given µ ⌫J ⌫ by each building block for eq.(60). The kinetic term is Lagrangian performed from SU (N ). At Apendice C, didactically, one studies each building block for eq.(60). The kinetic term is given by LK = block 2( L a= + )(@ G )I )(@ µ µJ 1 aIfor J eq.(60). JThe each building kinetic term is ⌫given ⌫ )(@ I )(@ ⌫ GµJ 2(⇠11a1⇠aI1Ia +⇠⇠µ11G G⌫J))by K Ia I J µGI⌫)(@ 2( Yang-Mills ashould 2( G 1The Ia J kinetic J )(@ µ JIJterm IIG µmore ⌫J)(@ Eq.(60) shows that should not be as the unique each building block forYang-Mills eq.(60). is given by ⌫ ) considered Eq.(60) shows that not be more considered as the unique ⌫ µJ I )(@ µG ⌫J )II (62) (62) 2( aaIIaaJJ + ⇠⇠11 a↵I G ⌫µ µJ ⌫J II aJJ )(@ µ LKperformed =from 2(8⇠1L )(@ G )(@ G ) J⌫⌫IApendice I )(@ µG ⌫J ) ↵I Jµ 1 µ 2( ⇠ )(@ G = + G Lagrangian from SU (N ). At C, didactically, one studies 1 I J 1 J K 1 I J 1 I J µ Lagrangian performed SU (N ). At Apendice C, didactically, one studies + ⇢ (@ G )(@ G ) ⌫ L1 KI+=J8⇠2(1↵↵I aJI a(@J ↵+G⇠JI1 )(@ )(@ G )(@ G ) ⌫ ) (62) I µJG µ I1 ⇢ ⌫ (62) I ⌫ ⌫J J µJ L = 2( a a + ⇠ )(@ G )(@ G ) + 8⇠ ⇢ (@ G )(@ G ) ↵I I ⌫ µJ K 1 I J 1 I J µ 1 I J ↵ ⌫G⌫)(@ each building eq.(60). The term is↵Jgiven 2(4⇠1 afor ⇠1g1kinetic )(@ )(@ G )I⌫by µ⌫ ↵I ↵J each building block forblock eq.(60). term is given ⌫ GµJ )by µ⌫ ↵I I a⇢ JThe II(@ J µkinetic + 8⇠ ⇢ (@ G G ) 2( a a ⇠ )(@ G )(@ I J ↵ 1 J 1 I J µ + ⇢ g G )(@ G ) 2( a a ⇠ )(@ G )(@ G ) + 4⇠ ⇢ ⇢ g g (@ G )(@ G ) 1 I J µ⌫ ↵↵I I1 I↵↵⌫J↵J µJµ ↵⌫ 1 1I IJ Jµ⌫ (62) (62) J↵I (62) 2( 8⇠ )(@ G)(@ )(@ G↵I )JJ⌫J I G + 4⇠1 11a⇢I IaI ⇢⇢JJJ+ g(@µ⌫⇠4⇠ g µ⌫⇢↵I GG ) )(@ µ⌫ ↵J I (@ J↵ µg ↵I GG ⌫↵I + )(@ ) µ ⌫J ↵1G µ ⇢ g (@ ) 8⇠ (@ G )(@ G ) 1 I J µ⌫ ↵ ↵ ↵ L = 2( a a + ⇠ )(@ G )(@ G ) + 8⇠ ⇢ (@ G )(@ G ) (62) L = 2( a a + ⇠ )(@ G )(@ G ) K 1 I J 1 I J µ 1 I J ↵ ⌫ K 1 I J 1 I J µ ⌫ ↵I J Working out eq.(62), one gets gets µ⌫ ↵I)µ⌫ ↵J↵I 8⇠14⇠eq.(62), ⇢ (@ Gone )(@ G ↵J µJ Working out eq.(62), oWorking ne gets + out + g↵2( (@ G ) )(@ µ⌫ (@ 1I ⇢IJ⇢ J+ µ⌫ ↵ ↵ IG ⌫↵I µJ 4⇠gone ⇢ ggµ⌫ gg⇠)(@ G G )) ) Working out eq.(62), gets 1 III⇢ J ↵ a a )(@ GI⌫)↵↵)(@ ⇢ ⇢ (@ G G↵J⌫ G 2(one ⇠ )(@ G )(@ Gµ)(@ 1 J 1 I J 1 J µ⌫ ↵ Working out eq.(62), gets 1 aI a+ J 4⇠ 1 I J µ ⌫ µ⌫ ↵I ↵J I 4⇠ µ⇢ ⇢ ⌫Jg µg ⌫J(@ GI )(@ ⌫ G µJ ↵I I I ⌫ µJ I (62) + ) I J @ µ⌫ B LK = A + G C GCIJ·@@↵ GGI↵II · @ (62) ↵I LIJ A⌫IJ·µ@1µone G GIJ @↵I + B⌫IJ·⌫@GµµJ ·+ @G GIJJ )@+ G(63) (63) µIG µ↵IG ↵↵I J↵ ⌫)(@ K @= ⌫J⌫ ·⇢ + 8⇠ Working out eq.(62), gets + 8⇠ (@ G )(@ G ) 1 I⇢ J·(@ ↵ 1 I J ↵ Working out eq.(62), one gets LK = AIJ @ G · @ G + B @ G @ G + C @ G · @ G (63) µ ⌫J I ⌫ µJ ↵I I IJ one µ ⌫gets @ G · @ IJG ↵ + C @ G ⌫ A out Working eq.(62), LµK = ·@ G (63) IJ @µ G⌫ · @ G µ⌫ + BIJ↵I µ ⌫ ↵J ↵I ↵JIJ ↵ out eq.(62), one gets + 4⇠ ⇢I transverse g µ⌫ G G ) Ioperators, Decomposing eq.(63) in terms longitudinal 4936 | P a g e Working ⇢of terms G ⌫ of transverse (@ ↵ ↵G and ) )(@ and longitudinal operators, + 4⇠ ⇢ ⇢ g g )(@ 1 I (@ J ↵g µ⌫ ↵ ↵I Decomposing eq.(63) in I µ ⌫J µJ 1 II J µµ⌫ ⌫J I ⌫ µJ ↵I I @ µK G + G + · @ @G I · B µ G I ·C ⌫longitudinal I µG ↵ G+ C L @ in + B@IJ @ofµµ G G @IJ G@µJ G↵I ·· @ @(63) Goperators, (63) of transverse and ⌫ ·A ⌫ ·B J u n e 2 0 1 7 Decomposing L K = A IJ L eq.(63) = @ IJ G @ G terms @ IJ @ ⌫J µ ⌫ ⌫ + @ + CIJ @↵↵Ioperators, G G (63) KI = A IJ⌫J µG IJµJ IJ Decomposing eq.(63) transverse longitudinal ⌫ · @ GinI terms ⌫ · @ G ↵Iand µ ⌫ I µ⌫ J I µ⌫ I 2 I µJ w w w . c i r w oLrL l d= .= cA o IJ mA µ G I µ⌫ J I µ⌫ I 2 I µJ @ · @ G + B @ G · @ G + C @ G · @ G (63) Working out eq.(62), one gets K IJ µ IJ ↵ G ⇤✓ G (A + B + C )G ⇤! G + M G G (64) ⌫ IJ Working one gets LKout =I ⌫eq.(62), GIJ MIJ Gµ G (64) K IJ µA ⌫ µCIJ )GIµ ⇤! ⌫ IJ⌫I+µµJ IJ G IJ + BIJ IJ I+ ⌫ of(A µ⌫ Jµ ⇤✓ 2G eq.(63) in(A terms transverse and longitudinal eq.(63) in terms and longitudinal operators, LKDecomposing = AL G= + C+IJof )Gtransverse ⇤!Cµ⌫ G Mµ⌫ Goperators, µ⌫ J B Iµ G 2(64) I µJ IJ Decomposing IJG+ IJ µ ⇤✓ A GG ⌫ I ⇤✓ µ+ ⌫ I+ IJG Decomposing eq.(63) in terms of transverse and longitudinal operators, (A B )G ⇤! + M G G (64) K IJI µ⌫J2I ⌫2⌫J I IJ ⌫ µJ ⌫ I↵I IJ µI µof ⌫ IJµJ µ↵I IA IJµ@ Decomposing in terms and longitudinal operators, 2@transverse 2IJ L = G · @ G + B @ G · @ G + C @ G · @ G (63) where AIJ =@ and M = m are the relationships that redefines the KIeq.(63) IJ µ µ IJ ↵ LK = AIJ · @ G + B G · @ G + C @ G · @ G (63) ⌫ ⌫ where A = and M = m are the relationships that redefines the IJ IJ µG IJ µ IJ ↵ µ⌫ J I µ⌫ I 2 I µJ II ⌫+II IJ⌫ G IJ2I IJ IJ)G ⇤! G I+ Mµ⌫ GI G J II µJ 2+ LK =AIJA= GIJ (Aµ⌫ BIJ C I ⇤✓ µ⌫ JIJ2IJ µ⌫ Iµ + M 2 2(64) µJ (64) IJK IJG IJB L = A G (A C )G G G where and M = m are the relationships that redefines the µ ⇤✓ ⌫ IJ µ+ ⌫ Iµ ⇤! IJG 2+ IJ IJ IJ IJ µ ⌫ ⌫ IJ µ II L = A G ⇤✓ G (A + B + C )G ⇤! G + M G G (64) constructor basis {D, X } in terms of the physical basis {G }. It diagonalises where = andXM m are physical the the K A IJ µi {D, IJ IJ relationships I constructor basis terms of IJ the {G }. It diagonalises ⌫} in= µ basis ⌫ that IJredefines µ

+ 4⇠1 ⇢I+ ⇢J4⇠ gµ⌫ g (@↵ G )(@↵↵GG↵I )(@ ) ↵ G↵J ) g µ⌫ (@ 1 ⇢I ⇢J gµ⌫ Working out eq.(62), one getsone gets Working out eq.(62),

I S S N 2 3 4 7 -‐ 3 4 8 7 V o l u m e 1 3 N u m b e r 6 o µJ u n ⌫aCl µJ o@f G A↵I d v· a@ nG c↵IeIs i n (63) PI h y s i c s I r+ @ ⌫J G

I GI · @ µ G⌫J µBIJ ⌫J@µ G⌫ · LK = ALIJ @= µ A ↵ C @ G ⌫ IJ @µ GI · + + BIJ @µ G⌫ · @ GIJ + K IJ ↵ ⌫ @ G

·@ G

(63)

knows that a quadrivector carries information about different spins states. For this understanding one has to separate in transverse and longitudinal I µ⌫ J I I 2 I µJ µ⌫ Jthat. I ⌫poles I Gµ G the 2 (64) Itransverse µJ LK =sectors. ⇤✓ (A + (A BIJIt+ CIJ )Gdifferent ⇤! µ⌫ G +µ⌫ MGIJ makes yields where LAKIJ=GµEq.(64) AIJ GGIµ⌫⇤✓ GIJ (64) IJ + BIJ +µCIJ )Gµ ⇤! ⌫ ⌫ + MIJ G µ G and longitudinal masses are related by the expression 2 2 and M =M m2II are thatXredefines 2that 2 the where AIJ = δIJwhere and MA relationships the relationships constructor basis {D, in terms othe f the physical = m=II2 δAIJIJIJ are IJ IJredefines IJ i } that IJ where =the IJ and redefines the IJ = mII IJ are the 2relationships knows that a quadrivector carries information about different spins states. constructor basis {D, X } in terms of the physical basis {G }. It diagonalises det m i I that a quadrivector carries information about different spins states. T 2 basis {GI } . Iknows t d iagonalises t he t ransverse s ector. constructor basis {D, X } in terms of the physical basis {G }. It diagonalises knows that a quadrivector carries information about different spins states. knows that a quadrivector carries information about different spins (65) i information I states. knows that a quadrivector carries about different spins L = For this understanding one has hasdet to m separate in + transverse and longitudinal thethis transverse sector. det(1 BIJ in + C ) longitudinal For understanding one to separate in transverse and longitudinal IJ the transverse sector. For this understanding one has to separate in transverse and longitudinal For this understanding one has to separate transverse and For this understanding one has to separate in transverse and Eq.(1) contains N-‐spin 1 and N-‐spin 0 quanta involving a diversity on quantum numbers. Eq.(64) splits the qlongitudinal uanta diversity. sectors. Eq.(64) makes that.N1It It yields different poles where involving thea transverse transverse Eq.(1) contains N -spin and N -spin 0 quanta involving diversity onone has on sectors. Eq.(64) makes that. yields different poles where the Eq.(1) contains -spin 1 and N -spin 0 quanta a diversity From group theory one knows that a q2makes uadrivector carries aIt bout different spins where states. For twhere his understanding to sectors. Eq.(64) makes that. It yields different poles where the transverse sectors. Eq.(64) makes that. yields different poles the sectors. Eq.(64) that. Itinformation yields different poles the transverse where m means the transverse and longitudinal masses. Notice that if m2T and longitudinal masses are related relatedthe byquanta thethe expression quantum numbers. Eq.(64)splits diversity. From group theory one T,L separate in transverse and longitudinal sectors. Eq.(64) makes that. It yields different poles where the transverse and longitudinal and longitudinal masses are by the expression quantum numbers. Eq.(64)splits quanta diversity. From group theory one and longitudinal masses are related by the expression and longitudinal masses are related by the expression and longitudinal masses are related by the expression masses are related by the contains expression zero mass it will yield zero masses on the longitudinal sector. det m222T det m The interaction Lagrangian is composed of22a trilinear term and 14 T 22 = detm m det m det m (65)a quadri2 TT 2 T14 det 2L = det m (65) L det m = det m = (65) det m = (65) det(1 + B + C ) linear term. They are IJ IJ L L L det(1 +B Bdet(1 C+IJBB ) IJ ++CCIJ)) IJ + C det(1 det(1 + IJ IJ + IJ )IJ (3) + (4) LI = Lint + Lint (66) where m m222T,L means means2 2the the transverse transverse and and longitudinal longitudinal masses. Notice Notice that that if if m m222T 2 2 where masses. T,L where means thetransverse transverse and longitudinal masses. Notice where mm means the and masses. Notice where mwhere means the transverse and longitudinal masses. Notice that if mTTthat if m2T where mT ,L means the transverse and longitudinal masses. Notice that if contains zero mass it will yield zero masses on the mTlongitudinal T,L T,L T,L contains zero mass it will yield zero masses on the longitudinal sector. contains zero mass it yield zero masses onmasses the longitudinal longitudinal sector. sector. contains zero mass willyield yield zero masses on the the ◆ longitudinal sector. mass ititwill zero on longitudinal contains contains zero masszero it will will yield zero masses on the sector. longitudinal sector. The interaction interaction✓Lagrangian Lagrangian is composed of a trilinear term and quadriThe is composed of a trilinear term and aa quadriquadri4 iga a + 2 a b + (3) 1 K 3 K (IJ) (IJ) The interaction Lagrangian is composed of a trilinear term and a]quadriThe interaction Lagrangian is composed of a trilinear term and The interaction Lagrangian is composed of a trilinear term and a K µI The ilinear nteraction Lagrangian is composed f a trilinear term and a quadrilinear term. They are @µ G [G , G⌫J LThey term. They are+4⇠ oi(g ⌫ int = are linear term. gJ I K(4)) + 2⇠3 K b[IJ] 1 are Iare J K linear term. linear term. They areThey linear term. They (3) (4) LII = =L L(3) +µJ L(4) (66)µJ ⌫K (4) (3) (3) (4) int int (3) L + L (66) I LL ⌫K int int + L = LLint = Lint I = I I int+ + 2 3 aI c[JK]L+ @µL Gint {G ,LG }++ 2 3 aI [JK] + (@µ GI⌫(66) )G G (66) int int ⌫ ✓ ◆ where where where 8⇠1 i(gJ I ⇢K + gJ K ⇢I )+ where where where ✓ ◆ @↵ G↵K [GI , G J ] + ✓ ◆ ✓ ◆ ✓ ◆µI ⌫J ✓ +2⇠ (⇢ b + u ) igaK aa(IJ) + 2[IJ]3aaK b(IJ) + 3 + K2 K + [IJ] (3) Kb (IJ) (IJ) K ◆ 44 11iga K µI , G⌫J ⌫J ] L(3) = @ G [G igaK 2Ka33a(IJ) a(IJ) 22+33aaKKbb(IJ) + 4 iga + + (3) µ (3)(3) 4 1 K a4(IJ) K b+ (IJ) 1 iga K 1+ K µI ⌫ (IJ) K int K ] µI µI ⌫J L = @ G [G G µ⌫ @µ G↵K I@,, G ↵J ⌫ [G +4⇠ i(g g ) + 2⇠ b Lint ][G L+4⇠ = @ G ] L = G G⌫J 1i(g I(8⇠ J 1 ig K J ⇢Ig JK I K 3 K [IJ] µ µ ⇢ 2⇠ ⇢ u )g g @ G [G , ] ,, G µ ⌫ ⌫⌫ [G int = ) + 2⇠ b int int 3+ K µ⌫2⇠ ↵bb [IJ] J K J I K 3 K [IJ] ↵ +4⇠11 i(gII +4⇠ g ) 2⇠ b i(g g ) + +4⇠ i(g g ) + 2⇠ J K K 1 1 IIIJJ JIµJ KK JJ I 3I KK [IJ] 33 KK [IJ] [IJ] ⌫K I µJ ⌫K J II⌫ {GµJ ⌫K } + 2 ↵I µJ G⌫K ⌫K + 2 33aaII cc+ +( @@IµµvG G GI⌫K aII [JK] +K(@ (@}µµ G GI )G )GµJ 3a [JK] [JK] µJ⇢,,IG I } )@ µJ↵2G⌫K µJ ⌫K + {G + + G [JK] ⌫ + 2✓ 2✓ 3 aI c+ + } µJ + a{G (@ GI⌫⌫⌫ )G 2 23+ cIG ++ @@µ, µGcG(JK) ,2,GG33⌫K + G +2⇠ c(JK) + {G +,22G+ + (@ (@G GI⌫I⌫)G )GµJ ◆ I}}+ [JK] [JK] 3a 33aa II µ[JK] µµG [JK] 3@Iaµ [JK] [JK] ⌫ {G ⌫ ⌫{G ◆ ✓ 8⇠ ◆ ✓ ◆ µ⌫ ↵I J ↵K ✓ ◆ 8⇠+1i(g i(g ⇢ + g ⇢ )+ J I K J K I ↵K I J 2⇠ ⇢ v g g @ G {G , G } ⇢⇢8⇠ + )+ ↵ (JK) K J ⇢ KK⇢ ↵K [GII , G↵K J] I + G 8⇠11 i(gJJ3 II8⇠ ⇢KII+ )+ g @@K⇢↵↵⇢↵IG i(gJggµ⌫ )+ K 1 i(g ↵K J 1+ JJ I u IK⇢ I)+ ↵K + [G@@↵,, G G +2⇠ (⇢K bb[IJ] + )+ gJJ K@ + G [G ]][G ++ 3(⇢ K u [IJ]) G [GI,,G G JJ]] ↵ ↵ +2⇠ + (67) K [IJ] [IJ] +2⇠33 (⇢K b + u ) +2⇠ (⇢ b + u ) +2⇠ (⇢ b + u ) K [IJ] 3 3 KK K K [IJ] K [IJ] [IJ] [IJ]↵K µ⌫ I ↵J µ⌫ @↵ G↵K ↵K [G I , G↵K ↵J ] I (8⇠1 igJ ⇢⇢I ⇢⇢K 2⇠ 2⇠3⇢⇢K u u[IJ])g )gµ⌫ ggµ⌫ ↵J µ⌫ µ⌫I↵ ↵J (8⇠ @↵↵)g G [G G K 3 µ⌫ (8⇠11ig igJJ ⇢II(8⇠ ⇢(8⇠ )g3↵I guu[IJ] @ G ⇢ 2⇠2⇠ ⇢3µ⌫ @↵↵@↵,,↵G ]] ⇢I ⇢I K ⇢Ku[IJ] ⇢KK )gKµ⌫gg[G G↵K]][G [G↵I↵,,G G↵J K 1 ig K [IJ] J2⇠ 1 ig J3 J[IJ] µ⌫ ↵I J K + 2⇠ ( v + ⇢ c )@ G {G , G } 3 I I ↵ (JK) (JK) ↵I J K ↵I J K + + )@ {G)@ GG↵I{G + 2⇠ 2⇠33(( II vv+(JK) )@ Gcc(JK) {G }}{GJ,,G (3 ( I⇢⇢µ⌫ vII (JK) +2⇠2⇠ vcc(JK) +⇢↵↵⇢G )@,, G G K}} (JK) (JK)+ 3+ ↵I I IJ(JK) ↵K↵↵ µ⌫ ↵I J ↵K + 2⇠ ⇢ v g g @ G {G , G } 3 I (JK)g µ⌫ g µ⌫ @ ↵G↵I {G ↵ G↵I ↵K } JJ µ⌫ J + ↵K} + 2⇠ 2⇠33⇢⇢II vv(JK) g3µ⌫ @↵↵ G }{G↵↵, ,GG↵K ++2⇠2⇠ ⇢3I⇢gvI (JK) ggµ⌫µ⌫g{G v(JK) g µ⌫@↵↵@↵,,↵G G↵I{G } µ⌫ (JK) (67) (67) and (67) (67)

Decomposing inand terms transverse operators, Decomposing eq.(63) iDecomposing n terms oeq.(63) f transverse longitudinal operators, and longitudinal eq.(63) in ofterms of transverse and longitudinal operators,

15

15 15 15

15 15


2

+

2 c[IJ] c[KL]

+ +2

and (4)

Lint

+

2 [IJ] b(KL)

3 ig [IJ] a(KL)

Gµ G⌫ [G , G ] I {Gµ , GJ⌫ }{GµK , G⌫L }

+ ⇠2 c(IJ) c(KL)

I S S N 2 3 4 7 -‐ 3 4 8 7 V o l u m e 1 3 N u m b e r 6 ⌫L J o u r n a l o f A d v a n c e s i n P h y s i c s

I J µK , G⌫L } 2 [IJ] c[KL] Gµ G⌫ {G

I J µK G 2 [IJ] [KL] Gµ G⌫ G

✓

◆ 8⇠ g g + 2⇠ b u + 1 J L I K 2 [IJ] [KL] I ↵J K L 0 [G1 2 + ↵ , G ][G , G ] +3 i(g b + ⇢I bb[KL] u ) 1 g a(IJ) a(KL)2⇠ 2 bJ(IJ) (KL)+ gJ3 iga (IJ) (KL) I [KL] ◆ A [GIµ , GJ⌫ ][GµK , G⌫L ] gK I4⇠Lg ggJ⇢gL⇢ I +K⇠) + ⇠ = @ +2⇠1 (gJ✓ 2 b[IJ] b[KL] + u u + I K 2 [IJ] [KL] ↵L + +⇠3 i(g1I JJ bL[KL] gµ⌫ g µ⌫ [GI↵ , G↵J ][GK b[KL] ) ↵ ,G ] 2⇠3 igJg⇢JI uI◆ [JK] ✓ ◆ 2 2 b(IJ) c✓[KL] 3 iga(IJ) c[KL] I )+ 2⇠2 (b[IJ] v(KL) + u[IJ] c[G + , GJ⌫ ]{GµK ,I G⌫L↵J } (KL) µ +I J c(KL) gJ I c(KL) ) [G↵ , G ]{GK , G L } +⇠3 i(g 2⇠3 i(gJ I v(JK) + gJ ⇢I c(JK) ) + 2 2 [IJ] b(KL) GIµ GJ⌫µK [GµK ⌫L , G⌫L ] 3 ig [IJ] a(KL) + 2⇠2 b[IJ] c(KL) gµ⌫ [GµI , G⌫J ]{G ,G } + {GI , GJ }{GµK , G⌫L } ↵L 2 c[IJ] c[KL] + ⇠2 c(IJ) c(KL) + (2⇠2 u[IJ] v(KL) 2⇠3 igJµ⇢I v⌫(JK) )gµ⌫ g µ⌫ [GI↵ , G↵J ]{GK ↵ ,G } I J µK ⌫L + 2 2 [IJ] c[KL] Gµ G⌫ {G , Gµ⌫ } I ↵J + ⇠2 v(IJ) v(KL) gµ⌫ g {G↵ , G }{GµK , G⌫L } + 2 [IJ] [KL] GIµ GJ⌫ GµK G⌫L + 2⇠2 v(IJ) c(KL) gµ⌫ {GI↵ , G↵J◆}{GµK , G⌫L } ✓ 8⇠1 gJ gL I K + 2⇠2 b[IJ] u[KL] + (68) + [GI↵ , G↵J ][GK , G L ] 2⇠3 i(gJ ⇢I b[KL] + gJ I u[KL] ) ✓ There is a wholeness principle ◆ to be understood classically. The first 4⇠1 gJ gL ⇢I ⇢K + ⇠2 u[IJ] u[KL] + ↵J ↵L + step is to understand how it provides gµ⌫ gaµ⌫whole [GI↵ , Ginterconnected ][GK ↵ , G ] dynamics. The 2⇠3 igJ ⇢I u[JK] ✓corresponding equations of motion ◆ derived from eq.(61) are 2⇠2 (b[IJ] v(KL) + u[IJ] c(KL) )+ + [GI↵ , G↵J ]{GK , G L } 2⇠3 i(gJ I v(JK) + gJ ⇢I c(JK) ) + 2⇠2 b[IJ] c(KL) gµ⌫ [GµI , G⌫J ]{GµK , G⌫L } + (2⇠2 u[IJ] v(KL)

↵L 2⇠3 igJ ⇢I v(JK) )gµ⌫ g µ⌫ [GI↵ , G↵J ]{GK ↵ ,G }

+ ⇠2 v(IJ) v(KL) gµ⌫ g µ⌫ {GI↵ , G↵J }{GµK , G⌫L } + 2⇠2 v(IJ) c(KL) gµ⌫ {GI↵ , G↵J }{GµK , G⌫L }

There is a wholeness principle to be understood classically. The first step is to understand how (68) it provides a whole interconnected dynamics. The corresponding equations of motion derived from eq.(61) are

[µ⌫] [µ⌫] There is [µ⌫] principle to be Jbunderstood classically. The first tra wholeness taJb 4iga [ta , tb ])+ 1 (4aI @⌫ Z [µ⌫] (IJ) G⌫ Z tr (4a @ Z t 4iga G Z [t , t ])+ 1 I ⌫ a a b (IJ) ⌫ step is to understand how itJbprovides a whole interconnected dynamics. Jb The [µ⌫] Jb [µ⌫] + tr [ta ,Jbtb ][µ⌫] + 4c[IJ] G + 2 [IJ] G⌫ z [µ⌫] [ta , tb ] 2 4b(IJ) G⌫ z a , tJb b } [µ⌫] 16 Jb [µ⌫] ⌫+z2 {tare corresponding of motion derived from eq.(61) + tr 2 4b(IJ) Gequations z [t , t ] + 4c G z {t , t } G z [t , t ] a b a b [IJ] ⌫ ✓ a b [µ⌫][IJ] ⌫ ◆ ⌫ ✓ ◆Z [µ⌫] [ta , tb ] 2aI @⌫ z taJb [µ⌫] 2iga(IJ) GJb z [µ⌫] [ta ,Jb tb ] + 2b(IJ) GJb [µ⌫] [µ⌫] ⌫ ⌫ 2aI+ @⌫tr z 3ta 2iga(IJ) G⌫ z Jb [ta[µ⌫] , tb ] + 2b(IJ) G⌫ Z Jb[ta[µ⌫] , tb ] + + tr 3 +2c[IJ] G⌫ Z {ta , Jb tb } + 2 [IJ] G⌫ Z {ta ,+ tb } Jb [µ⌫] [µ⌫] +2c0 {ta , tb } + 2 [IJ] G⌫ Z {ta , t1 b} [IJ] G⌫ Z (µ⌫) 0 1µ (⇢ ) ta + (µ⌫) 4 I @⌫ Z µta (⇢ 4⇢ ) I g⇢ @ Z 4 I @⌫ Z ta 4⇢I g⇢ @ Z taJb + (µ⌫) @ +4i(gIJbJ (µ⌫) + tr ⇠ g )G Z A +[ta , tb ]+ A + 1 J I ⌫ @ + tr ⇠1 +4i(gI J gJ I )G⌫ Z [ta , tb ]+ µJb (⇢ ) ⇢I )g ) ⇢ G I ⇢J µJbg +4i(gI ⇢J ✓ g+4i(g ZJ(⇢ [ta , tb ] Z [ta , tb ] J ⇢I )g⇢ G ◆ ✓ Jb (µ⌫) +4b G z4c [tG tbz] (µ⌫) + 4c G}Jb z (µ⌫)◆ {ta , tb } Jb (µ⌫) a ,Jb [IJ] (IJ) ⌫ ⌫ +4b G z [t , t ] + {t , t aµJbb a b µJb + tr ⇠[IJ] + ⌫ ⌫ 2 (⇢(IJ) ) + tr ⇠2 +4u g+ [taG , tµJb 4u g}⇢ z (⇢+ ) {ta , tb } (⇢[IJ] ) G (⇢ ) G ⇢ z b] + (IJ]) +4u[IJ] GµJb g z [t , t ] 4u g z {t , t ⇢ a b ⇢ a b (IJ]) 0 0 1 ) Jb (µ⌫) zµ(µ⌫) @ µ2b g⇢ z (⇢ 2b[t[IJ] GJb Z (µ⌫) [ta , Jb tb ] + 2c{t (µ⌫) Jbta + (µ⌫) (µ⌫) I @⌫@ (IJ), G 2 I @⌫ z ta 2 2⇢ g⇢ tza (⇢ )2⇢ ta I+ }+Z {ta , tb I a , tb ]⌫+ 2c(IJ) G⌫ Z a tb ⌫ [IJ] G⌫ Z (⇢ ) µJb (⇢ ) (⇢ (⇢ )G + tr ⇠3 @ +2u[IJ] GµJb g+2u GaµJb [tG tb ] g+⇢ 2v {ta , tb }+ A a , µJb ⇢ Z [IJ]) [t + tr ⇠3 @ , tbg]⇢+Z2v(IJ) Z(IJ) {ta , tgb }+ ⇢ Z Jb (µ⌫) µJb Jb 16 (µ⌫)gJ I )G⌫ z µJb ⇢ g (⇢⇢ ))G g z (⇢ ) [ta , tb ] +2i(gI J g+2i(g [ta , tb ] + 2i(gI[t⇢aJ, tb ] g+J ⇢2i(g J I )GI Jz I )GI J g⇢ zJ I [ta , tb ] ⇢ ⌫

=0

=0

(69)

(69)

Eq.(69) shows the systemic set } evo{Gµ I } evolution Eq.(69) shows the systemic dynamics involving the fields involving set dynamics at space-‐time. Notice that {G the µI term λ1 Eq.(69) shows the systemic dynamics theinvolving fields setthe {Gafields } evoa

a

µI

lution at space-time. that1athe term the for Yang-Mills for rewrites the Yang-‐Mills for N-‐potential fields. Nevertheless tNotice he other terms re adding on new contributions. They are showing that 1 rewrites lution at space-time. Notice that the term rewrites the Yang-Mills

potential theinclusion other arenew new consymmetry should N not be limited to fields. Yang-‐Mills Nevertheless theory. Beyond the of terms other fields it adding develops an expressian in SU(N ) N potential fields. Nevertheless the other terms are adding on con-on are showing that SU (N ) symmetry should not be limited that SU (N ) symmetry should not be limited totheory. theory. Beyond of it other fields an it develops an Taking the trace from tYang-Mills he above eBeyond quations, one gets to Yang-Mills the inclusionthe of inclusion other fields develops in terms of group generators. expressian inexpressian terms of group generators.

terms of group generators. They tributions. They tributions. are showing



Taking the trace from the above equations, one gets Taking the trace from the above equations, one gets [µ⌫]a [µ⌫]c + 2ga(IJ) fabc GJb + 1 2aI @⌫ Z ⌫ Z [µ⌫]a [µ⌫]c + 2ga(IJ) fabc GJb Z + 1 2aI @⌫ Z Jb [µ⌫]c Jb [µ⌫]c + 2 2ib(IJ) fabc G⌫ z + 2c[IJ] dabc⌫G⌫ z [µ⌫]c + [IJ] dabc GJb + ⌫ z Jb [µ⌫]c Jb [µ⌫]c ✓+ 2 2ib ◆ Jb [µ⌫]c + + 2c + [µ⌫]c (IJ) fabc G⌫ z [IJ] dabc G⌫ z [IJ] dabc G⌫ z [µ⌫]a Jb [µ⌫]c Jb aI @✓ + ga(IJ) fabc G⌫ z + ib(IJ) fabc G⌫ Z ⌫z ◆ + 3 [µ⌫]a Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c a+c @ z + ga f G + ib f G Z I [IJ] ⌫ dabc G⌫ Z (IJ) + abc [IJ] abc (IJ) d G Z ⌫ z ⌫ abc ⌫ + 3 Jb [µ⌫]c [µ⌫]c ! + [IJ] dabc GJb ⌫)a ⌫ Z ⌫ Z Jb (µ⌫)c (µ⌫)a +c[IJ]µdabc G @ Z 2⇢ @ Z 2(g + g )f G Z ! I ⌫ I I J J I abc ⌫ (⌫ ⌫)a (µ⌫)a µ Jb (µ⌫)c + ⇠1 ⌫)c @ Z 2⇢ @ Z 2(g + g )f G Z µJb I J I ⌫ I g ⇢(⌫)f J I abc 2(gby I ⇢Jtk+and J Itaking abc GtheZtrace Now one derives⌫ the relation (⌫ + ⇠1 multiplying µJb ! ⌫)c (70) 2(gI ⇢J + gJ ⇢IJb )fabc G Z(⌫ (µ⌫)c (µ⌫)c 2ib[IJ] fabc GJb z + 2c d G z (70) ! (IJ) abc ⌫ [µ⌫]e ⌫ Z [µ⌫]e ifekc )GJb + aek )@⌫)c ⌫Jb abc (dekc (µ⌫)c+ ga(IJ) fµJb Jb + ⇠2 1 aI (daek2ib ifµJb ⌫ Z ⌫)cz (µ⌫)c f G z + 2c d G [IJ] (IJ) 0 1 G abc z(⌫ ⌫ + 2u(IJ) dJb G abcz(⌫⌫2 abc Ja [µ⌫]k ++2iu ⇠ [IJ] fbabc (id z [µ⌫]e +GNµJb c1 G⌫)c z µJb f⌫)c eck eck )G (IJ) fabc [IJ] ⌫ ⌫ 0 2 +2iu f G z + 2u d z abc 1 [IJ] abc (IJ) (⌫ (⌫ (µ⌫)a Jb JbG [µ⌫]e [µ⌫]k A @⌫ z+c (g(d z (µ⌫)c + 20 + )G + N [IJ] GJa I@ I eck J +gif J eck I )f abc [IJ] dabc ⌫ z⌫ ⌫1z (µ⌫)a Jb (µ⌫)c ⌫)c 1 B ⇢ @ µ z ⌫)a C Jbz [µ⌫]e µJb @ z (g g )f G I ⌫ I J J I abc +I2⇢J[IJ] dgabc z (g G ifeck z(⌫)G⌫⌫ C eck I J ⇢(d I )f abc+ B B0 1(⌫⇢ @ µ zJb ⌫)a + ⇠3 B 1 µJbC ⌫)c C (µ⌫)c Jb (µ⌫)c g (g ⇢ g ⇢ )f G z(⌫ [µ⌫]e I (d abc f B[IJ] faabc C +@Ic⌫(IJ) GI⌫ aZ(IJ) @+ ⇠+ib A zJ dabcJ+ ifekc )GJb z [µ⌫]e ⌫ (⌫Z ifaek I Gaek abc (d ekc ⌫ 2 2 B C 3 Jb (µ⌫)c Jb (µ⌫)c1 ⌫)c ⌫)c 1µJb µJb Jb G[µ⌫]e [µ⌫]k B f+ib C Z +fceck dabc A GJa abc [IJ] (IJ) +iu@[IJ] G Z(⌫fGabc + veck G)G bf(IJ) ⌫ (id ⌫ Z + N c[IJ] abc+ (IJ) dabc ⌫ZZ ⌫ Z (⌫ 2 B C+ + 3@ ⌫)c ⌫)c µJb µJb 1 1 Jb [µ⌫]e Ja [µ⌫]k A +iu+ fcabc G Zeck dabc (d ifveck )Gd⌫abcZG Z+(⌫N [IJ] G⌫ Z [IJ] (IJ) (⌫ ++ 2 [IJ] 1 Jb [µ⌫]e Comparing with the usual Yang-‐Mills dynamics, eq.(70) is showing how from symmetry one can enlarge the SU(N ) Comparing with the usual+Yang-Mills eq.(70) is showing how d (ddynamics, + ifeck )G ⌫ Z 2 [IJ] abc eck 0 1 dynamics. As consequence, would say that just from symmetry one the can not defines physics. is necessary to implement an Comparing with the usual Yang-Mills dynamics, from SUwe (N ) symmetry one can enlarge dynamics. As It eq.(70) consequence, we how ⌫)e is showing daek )@⌫ Z (µ⌫)e + ⇢I (ifaek daek )@ µ Z(⌫ observational physical principle. I (ifaek SUB (N ) symmetry one can dynamics. AsItconsequence, we wouldfrom say that just from symmetry oneenlarge can notthe defines physics. isCnecesJb (µ⌫)e + a⇠nd + +(g +erives gJ tIhe )frelation (ifaekonedderives A 1@ Ione J taking abctrace aek )G⌫ Z Now multiplying by t and the the relation t Now multiplying b y t aking t he race d k say that from symmetry one can not µJb definesµ)ephysics. It is necessary towould implement an just observational physical principle. k +(gIan ⇢J observational + gJ ⇢I )fabc (ifphysical daekprinciple. )G Z(µ aek sary to implement Jb [µ⌫]e 0 1 ifaek )@⌫ Z [µ⌫]e + ga(IJ) fabc (dJb ifekc )G 1 aI (daek ekc (µ⌫)e 2 ⌫ Z Ja + (µ⌫)k b f (id + f )G z + c ekc ekc [IJ] abc (IJ) G 0 1 ⌫ ⌫ z N Jb [µ⌫]e [µ⌫]k Jb (µ⌫)e C b(IJ)B fabc (ideck feck + N2 cif GJa [IJ] +c)G ⌫ dzabc (dekc ⌫ z ekc )G (IJ) ⌫ z B C 1 Jb [µ⌫]e Ja 2[µ⌫]k A @ ⌫)e ⌫)k + ⇠ B C+ + 2 + z +c[IJ] +µJb 2 dabc (deck + ifeck )G⌫ z [IJ] G+ ⌫ z v(IJ) GµJa N z(⌫ @ +u1[IJ] fabc (idekc + fekc )G Jb A (⌫ N [µ⌫]e + 2 [IJ] dabc (deck + ifeck )G⌫ z ⌫)e µJb +v(IJ) dabc (dekc ifekc )G z(⌫ 0 1 1 [µ⌫]e Jb [µ⌫]e 0 1 a (d if @ + g2 a(IJ) f (d if )G z aek ⌫ z abc ekc ekc ⌫ ⌫)e 2 I aek 1 1 (µ⌫)e µ 1 1I (ifaek Ja daek (if d )@ z + ⇢ )@ z Jb [µ⌫]e [µ⌫]k I aek aek ⌫ B C (⌫ + 2Bb(IJ)2fabc (ideck feck )G⌫ Z +2 N c[IJ] G⌫ Z 1 Jb (µ⌫)e C + C + 3B 1 Jb [µ⌫]e Ja [µ⌫]k + (g + g )f (if d )G z @ A C I J J I abc aek aek ⌫ + 12B c d (d + if )G Z + G Z 2 eck ⌫ N [IJ] ⌫ B[IJ] abc 1 1eck C µ)e µJb Jb [µ⌫]e B C (gId⇢J + daek )G z(µ I )f ++ (dgJ ⇢+ ifabc )Gaek eck(if ⌫ Z B C 2 2[IJ] abc eck 0 + ⇠ B + 1 b f (id + f )GJb Z (µ⌫)e + 1 ⌫)e 1 Jb (µ⌫)k C c G Z 3B abc ekc ekc [IJ] (IJ) (µ⌫)e µ ⌫ ⌫ 2 N C + ⇢I (ifaek daek )@ ZJb(⌫ I (ifaek B daek )@⌫ Z+ 1 c(IJ) C dabc (dekc if )G⌫ Z (µ⌫)e C B Jbekc(µ⌫)e B C 2 + ⇠1 @ + +(g + g )f (if d )G Z A abc aek aek ⌫)k C B I+J1 u Jf I (id 1 µJb ⌫ ⌫)e µJb + f )G Z + v G Z abc ekc ekc (IJ) µ)e @ ⇢ 2 +[IJ] A (⌫ (⌫ +(g gJ ⇢I )fabc (ifaek daek )GµJb Z(µ µJbN ⌫)e I J 1 + 2 v(IJ) dabc (dekc 2 ifekc )G Z(⌫ 1 0 (µ⌫)e (µ⌫)k b[IJ] fabc (idekc + fekc )GJb + N c(IJ) GJa ⌫ z ⌫ z (71) (µ⌫)e B C +c(IJ) dabc (dekc ifekc )GJb ⌫ z B C + ⇠2 B +u f (id + f )GµJb z ⌫)e + 2 v GµJa z ⌫)k C + equations(⌫ associated SU (N abc Noether ekc ekc @ The[IJ]local A) symmetry are N (IJ)to the (⌫ ⌫)e µJb +v(IJ) dabc (dekc ifekc )G z @µ JNµa (G)(⌫= 0 (72) 0 1 18 ⌫)e 1 1 (µ⌫)e µ (if d )@ z + ⇢ (if d )@ z I aek aek ⌫ I aek aek (⌫ 2 18 B 2 C Jb (µ⌫)e L µa + 12 (gI J + gJ I )fabci(if d )G z B C aek aek ⌫ JN @ = (73) µ a B C µ)e 1 g (@ G ) µJb I µ ⌫I B C (g ⇢ + g ⇢ )f (if d )G z + J I abc aek aek (µ 2 I J B C 1 1 Jb (µ⌫)e Jb (µ⌫)k C i L + ⇠3 B + b f (id + f )G Z + c G µ ⌫ a ekc ekc (IJ) ⌫ Z ⌫ 2 [IJ] abc N C B @Jb @ (µ⌫)e ! =0 (74) 1 a B C g (@ G ) + c d (d if )G Z I µ abc ekc ekc (IJ) ⌫I ⌫ B C 2 B + 1 u f (id + f )GµJb Z ⌫)e + 1 v GµJb Z ⌫)k C ekc @ 2 [IJ] abc ekc A (⌫ (⌫ N (IJ) ⌫)e 1 µJb + 2 v(IJ) dabc (dekc ifekc )G19 Z(⌫ (71) 4939 | P a g e The local Noether equations associated to the SU (N ) symmetry are J u n e 2 0 1 7 w w w . c i r w o r l d . c o m µa

i

@µ JN (G) = 0

(72)

L

(73)

@µ

a

= JNµa

B C 1 Jb 1 (µ⌫)k Jb 1 ekc + fekc )GJb Z (µ⌫)e +Jb1 c(µ⌫)e Bf +(id + ⇠3 B b[IJ] fabc (idekc⌫+ fekc )G⌫ NZ (IJ) G+⌫ NZ c(IJ) G⌫C Z (µ⌫)k C C B ++2 ⇠b[IJ] 3 B abc 2 C B 1 ekc Jb (µ⌫)e C B + 12 c(IJ) dabc C (d ifekc )GJb Zif(µ⌫)e ⌫ + c d (d )G Z B C abc ekc ekc (IJ) ⌫ I SCS N 2 3 4C 7 -‐ 3 4 8 7 2 ⌫)e ⌫)k B +1u B 1 µJb µJb ⌫)eG 1 Z ⌫)k C 1 ekc + fekc )G µJb v(IJ) µJb f (id Z + V o l u m e 1 3 N u m b e r 6 abc @ 2 [IJ]B A + u f (id + f )G Z + v G Z (⌫ (⌫ N ekc (IJ) @ 2 [IJ] abc ekc A (⌫ (⌫ J oµJb u r n a ⌫)e l o f N A d v⌫)e a n c e s i n P h y s i c s 1 1 µJb + 2 v(IJ) dabc Zekc +(d vekc d ifekc (d )G if Z(⌫ (⌫ )G 2 (IJ) abc ekc (71) (71)

The local Noether equations associated toare the SUto (Nthe ) symmetry are The local N oether equations alocal ssociated to the SU(N ) symmetry The Noether equations associated SU (N ) symmetry @µ JNµa (G) @=µ J0µa (G) = 0 N

(72)

are (72)

i Li µa @µ = JNL (73) = JNµa (73) a@µ gI (@µ G ) gI⌫I (@µ Ga⌫I ) i L i µ ⌫L a @ @ ! = 0µ ⌫ a (74) (74) gI (@µ Ga⌫Ig)I (@µ Ga ) @ @ ! = 0 ⌫I where where where L where µa where where L Jµa =[ IL , GII⌫ ]aa (75) µa I a L N a where L where where µa a µa J = [ , G ] (75) (75) I a (@ G ) ⌫ µ N J = [ , G ] JNµa = [ , G ] (75) L a , ⌫G ⌫I JN = [ 19 ] (75) µa ⌫ L N I a L a a (@ G[)a⌫I ))(@19 µ a⌫ ), G I ] a G JJNµa = (75) (@ G µG = [ , GI⌫µ]G (75) a⌫I , G⌫⌫ ] JN = [ (@ (75) ⌫I µ N symmetry a )) a ⌫I (@ Eq.(73) is understood as the equation involving a set of fields. µ (@ G (@ G ) ⌫I µ ⌫I involving a set of fields. µ ⌫Iequation Eq.(73) is understood as the symmetry involving Eq.(73) isEq.(73) understood asas the symmetry equation involving a set of of fields. isgranular understood as the symmetry equation involving afields. set of fields. Eq.(73) is understood the symmetry equation a set The Bianchi identities are Eq.(73) is understood as the symmetry equation involving a set The granular Bianchi identities are Eq.(73) is understood as the symmetry equation involving a set of fields. Eq.(73) is understood as the symmetry equation involving a set of of fields. fields. Eq.(73) is understood aThe s the sgranular ymmetry equation involving a set identities oare f fare ields. The granular Bianchi identities The granular Bianchi are Bianchi identities The granular Bianchi identities are The granular Bianchi identities are + D G granular Bianchi identities are + D G The granular Bianchi iThe dentities are D+⇢ GD (76) ⌫ D µ 0[⌫⇢]I = 0 [µ⌫]I [⇢µ]I D G G + G+ = (76) (76) ⇢ ⌫ µ[⌫⇢]I [µ⌫]I [⇢µ]I [⌫⇢]I D⇢DG⇢[µ⌫]I + D G + D G = 0 (76) D G + D G D G = 0 G[µ⌫]I + D G + D G = 0 (76) ⌫ µ [⇢µ]I ⇢ ⌫ µ [µ⌫]I [⇢µ]I [⌫⇢]I ⌫ µ [⇢µ]I [⌫⇢]I D G[µ⌫]I +D G[⇢µ]I +D G[⌫⇢]I = 00 (76) [µ⌫]I [⇢µ]I G + G + = (76) D⇢ G[µ⌫]I D +⇢⇢⇢D G[⇢µ]I +⌫⌫⌫D G[⌫⇢]I =µµµ0G (76) D Gcollectives +D D G +D D G[⌫⇢]I (76) ⌫[µ⌫]I µ[⇢µ]I [⌫⇢]I = 0 The antisymmetric ones are The antisymmetric collectives ones areones are The antisymmetric collectives ones areare The antisymmetric antisymmetric collectives The antisymmetric collectives ones The collectives ones are are The antisymmetric c ollectives o nes a re The antisymmetric collectives The antisymmetric collectives ones are ones The antisymmetric collectives ones I Jare I J I J I J I J J G G @@µ⌫zz[⌫⇢] ++@@⌫ z⇢ z[⇢µ] +=@⇢ z[IJ] +II JG[IJ] [µ⌫]IG= [IJ] G [IJ] G JGJ JGJ [⇢⌫] (77) I⌫ G J[µ⇢]IG+ I⇢ G[IJ] J[⌫µ]IG IIµ(77) JJ @ z + + + I I J µ [⌫⇢] [⇢µ] [µ⌫] [IJ] ⌫ [µ⇢] ⇢ [⌫µ] µ [⇢⌫] I J I J @µ@zµ[⌫⇢] +@+ @⌫µz@z⌫[⇢µ] +@+ @⇢⌫z@z[µ⌫] =@= G⌫IG G⌫[µ⇢] +G G⇢IG G⇢[⌫µ] +G GµIG+ Gµ[⇢⌫] (77) + @⇢⇢[IJ] z[µ⌫] = G+ GJ[IJ] + G+I⇢[IJ] GJ[IJ] + G G (77) z[⌫⇢] z[⇢µ] z[µ⌫] G[IJ] G[IJ] G[IJ] (77) [⌫⇢] [µ⌫] [IJ] [IJ] I⌫I⌫[IJ] Iµ J ⇢[⇢µ] [IJ] [µ⇢] [⌫µ] [⇢⌫] + + z = G + G G G (77) [µ⇢] [⌫µ] [⇢⌫] J J J µ ⌫ [⌫⇢] [⇢µ] [IJ] J I J I J [µ⇢] ⇢ [⌫µ] µ [⇢⌫] ++@@@⌫⌫⇢zz[⇢µ] +=@@⇢⇢[IJ] zz[µ⌫] G G G (77) G (77) [IJ] [IJ] [IJ] @µ z[⌫⇢] +@@@µµ⌫zzz[⌫⇢] G⌫= G[µ⇢] G⇢+ G[⌫µ] +G Gµ+ G[⇢⌫] µ = G⌫⌫G G[µ⇢] + G⇢⇢[IJ] G[⌫µ] + G[⇢⌫] (77) [⇢µ] + [µ⌫] + [IJ] [⌫⇢] [⇢µ] [µ⌫] [IJ]+ [IJ] [IJ] G [µ⇢] [⌫µ] µ [⇢⌫] The collective symmetric Bianchi identity is The collective symmetric Bianchi identity is The collective symmetric Bianchi identity is The collective symmetric Bianchi identity isymmetric s symmetric The collective Bianchi identity is The collective symmetric Bianchi identity is The collective Bianchi identity is The collective symmetric Bianchi identity is The collective symmetric Bianchi identity is The collective symmetric Bianchi identity is J GI GJ J GI GJ J GI GJ @@µ⌫zz(⌫⇢) ++@@⌫ z⇢(⇢µ) +=@⇢ z(IJ) =IIµJG(IJ) +II⌫JG(IJ) +II⇢JG(IJ) (µ⌫)IG @zµ zz(⌫⇢) + + + J(⌫⇢)IG J(⇢µ)IG II(78) ⇢ JJ (µ⌫) (78) J J J (⇢µ) (µ⌫) (IJ) (IJ) IIµ J II⌫(IJ) J (⌫⇢) (⇢µ) (µ⌫) @µ@ +@+ @µ⌫µz@ z(⌫⇢) +@+ @⌫⇢⌫z@z(⇢µ) =@@= G G + G G + G G (78) + z = G G + G G + G G (78) z(⇢µ) zz(µ⌫) G G + G G + G G (78) (⌫⇢) (⇢µ) (µ⌫) (IJ) (IJ) ⇢ (⌫⇢) (µ⌫) (IJ) (IJ) (IJ) µ (⌫⇢) ⌫ (⇢µ) ⇢ (µ⌫) + + z = G G + G G + G G (78) I J I J II⇢ J µ ⌫ ⇢(⇢µ) (⌫⇢) (IJ) (IJ) (IJ) µ (⌫⇢) (⌫⇢)I ⌫ J(IJ) ⌫ (⇢µ) (⇢µ)I ⇢J (IJ) ⇢ (µ⌫) µ (⌫⇢) (⇢µ) (µ⌫) ⇢ (µ⌫) (IJ) I J µ ⌫ (µ⌫) I J I J J G G G G (78) (IJ) (IJ) (IJ) @µ z(⌫⇢) +@@@µµ⌫zzz(⌫⇢) +@@@⌫⌫⇢zzz(⇢µ) =@@⇢⇢z(IJ) G= G⌫+ G(⇢µ) G⇢+ G(µ⌫) (⇢µ)+ (µ⌫)+ (IJ) + + z(µ⌫) =G(⌫⇢) GµµG G(⌫⇢) + G⌫⌫(IJ) G(⇢µ) + G⇢⇢G G(µ⌫) (78) µ (⌫⇢) (⇢µ) (µ⌫) (IJ)+ (IJ)+ (IJ) (78) (⌫⇢) (⇢µ) (µ⌫) and and and and and and and ⌫) ⌫) ⌫) I ⌫J ⌫J ⌫) I2 ⌫J GI G⌫J and @µ2@ zz(⌫⌫) +⌫) 2@ = GIIIµ2G⌫J (79) ⌫) IG IGI⌫ ⌫J and @zµ zz⌫)(⌫ ⌫) + =⌫zz(IJ) G(IJ) + G(IJ) (79) (79) ⌫) ⌫J Iµ⌫J ⌫J ⌫J ⌫ G+ µ (79) ⌫⌫)z⌫) (IJ) and @µ@ (⌫ ⌫+G µ GI⌫G⌫J (⌫ + 2@ = G G 2 G @ z + 2@ = G + 2@ z = G G + 2 G G (79) ⌫ (IJ) (IJ) @ z + 2@ z = G + 2 (79) ⌫) ⌫) µ ⌫ µ ⌫ ⌫ (IJ) (IJ) I ⌫J I⌫ ⌫J µ ⌫(⌫ ⌫) (IJ) (IJ)II ⌫µ (⌫ ⌫) ⌫ (⌫ (IJ) µ ⌫ µ µ ⌫ µ I ⌫J ⌫J µ ⌫ (⌫⌫) (⌫ (⌫ @ µz (⌫ (⌫ ⌫) ⌫) ⌫) I ⌫J ⌫J I G⌫⌫JG ⌫J (79) G⌫I Gµ + 2@ = G + 22 µ(IJ) (79) @µµz(⌫ + 2@ z = G G + 2 G µ z(⌫ ⌫z (IJ) µ ⌫ (IJ) (IJ) (⌫ @ + 2@ z = G G + G G (79) µ ⌫ ⌫ ⌫ (IJ) (IJ) (⌫ µ ⌫ (IJ) (IJ) µ ⌫ ⌫ µ (⌫ µ from ⌫ that ⌫symmetry µ(N ) symmetry (⌫ (⌫ (⌫ showing The above classical equations arethat showing from it The above classical equations are that SU (N ) SU it The above classical equations areare showing from SUSU (N )generate symmetry it it The above classical classical equations are showing (Na )physics symmetry it The above classical equations showing that (N )SU symmetry The above equations showing that from symmetry it The above classical equations are showing that from symmetry it from is possible to with Yang-‐ SU(N )are The above classical equations are showing that from SU (N ) symmetry it ispossible possible togenerate generate awith physics with Yang-Mills families interlaced. The above classical equations are showing that from SUinterlaced. (N ) symmetry it Someis possible to generate a physics with Yang-Mills families interlaced. SomeThe above classical equations are showing that from SUinterlaced. (N )Somesymmetry it is possible to generate a physics Yang-Mills families Someis to a physics with Yang-Mills families Someis possible to generate a physics with Yang-Mills Someis possible to generate a physics with Yang-Mills families interlaced. Mills families interlaced. Sis omething that m ove s to understand that QCD is not Yang-Mills the only one theory related tinterlaced. o SU(3)c [29, Some30]. possible to generate aawith physics with families is thing possible tomove generate auunderstand physics Yang-Mills families interlaced. Somethat us to that QCD is not the only one theory related thing that move usto tounderstand understand that QCD not the only one theory related isthing possible to generate physics with Yang-Mills families interlaced. Something that move usus to understand that QCD is is not the only one theory related thing that move us that QCD isis not the only related thing that move to understand that QCD not the only one theory related that move usto understand that QCD one theory related thing that move us that QCD not the only theory thing that move us to understand that QCD is not theis related to SU (3) 30]. to SU (3) [29, 30]. c [29, cmove thing that us to to understand understand that QCD isonly not one the theory only one one theory related related to SUSU (3)(3) 30]. to (3) 30]. to [29, 30]. SU (3) [29, 30]. cto[29, cc [29, cSU to SU (3) [29, 30]. to SU (3) [29, 30]. c [29, 30]. c to SU (3) c c 5 Conclusion

5 Conclusion Conclusion Conclusion 55 Conclusion 55 Conclusion Conclusion 555 Conclusion 5 Conclusion Conclusion

Physical laws are being derived from symmetry. Three performances classify the symmetry behavior. They are the phase factor–gauge transformation [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry of difference [34, 35]. Another possibility is the dynamical symmetry breaking [16]. Most of present physical models are developed through the first two Physical laws are being derived from symmetry. Three performances classify Physical laws arederived being derived from symmetry. Threethe performances classify aspects. A mechanism where tlaws he fare irst olaws ne generates interactions symmetry. and tfrom he SSB symmetry. m asses. At this cThree ontext standard model uclassify nifies the Physical are being derived performances Physical laws being derived from Three performances classify Physical are being from symmetry. Three performances classify Physical laws are being derived from symmetry. Three performances laws are being derived from symmetry. Three performances classify Physical laws are being derived from symmetry. Three performances classify electroweak interaction wPhysical ith H iggs [ 36]. the symmetry behavior. They are the phase factorâĂŞgauge transformation thesymmetry symmetry behavior. Theyare are the phase factorâĂŞgauge factorâĂŞgauge the behavior. They the phase transformation Physical laws are being derived from symmetry. Three performances classify

thethe symmetry behavior. They areare thethe phase factorâĂŞgauge transformation symmetry behavior. They phase factorâĂŞgauge transformation the symmetry behavior. They are the phase factorâĂŞgauge transformation

the symmetry behavior. They are the phase factorâĂŞgauge transformation the symmetry behavior. They are the phase factorâĂŞgauge transformation [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry oflook for ofa the symmetry behavior. They arecontains the phase factorâĂŞgauge transformation Eq.(1) assumes as physical principle that nature diversity and systemic behaviors. This leads us [31, 32, 33],spontaneous spontaneous symmetry breaking [13, 14,the 15] and [31, 32, 33], symmetry breaking [13, 14, 15] and the symmetry the symmetry behavior. They are the phase factorâĂŞgauge transformation [31, 32,32, 33], spontaneous symmetry breaking [13, 14, 15] and symmetry ofto of [31, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry of [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry of symmetry performance able to sustain both aspects. So the corresponding fields set introduces a step forward in the symmetry [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry of difference[34, 35]. Another possibility is the dynamical symmetry breaking [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry of difference[34, 35]. Another possibility is the dynamical symmetry breaking difference[34, 35]. Another possibility is the dynamical symmetry [31, 32, 33], spontaneous symmetry breaking [13, 14, 15] and the symmetry of difference[34, 35]. Another possibility iscomes dynamical symmetry breaking difference[34, 35]. Another possibility isthethe dynamical symmetry breaking difference[34, 35]. Another possibility is the tdynamical symmetry breaking notion. It provides t he s ymmetry o f d ifference. H istorically, s ymmetry o ut t hrough he m ultiplet d egeneracy [ 6]. F or i nstance, difference[34, 35].physical Another possibility is the dynamical symmetry breaking difference[34, 35].present Another possibility is the symmetry breaking [16]. Most of models are dynamical developed through the first two

difference[34, Another possibility ismodels the dynamical symmetry breaking [16]. Most ofof35]. present physical through the breaking first two [16].35]. Most present physical models are developed through difference[34, Another possibility isare thedeveloped dynamical symmetry [16]. Most of present physical models are developed through the first two aspects. A mechanism where the first one generates interactions and the [16]. Most of present physical models are developed through the first two aspects. AAAof mechanism where the first one generates interactions and two the aspects. mechanism where the first onedeveloped generates interactions [16]. Most present physical models are through the first aspects. A mechanism where first one generates interactions and thethe aspects. A mechanism wherethe the first one generates interactions and aspects. mechanism where the first one generates interactions and the neutron associated through a same multiplet. Nevertheless, eq.(1) incorporates a quanta diversity. It says that it is also possible to aspects. A mechanism where the first one generates interactions and the aspects. A mechanism where the first one generates interactions and the SSB masses. At this context the standard model unifies the electroweak aspects. A mechanism where the one interactions the SSB masses. this context the standard model unifies the electroweak aspects. A mechanism where the one generates andcalled the SSB masses. At context theofirst standard model unifies the electroweak introduce different quantum numbers uthis nder acontext At sAt ame sthis ymmetry. So, standard instead f generates Weyl degeneracy, it the develops aand nthe on-‐degeneracy SSB masses. At the standard model unifies electroweak SSB masses. At this context thefirst model unifies theinteractions electroweak SSB masses. this context the standard model unifies electroweak SSB masses. At this context the standard model unifies the electroweak SSB masses. At this context the standard model unifies the electroweak interaction with Higgs [36]. as pluriformity. interaction with Higgs [36]. SSB masses. At this context the standard model unifies the electroweak SSB masses. At[36]. this context interaction with Higgs [36]. the standard model unifies the electroweak interaction with Higgs [36]. interaction with Higgs interaction with Higgs [36]. interaction with Higgs [36]. interaction with Higgs [36]. Eq.(1) assume as physical principle that nnature contains diversity and Eq.(1) assume as physical that nature contains diversity and interaction with Higgs [36]. A new approach t o m ake p hysics u nder s ymmetry aprinciple ppears. Iprinciple t opens ew relationship between physics diversity adiversity nd and group theory. interaction with Higgs [36]. Eq.(1)asassume assume as principle physical principle that nature contains and Eq.(1) that contains diversity and Eq.(1)assume assume asphysical physical thatanature nature contains diversity Eq.(1) as physical principle that nature contains and Eq.(1) assume as physical principle that nature contains diversity and Eq.(1) assume as physical principle that nature contains diversity and systemic behaviors. This leads us to look for a symmetry performance able systemic behaviors. This leads us to look for a symmetry performance able Eq.(1) assume as physical principle that nature contains diversity and Based on a fields set gauge transformations, eq.(1) introduces different fields families under a common symmetry group. SU(N )diversity Eq.(1)behaviors. assume as physical principle contains and systemic behaviors. This leads usus to look alook symmetry performance able systemic behaviors. This leads usfor to look for nature symmetry performance able systemic behaviors. This leads to look for a that symmetry performance able able systemic This leads us to for aa symmetry symmetry performance systemic behaviors. This leads us to look for aalook symmetry performance able systemic behaviors. This leads us to for a performance able to sustain both aspects. So the corresponding fields set introduces a step to sustain both aspects. So the corresponding fields set introduces a step systemic behaviors. This leads us to look for symmetry performance able It says that group theory is not anymore enough for defining the physicity. From a common gauge parameter one can define systemic behaviors. This leads uscorresponding to look for set a set symmetry performance able totosustain both aspects. So thethecorresponding fields introduces a astep sustain both aspects. aspects. So the the corresponding fields set introduces introduces sustain both aspects. So corresponding fields introduces step a step toto sustain both So fields set different Lagrangians involving an undetermined number of potential fields. As consequence, the relationship between physics and to sustain both aspects. So the corresponding fields set introduces aa step to sustain both aspects. So the corresponding fields set introduces aa step step forward in the symmetry notion. It provides the symmetry of difference. forward in the symmetry notion. It provides the symmetry of difference. to sustain both aspects. So the corresponding fields set introduces step to sustain both aspects. So the corresponding fields set introduces forward in symmetry It the of forward in the symmetry notion. It provides the symmetry symmetry inthe symmetry notion. Itprovides provides thesymmetry symmetry ofdifference. difference. forward in symmetry notion. It provides the ofwFor difference. symmetry must bforward e restructured. Sthe omething sthe aying tnotion. hat the physical principle precedes the symmetry. We follow the hole principle forward in the symmetry notion. It provides the symmetry of difference. forward in the symmetry notion. It provides the symmetry of difference. Historically, symmetry comes out through the multiplet degeneracy [6]. Historically, symmetry comes out the through the multiplet [6]. For forward inHistorically, the symmetry notion. It provides the symmetry of degeneracy difference. forward in the symmetry notion. It provides the symmetry of difference. Historically, symmetry comes out through multiplet degeneracy [6]. For Historically, symmetry comes out through the multiplet degeneracy [6]. For symmetry comes out through the multiplet degeneracy [6]. For Historically, symmetry comes out through the multiplet degeneracy Historically, symmetry comes out through the multiplet degeneracy [6]. For Historically, symmetry comes out through the multiplet degeneracy [6]. For Historically, symmetry comes out through the multiplet degeneracy [6]. For [6]. For Historically, symmetry comes 4940 | P a g e out through the multiplet degeneracy 20 20 J u n e 2 0 1 7 2020 20 20 20 w w w . c i r w o r l d . c o m 20 20 20

of ofpresent physical models are developed through two [16].Most Most present physical models are developed through thefirst first two [16]. Most present physical models are developed through the first as degeneracy [16]. examples on energy and mass, take the hydrogen atom described through symmetry or the proton two and SU(4) the [16]. Most ofof present physical models are developed through the first two


as basis to construct physical laws. After that depending on the whole physicity to be described one defines the group and number of fields associated. For instance, an electromagnetism defined by four bósons [37]. Our viewpoint is that nature laws are antireductionist. For this, we should center the physics behavior under a same single group. Given a certain number of fields instead of establishing relationships through a symmetry group decomposition as

SU(M ) × SU(N ) ×U(1) or even by introducing a bigger group as SU(5) , it is possible to associate any branch of fields under a common SU(N ) group. Make them to share the same conservation laws through Noether identities and other interlaced equations. A first consequence is on QCD. Being a model derived from the experimental result where quarks are derived with three colours, it is associated to the

SU(3)c group. Then, considering the Yang-‐Mills interpretation, one associates eight gluons the asymptotic freedom property be preserved. Given theGiven increasing number number the asymptotic freedom property be preserved. the increasing SU(3) interacting with quarks. Nevertheless, without violating c experimental results, eq.(1) introduces massive gluons. Physically of non-linear terms in eq.(59) with respect to QCD, its asymptotic freedom of non-linear terms in eq.(59) with to QCD, the asymptotic freedom property berespect preserved. Givenits theasymptotic increasing freedom number massive gluons can be responsible for short interactions. theproperty asymptotic freedom property be preserved. Given the increasing number isof expected. Another possible physical situation is on the possibility property is expected. Another possible physical situation is on the possibility terms inbebeq.(59) with Given respect toincreasing itsa mnumber asymptotic freedom A perspective fterms or non-linear studying teq.(59) he quarks w orld ecomes available. Quarks bring pQCD, hysics for ysterious nature behavior the asymptotic freedom property preserved. the ofnew non-linear incharged with respect tobeQCD, its asymptotic the asymptotic freedom property preserved. Given thefreedom increasing number of having massive gluons intermediating charged quarks[42]. of having massive charged gluons intermediating charged quarks[42]. property is expected. Another possible physical situation is on the possibility the asymptotic freedom property be preserved. Given the increasing number which is confinement. F ifty y ears a fter q uarks d iscovery, u p t o n ow p hysics d oes n ot h ave a n a nswer t o i t. Q CD i s b eing t he only one the asymptotic freedom property be preserved. Given the increasing number of non-linear terms inAnother eq.(59) with respect to QCD, itstoasymptotic freedom property is of expected. possible situation isQCD, onantireductionistic the possibility non-linear terms inIt eq.(59) with respect its asymptotic freedom new tophysical physics is physics through field theory mnon-linear odel Concluding, for describing the aqin uarks popportunity roperties. iopportunity s respect main rdo esult is ointermediating n asymptotic freedom [38, 39], bquarks[42]. ut as counterpart it has Concluding, a new torespect do ischarged through antireductionistic of having massive charged gluons of terms eq.(59) with to QCD, its asymptotic freedom of non-linear terms in eq.(59) with to QCD, its asymptotic freedom property ismassive expected. Another possible physical situation isquarks[42]. on the possibility of having charged gluons intermediating charged infrared problems [40], including at high energies [41]. Thus the introduction of massive gluons are welcomed for solving infrared property is expected. Another possible physical situation is on the possibility gauge isfields [25, 34, 42]. Instead of looking for newisspace and geometry gauge fields [25, 34, 42]. Instead ofcharged looking for new space and geometry Concluding, abe new opportunity to do physics is of through antireductionistic property expected. Another possible physical situation on the possibility property is expected. Another possible physical situation is on thein possibility problems since the asymptotic freedom property preserved. Given the increasing number non-‐linear terms eq.(59) with of having massive charged gluons intermediating quarks[42]. Concluding, aextra new opportunity to do physics is through antireductionistic ofcategories having massive charged gluons intermediating charged quarks[42]. categories as dimensions [43], supersymmetry [10], quantum gravity as[25, extra dimensions [43], supersymmetry [10], quantum gravity gauge fields 34, 42]. Instead of looking forcharged newis space and geometry having massive charged gluons intermediating charged respect to of QCD, its asymptotic freedom property is expected. Another possible physical quarks[42]. situation on the possibility of having of having massive charged gluons intermediating quarks[42]. Concluding, a new opportunity to do physics is through antireductionistic gauge fields [25, 34, charged 42]. Instead ofunification looking to for[45], new space andphysical geometry Concluding, a new opportunity do physics is through antireductionistic [44] or even through SU (5) grand understand laws massive charged g luons i ntermediating q uarks [ 42]. [44] or even through SU (5) grand unification [45], understand physical laws categories as extra dimensions [43], supersymmetry [10], quantum gravity Concluding, a new opportunity to do physics is through antireductionistic Concluding, a new opportunity to do physics is through antireductionistic gauge fieldsas [25, 34, 42].[25,Instead looking new[10], space and categories extra dimensions quantum gravity gauge fields 34, [43], 42].ofsupersymmetry Instead for of looking for new geometry space and geometry through the wholeness principle. through the wholeness principle. [44] or even through SU (5) grand unification [45], understand physical laws gauge fields [25, 34, 42]. Instead of looking for new space and geometry Concluding, a new opportunity to do physics is through antireductionistic gauge fields [25, 34, 42]. Instead of looking for gauge [25, 34, 42]. Instead ofsupersymmetry looking for physical new space and geometry categories extra fields dimensions [43], supersymmetry [10], quantum gravity [44] or evenas through SU grand unification [45], understand laws categories as (5) extra dimensions [43], [10], quantum gravity new space and geometry categories as extra dimensions [43], supersymmetry [10], quantum gravity [44] or even through SU(5) through theas wholeness principle. categories asthrough extra dimensions [43], supersymmetry [10], quantum gravity categories extra dimensions [43], supersymmetry [10], quantum gravity [44] or even SU (5) grand unification [45], understand physical laws through the[44] wholeness principle. orpeven through SU (5) grand unification [45], understand physical laws grand unification [ 45], u nderstand hysical l aws t hrough t he w holeness p rinciple. [44] or even through SUprinciple. (5) grandSU unification understand lawsphysical laws [44] or the even through (5) grand[45], unification [45], physical understand through the wholeness through wholeness principle. A Group theory relationships A Group theory relationships through the wholeness principle. through the wholeness principle.

A

Group theory relationships

ATheGroup theory relationships following useful identities are relatedare torelated the SUto (Nthe ) generators ta : following identities SU (N ) generators ta : Appendix A Group theoryuseful relationships AThe Group theory relationships

The following useful identities are related to the SU (N ) generators ta : A Theory theory Auseful Group theory (i)Group Commutation The following identitiesrelationships are related relationships to the SU (N ) generators ta : 6 Group Relationships (i) Commutation c (N The following identities are related to SU generators Theuseful following useful identities are related to) the [t if[t (80) ta : (80) a: a , tb ] = the abc (i) Commutation ttb ] = tc SU (N )tgenerators a , SU ta : if The following useful identities are related to tidentities he generators SU(N )to are The following useful identities are related the (N ) abc generators : (i) Commutation The following useful related to thec SU (N )tagenerators ta : [tac , tb ] = ifabc t (80) (i) Commutation (i) Commutation [ta , tb ] = ifabc t (80) (i)Commutation Anti-commutation (i) (i) Anti-commutation c c (i) Commutation (i) Commutation [ta , tb ] = ifabc [tat, tb ] = ifabc t (80) (80) (i) Anti-commutation [ta , tb ] =1ifabc[ttca , tb1] = ifc abc tc (80) (80) (i) Anti-commutation {ta , tb } = {ta , ab id=+ dabcabt id + dabc tc (81) t } (81) (i) Anti-commutation b (i) Anti-commutation 1 N (ii) Anti-‐commutation (i) Anti-commutation (81) (i) Anti-commutation 1 {ta , tb } = N c ab id + dabc tc {ta , tb } = 1 ab id + dabc (81) 1Ntc c N (i) Trace (i) Trace =dabc1t ab id + dabc t (81) {ta , tb } = 1{taab, tid (81) b }+ N tc id + d tc N {t , t } = id + d (81) {t , t } = (81) a b aba b abc ab abc (i) Trace N N (i) Trace tr t = 0 (iii) Trace a tr ta = 0 (i) Trace (i) Trace tr ta = 0 1 (i) Trace (i) tr ttr = aTrace ta t0b = trabta tb = 1 ab 2 ta = 0 12 tr ta = 0 1 tr tr1 ta tb = ab (82) 0 abtr (82) tr ttr ta f= 01+ 2 1d ) a t= b = ta tb t12c = (i abc abc tr t t = tr t t t = (i f + d ) (82) abc tr ta tb = 1 ab 4a ab b c 2 1ab 14 abc (82) 2 1abtr ttaa1ttbbtc== ab (i f1abc + dabc ) tr t t = tr a b (82) tr ta tb tc =2 1 (i fabc + dabc 4) 1 (i f + 1d )(i f e + d e ) e (82) e 21= tr t t t t = tabttbctc= ab cd(i+ abc abcf f + d ) cd dabc )(i cdf t + (i + + d ) abc abc d ab cd abc tr ta tb tac b=c 41dtr(itrtfatabc + d ) cd cd 1 1 (82) abc (82) e e 4=114N8 8 (i fabc e+ dabc )(i 1ftt4N 4 (itr e fcd + dcd ) aattbbt cctd ab+cdd+ tr t t t = + ) tr t = (i f ) a b c abc abc abc abc tr ta tb tc td =4 1 ab cd +4114N (i fabc + dabc 18)(i fcd + dcd ) e e e dabc )(i e fcd + dcd ) tr4N ta tb tc td =+ 8 (i fab cd++d (i tr t t t t = )(ifabc fcd + + dcdconstruc) a b c d ab cd abc abc 1 1 1 1 4N 8 B Non-abelian vectorial equations at e e 8 (i fabc Btr taNon-abelian equations ate +constructb tc td = 4N +ddabc ) fcd tr ta tabb tccdtd+=vectorial +abc )(i (i ffabc dcde ) ab + cd d cd + cd )(i 4N 84N B Non-abelian vectorial at construc8 equations B Non-abelian vectorial equations at constructor 7 Non-abelian Vectorial Equations at Constructor Basis tor basis B basis Non-abelian vectorial equations at construc-

B Non-abelian vectorial equations at constructor basis tor basis B Defining Non-abelian vectorialvectorial equations at construcB Non-abelian equations at constructor basis Defining tor basis I I I I ~ ~ ⌘ Dbasis ⌘ DD ⌘D X0, , IX⌘ X ⌘ I ,XX Defining 0, D ⌘ i ,~ i~ I ⌘ torDefining basis Xi II (83) (83) tor 0, D ⌘ i Defining 0I I ~ ~I

Defining

⌘ D0 , D D Xi fields: (83) I ⌘ I i, I ⌘ X0 ,I X ⌘ electromagnetic X I i ~ I electromagnetic I (83) following fields: ⌘ D , D ⌘ D , ⌘ X , X ⌘ X (83) I I I I 0 i 0X i (83) ~ ⌘anti-symmetric ~ the Defining Defining ⌘ Dfor D Di , ⌘ X0sector , X ⌘ following one gets electromagnetic fields: 0 , the i II ~I I I~ I II electromagnetic one gets forone thegets anti-symmetric sector the following fields: ~ ~ ⌘ D , D ⌘ D , ⌘ X , X ⌘ X (83) ⌘ D , D ⌘ D , ⌘ X , X ⌘ X for0 the anti-symmetric fields: i 0 22 4941 | P a g e sector 0 i the following 0 i electromagnetic i (83) one gets for the anti-symmetric sector the following electromagnetic fields: 22 J u n e 2 0 1 7 22the one gets for the anti-symmetric sector following electromagnetic fields: sector22the following electromagnetic fields: w w w . c one i r w gets o r l dfor . c othe m anti-symmetric 22 22 22 22

~ ⌘ electromagnetic ~I ⌘ Defining one gets for following one gets for the anti-‐symmetric sector tD he fX ields: ⌘anti-symmetric D Di ,sector , X 0f,ollowing 0sector ~ ⌘the onethe gets for the anti-symmetric the

where


••• •• I ~~ ⌘ ~~ ⌘ 11 ✏Iimn ~I III ⌘ IX ~ II ⌘ 11 ✏Iijk X[kj] II 1 B • ~ ~ ~ ~ E D Dnm E B 1 • I D I[0i]1I ,, 1 B 0i,,, B nmE ~ ~ ~ ~ E ⌘ D ⌘ ✏ ⌘ X ⌘ I2 ✏ijk X(84) I I I E ⌘ D B 0i imn ~ ~ ~ ~ E⌘D ,B B ✏imn D XI[0i], ,BI B⌘ ✏ijk X (84) [0i] imn 1⌘I ✏ ijk [kj] 2I E ⌘ 1⌘0i ✏2imn ⌘⌘ X[0i] X 0i,0i~ imn nmnm [0i] nm ~E⌘⌘DD0i0i ~12E ~ 21 ✏ X IB I [kj]I[kj] 2 ~Dnm ~ ~ E , B D E ⌘ X , ⌘ ✏ X (84)I 2✏0iimn 2 ijk E⌘D , B ⌘ ✏ D E ⌘ X , B ⌘ [kj] imn [0i] nm ijk [kj] [0i] 2 2 2 2 ••• •• 1 1 , ~~b ⌘ 1 ✏ijk z[kj] • ~e ⌘ • [0i] ~b 1⌘ zz1[0i] ,✏ijk ~e ⌘ z[0i] z⌘[kj]12 ✏ijk z[kj] (85) ~e ⌘ z[0i] , ~~,e~b ⌘ ⌘ ✏ijk zb[kj] (85) [0i] 2✏ z 2ijk ~b[kj] ~e ⌘ z[0i] , b~e⌘⌘ z2✏[0i] z[kj] (85) ijk , ⌘ ijk [kj] 2 2 For symmetric sector: For symmetric sector: For symmetric sector: For symmetric sector: For ssymmetric sector: For symmetric ector: For symmetric sector: For symmetric sector: I I I ij I⌘I I X(ij) I I I X(00)II , ~ II⌘I I X(0i) I , I⌘ I ⌘ ⌘ X , ~ ⌘ X , ⌘ X X , ~ ⌘ X , ⌘ X I I I ij I I ij I (00) ⌘ ⌘IIXI(00) , ~ ⌘ X , ⌘ X (00) (0i) (ij) ij I (ij)I(ij) ↵)i (00) (00) (0i) III (0i) I I(0i)↵)i ij IIX I ⌘ I X(00) , ~✓I⌘ II X ⌘ X↵)i ,(0i) ,⇢↵)i ⌘(↵ ↵)i (ij) ⌘ ⇢↵)i g00 X~(↵,⌘↵)i ,ijI X ✓⌘ij(0i)⌘I↵)i I X(ij) I gij ij X (00) ↵)i ↵)i ✓ ✓⌘ g X , ✓ ⌘ ⇢ g X ✓ ⌘ ⇢ g X , ✓ ⌘ ⇢ g X (86) ✓⌘ ⇢II⇢g00 X , ✓ ⌘ ⇢ g X I 00 ij I ij I00 00 (↵ Iij (↵↵)i ij ij (↵↵)iII ij (86) (↵ I 00ij (↵ (↵ ↵)i (↵↵)i (↵ (↵ µ) ✓ ⌘ ⇢⇤ , ✓ ⌘ ⇢ g X ~I g⌘ (86) 00 X ij I ij ✓ ⌘ ⇢ g X , ✓ ⌘ ⇢ g X z , ⌘ z , ⌧ ⌘ z , ⌧ + ⌘ z (↵ (↵ I 00 ij I ij ij (0i) (ij) (00) (↵ (↵ ii µ) µ) (µµ) ~, ⌘ ~⌘ ⌘ ~⇤ ~⇤ ⌘ ,z(ij) , ⌧ ⌘⌧z+ ,ii ⌘ ⌧z (00) + z(0i) ⌘ ⌧⌘⌘ zµ) z(00)ii ⌘ z(00) zz(0i) (0i) (ij) ij ⇤ z(0i) ,⇤ z(ij) , ⌧ijij, ⌘ zz(µ ⌘ (ij) ij ⌘ ii (0i) (ij) µ) (µii (µ, ,⌧ + ij (µ µ) ~ ~ ij⌘⌘z(0i) ⇤ ⌘ z(0i) , ⇤ z(ij) + z ii ⌘, z⌧(00) , , ⌧ ij⌘⌘z(µ z(ij),, ⌧⌧ ⌘ + ii ⌘ z(00) (µ where where where where where I = 2, . . . , N i = 1, 2, 3 where where I= .= , N2, . . . ,iN=i = (87) I= 2, 2, . . .. ,.IN 1, 1, 2, 2, 3 i 3= 1, 2, 3 (87) I = 2, . . . , N (87) I = 2, . . .i, = N 1, 2, 3 i = 1, 2, 3

B.1 Non-abelian Maxwell equations B.1 Non-abelian Maxwell equations B.1 Non-abelian Non-abelian B.1 Non-abelian Maxwell equations B.1 Maxwell equations 7.1 Non-abelian Maxwell Equations B.1 Non-abelian Maxwell Considering the field DµaMaxwell , equations one derivesequations the following equations: B.1 Non-abelian

Considering thefield D , fone the following equations: Considering the field Dderives , one one derives the following following equations: Considering the ,the one derives the following equations: Considering the field D derives the equations: For µfield =D 0, µ D Considering the field ollowing a µ µ µµa ,equations: µ , one derives a

aa a

aa

(84) (84) (84) (85) (85) (85)

(86) (86) (86) (87) (87) (87)

Considering the thederives following the fieldderives Dµ , one theequations: following equations: µ , one For µConsidering = 0,field For µ= =D0, 0, For µ= 0,For µ For µ = 0, For µ = 0, For µ = 0 , ~ · (2dE ~ a + 2↵I E ~ Ia + ~ea ) d g fabc D ~b ~ c + 2↵I E ~ Ic + ~ec ) dr g ~ cc~ cb · (2d~E a Ia aa a~ Ia g g a ~ bb~ N bg Ic c c~ Ic a Ia a c ~ ~ ~ ~ a Ia Ic ~ ~ ~ ~ ~ ~ a ++ a c + · (2d E 2↵ ~e E d f~e~eabc f)abc · ff(2d E ++ dr r+ (2d E + D (2d E 2↵ + ~~eeccc)) IE IE ~+ ~Ia ~) Ia ~+bb + ~Ic ~) Ic d~rdr · (2d ~e2↵ dg+ Db D Ec D ~ec )~eII E abc abc g(2d II~E I~E a )IIE ·2↵ E +~~eIb 2↵ d·d~·(2d ··2↵ (2d E + 2↵ E g· (2d g2↵ ~Ead ~ ~ abc a+ Ia~N a )D c+ Ic + N c Ib Jc c ~ ~ ~ ~ ~ ~ N dr · (2dE + 2↵ E ) d f E + 2↵ E ~ e ) ~ ~ c) I↵I E I E d+ r · (2d E E ++ ~eabc2↵ ) JE d N f+abc +X 2↵I E e )⇤ fabc+ X 2↵·I(2d ~eD) · (2d · (2~+c ~+ I fabc g N gg g g g N Ib c Ib c c Jc c N N Ib c Ib c c Jc c Ib c Ib c Jc c ~ ~ ~ ~ ~ g g Ib · (2d ~Jc ~~ cc) ~IbgX ~cX ~E ~Ib ~⇤c·⇤(2~ f+ (2d 2↵ E + fabc X · I(2~ (2~ + c + ++ X E ++ 2↵ +2↵ ~ec c ~)eJ E X +Ib ) ) cc + + ↵IIf·fabc · (2d (2d 2↵ E +jcI~Ie~efcc Iabc )abc f· abc X ⇤ g ↵II↵fabc g+ Iabc abc ~c~EIbIb ~JJc ~~) Jc ~~Ib abc abc g JJ~E c X g g ↵ X · E + ) f · (2~ + ~ ~ ~ ~ Ib c Ib Jc J I c c c c c N N ~ ~ ~ ~ ~ c )) (88) For µ = i, + N↵I fabc+ X E + 2↵ ~e2↵)⌧J E+ N+I ~fe)abc + ⇤ ) + ✓ )+ ⇤ N·↵(2d N⇢· I(2~ JE · (2d ++ )X ⇤ +E2✓+ + 2N +N II fabc X (2 Iffabc abc X ( · (2~ jg Ng g N g g g jc N N Ib c c c Ib c c N N jc jc f Ib Ib Ib Ib cc cc⌧ +cc jc )cc + 2 jc g (2ccfabc f+ 2✓ ✓Ib () cc + ✓cc(88) g Ifabc (2 ++ 2✓jj + +ajjc +abc 2bIbIbgg( ⇢⇢c(Ic ff+abc+ ) (88) (88) Iabc abc +a+ 2✓Ib ++ ⌧ccccjc ) ⌧⌧+c + 2+g gjcN ⇢jjcII⇢f))Iabc ✓cc c )Ib ++ abc (88) Ib gg(2 IcIf c (2 + + 2 ✓ + (88) Ib Ib ( c + c) ~c Ib Ia2✓ cjc Ic abc I abc (88) ~ ~ ~ ~ ~ ~ ~ j (2 + 2✓ + ⌧ + ) + 2 ⇢ f ( + ✓ ) + N NI Ifabc N N N I abc f (2 + 2✓ + ⌧ + ) + 2 ⇢ f ( + ✓ ) + + ⇢ + 2✓ + ) d r ⇥ (2d B + 2↵ B + ) d f D ⇥ (2d B + 2↵ B + b ) j b I abc I abc I abc I abc I N N j j j j Ng g NN g jc Ib jc jc jcjc N N N Ib jc jc jc Ib jc jc jc jc Ib fabc + 2✓ ) + jjc ) jc + g ⇢II⇢fabc Iabc +Ibggg(2⇢⇢(2 +jc 2✓ ++ 2✓(2 j + Ib j) jjc jfabc j jj+ jjc jj 2✓ jc jc + IbIb(2 = (2 0 ⇢IIjIffjjc abc + N⇢N + 2✓ + j jc + N c) j + I fabc + (2 + 2✓ +J B j j ~ ~ ~jjJc))+ ~bc ) d @ (2dE ~ a + 2↵I E ~ Ia + ~ea ) abc j(2dB + j 2↵ + N ↵ f X ⇥ I abc N N N @t = 0 == 0 0 = =0 g = 00 g b c Ic c Ib c ~ ~ ~ ~ Jc + ~ec ) For µ = i, + d f (2d E + 2↵ E + ~ e ) ↵ f (2d E + 2↵ For µ = i , abc I I abc JE N N g g g ijc c ~c ~ Ib ~ba )ijc + ~ ⇥ (2dB ~+a + 2↵ ~ Ia ~b ⇥ ~ c +I f2↵ ~IbIc(2~ X )(2dB dr B +(2 d 2✓fijc D + c~b+ )⇤ ) I fIabc abcI B abc+ N N N g g c g ~cIb ( c + ~✓Jc ~ Ib (2~ a jc + 2✓~jcIa+ 2 a jc ) +X ⇢I (2d fabcB )+ ~bc )⇢I fdabc@X ~2Ib ⇥ ~X + ↵I fabc + 2↵ B + (2dE j + 2↵I Ej + ~e j) J N N N @t g g = 0 ~c ~ Ic + ~ec ) ~ c + 2↵J E ~ Jc + ~ec ) + d fabc b (2dE + 2↵I E ↵I fabc Ib (2dE N N (89) g g Ib ijc ijc ijc Ib c c ~ ~ + + 2✓ + ) (2~ + ⇤ ) I fabc X (2 I fabc N Multiplying by tk , one derives the N non-abelian Gauss law: 23 g g jc jc Ib c c Ib ~ ~ + 2 ⇢I fabc X ( + ✓ ) + ⇢I fabc23 X (2g j + 2✓j + 2 j jc ) a Ja 23a N ↵r N 23 ~ ~ ~ ~b ~c ~ Jc + ~ec ) · (2dE + 2↵J E + I fabc D · (2dE + 2↵J E I 23~e ) + ↵23 N =0 1 Jb c Kc c ~he non-‐abelian ~Gauss ~ + 2dabc~e ) (89) · (dfabc E +↵ Multiplying by tk , one dierives law: K fabc E [IJ] tX 2 ~ Jb ~ cthe ~ Kc + 2~ec )Gauss ~ Jb · (dE ~ c + ↵K E ~ Kc + 2~ec ) Multiplying ia by[IJ] tkf, abc one derives X · (dE + ↵non-abelian b[IJ]law: dabc X KE ~JaJb · ~eca b(IJ) ~ Jb b· (dE ~ c +c ↵K E ~ Kc + g dabc X 2ia f ~X ec )c Jc 2~ ~ · (2dE ~ a (IJ) ~ ~ ~ ↵I r + 2↵abc E + ~ e ) + ↵ f D · (2d E + 2↵ E + ~e ) I abc J J @ a @N a ja a 2 ( + ✓ )+ (⌧ + j )c 1 ~ Jb · I(df ~ c + ↵K fabcI E ~ Kc + 2dabc @tabc E @t i [IJ] X ~e ) g 2 a a b c ~ ~ ~ ~ c) +~ Jb )~ Kc ⇤ I r · (2~ I fcabc D · (2~ + ~ c ++↵⇤K E ~ Jb ~c + ↵ E ~ Kc + 2~ec ) ia f X · (d E + 2~ e ) b d X · (dE N abc abc [IJ] [IJ] 4942 | P a g e K g g b c Kc c b ( c + Jb J u n e 2 0 1 7 + 2 ~ Jb ✓ c ~ ~I fcabc ~ + +j jc2~ ) + (⌧E )ec ) 2ia(IJ) fabc XN I· f~e abc b(IJ) d abc X ·N(dE + ↵K w w w . c i r w o r l d . c o m @ @a a @ @ja ja 2 I ( a 2⇢ + I✓a ) (+ a + ) j + 2✓j ja + j ja ) ) ++⇢I j (2 I ✓ (⌧ @t @t @t @t g c g c ~cb a g ~a ~ · (2~ b c~ b · (2~ + Ir + ⇤ ) f D +2 ⇢ f ( I +abc✓ ) + ⇢ + f ⇤ ) (2 jc + 2✓ jc + jc )

+2

⇢I fabc X ( ~+Jb✓ ) +~ c ⇢I fabc~X (2 + 2✓j + 2~ Jb ) ~c ~ Kc + 2~ec ) ia[IJ] fabc X · (dEN+ ↵K E Kc + 2~ej c ) b[IJ] dabc X j · (dE + ↵K E c ~ Jb · ~ec b(IJ) dabc X ~ Jb · (dE ~ c + ↵K E ~ Kc + 2~ 2ia(IJ) fabc X IeS )S N 2 3 4 7 -‐ 3 4 8 7 V o l u m e 1 3 (89) N u m b e r 6 @ a @ a jaJ o u r n a l o f A d v a n c e s i n P h y s i c s a 2 I ( + ✓ ) + I (⌧ + j ) Multiplying by tk@t , one derives the@tnon-abelian Gauss law: ~ · (2~ a + ⇤ ~ a ) g I fabc D ~ b · (2~ c + ⇤ ~ c) + Ir g a Ja a b c ~ · (2dE ~ + 2↵J E ~ + ~e ) +N ↵I fabc D ~ · (2dE ~ + 2↵J E ~ Jc + ~ec ) ↵I r g g N jc b c b c +2 + ✓c ) + I fabc ( I fabc (⌧ + j ) 1 N N Jb c Kc c ~ · (dfabc E ~ + ↵K fabc E ~ + 2dabc~e ) i [IJ] X @ a @ 2 a ( + ✓ ) + ⇢ (2c j ja + 2✓j ja ~+Jb j ja ) ~ c 2⇢ I I Jb c Kc ~ · @t ~ + ↵K E ~ +@t2~ ~ Kc + 2~ec ) ia[IJ] fabc X (dE e ) b[IJ] dabc X · (dE + ↵K E g g c jc jc jc b ~ Kc ~c Jb ~I fabc +X 2~ Jb ⇢· I~efcabc bb(IJ) ( cd+ +2~ 2✓ 2ia(IJ) fabc E + ↵K(2E ecj) + j ) j + abc✓X) +· (d⇢ N N @ + @( ca+ ✓c +ja4⌧ c ) iu f Jb a iu[IJ] a fabc Jb ( j jc + ✓j jc + 2 j jc ) [IJ] abc 2 I ( + ✓ ) + I (⌧ + j ) @t @t Jb Jb +a vAmpère dabc law: ( j jc + ✓j jc + 2 j jc ) g ( c + ✓~c b+ 4⌧ c )c v~(IJ) (IJ) a c dabc ~ ~ + I r · (2~ + ⇤ ) I fabc D · (2~ + ⇤ ) N~ a =↵0 r ~ ~ Ja + ~ba ) + g ↵I fabc D ~ b ⇥ (2dB ~ c + 2↵J B ~ Jc + ~bc ) ⇥ (2d B + g g 2↵J B jc bI c c b c N +2 f ( + ✓ ) + f (⌧ + ) (90) I abc I abc j N N 1 Jb c Kc c ~ ~ ~ ~ i [IJ] fabc X @ @ ⇥ (dB + ↵K B + 2b ) 2 ✓a ) + ⇢I (2 j ja + 2✓j ja + j ja ) 2⇢I ( a + @t Jb ~ @t ~ c + ↵K B ~ Kc + 2~bc ) b[IJ] dabc X ~ Jb ⇥ (dB ~ c + ↵K B ~ Kc + 2~bc ) ia[IJ] fabc X ⇥g (dB g jc jc jc b c c b + 2 ⇢I fabc ( + ✓ )~+Jb ⇢~Icfabc (2 j ~+Jb2✓j +~ c j ) ~ Kc c N + 2ia(IJ) fabc X N⇥ b b(IJ) dabc X ⇥ 24(dB + ↵K B + 2~b ) jc jc jc Jb c c c Jb + iu[IJ] fabc ( @ + ✓ a+ 4⌧ ) Ja iu faabc g( j + ✓jb + 2c j ) ~ + 2↵J E ~ [IJ] ~ + 2↵J E ~ Jc + ~ec ) +Jb ↵I c (2dE + ~e ) ↵I fabc (2dE c c + v(IJ) dabc ( @t + ✓ + 4⌧ ) v(IJ) dabc Jb (N j jc + ✓j jc + 2 j jc ) 1 ~ c + ↵K E ~ Kc + 2~ec ) + ia[IJ] fabc Jb (dE ~ c + ↵K E ~ Kc + 2~ec ) + i [IJ] fabc Jb (dE =0 2 Ampère law: (90) Ampère law: ~ c + ↵K E ~ Kc + 2~ec ) + 2ia(IJ) fabc Jb (~ec ) + b[IJ] dabc Jb (dE ~ ⇥ (2dB ~ a + 2↵J B ~JbJa +c~ba ) + g Kc ~ b ⇥ (2d@B ~ c + 2↵ B ~ Jc ~ c ↵I r ~ a) + b ) ~ + ↵K N ~ ↵I f+abc2~D + b(IJ) dabc (dE E ec ) + I (2~ a +J⇤ @t 1 c g ~ Jb~⇥ (dija ~ c + ↵K ~ Kc + ijaB ija2~ i [IJ] fabc X B b ) ~ b ijc + 2✓ijc + ijc ) + 2✓ + )24 + I r(2 I fabc D (2 2 N c ~c ~ Jbg⇥ (dB ~c + ~ Kc ~ Jb ~c ~ Kc ia[IJ] fabc X ) r( b ↵Kc B a dabc X ~ c+ 2~b2⇢ ~ b[IJ] + ✓a ) ⇥ (dB + ↵K B + 2b ) I fabc (2~ + ⇤ ) I Jb ~N ~ Jb ⇥ (dB ~ c +g ↵K B ~ Kc + 2~bc ) + 2ia(IJ) fabc X ⇥ ~bc jab(IJ) dabcjaX ja ~ ~ b ( c + ✓c ) 2⇢I r(2 j + 2✓j + ⇢I fabc D j )+2 @ g N a Ja a b c ~ + 2↵J E ~ + ~e ) ~ + 2↵J E ~ Jc + ~ec ) + ↵I (2dE ↵jcI fabc jc(2dE g jc b @t N ~ (2 ~ Jb c + ✓c + 4⌧ c ) + ⇢I fabc D j + 2✓j + j ) + iu[IJ] fabc X ( N ~c 1 c Jb ~ Kc ~ c +Jb↵K E ~ Kc + + i [IJ] fabc Jb (dE +~↵Jb ejc ) + ia[IJ] (dE ec )c jc fabc K E jc + 2~ c c 2~ ~ iu f X ( + ✓ + 2 ) + v d X ( + ✓ + 4⌧ ) 2 [IJ] abc (IJ) abc j j j Jb c Kc c Jb c ~ + ↵~KJb ~ jc 2~e ) jc+ 2ia(IJ) + b[IJ] dabc (d E v(IJ) dabc X E( j + + ✓j + 2 j jcf) abc (~e ) Jb 0 ~ c ~ a) ~ Kc + 2~ec ) + I @ (2~ a + ⇤ + b(IJ) dabc = (dE + ↵K E @t (91) g ija ija ija b ijc ijc ijc ~ ~ r(2 + 2✓ + ) + f D (2 + 2✓ + ) I I abc N g b c ~c ~ a + ✓a ) 2⇢I r( I fabc (2~ + ⇤ ) N g ja ~ ~ b ( c + ✓c ) 2⇢I r(2 + 2✓j ja + j ja ) + 2 ⇢I fabc D j N g ~ b (2 jc + 2✓ jc + jc ) + iu[IJ] fabc X ~ Jb ( c + ✓c + 4⌧ c ) + ⇢I fabc D j j j N ~ Jb ( jc + ✓ jc + 2 jc ) + v(IJ) dabc X ~ Jb ( c + ✓c + 4⌧ c ) iu[IJ] fabc X j j j ~ Jb ( jc + ✓ jc + 2 jc ) v(IJ) dabc X N =0

j

j

j

=0 ia

For X µ , µ

= 0 :

(91) 25


g ~ b · (dE ~ e + ↵J E ~ Je + 1 ~ee ) + id g fekl fabl D ~ b · (dE ~ e + ↵J E ~ Je + 1 ~ee ) dekl fabl D N 2 N 2 I S S N 2 3 4 7 -‐ 3 4 8 7 g 1 g Ib e Je e Ib e Je mX e~ 1 ·3 (d NE ~ · (dE ~ + ↵J E ~ + ~e ) + i ↵IVfoekll fu abl ~u m+b↵eJrE ~ 6 + 1 ~e) ↵I dekl fabl X J o u r n a l o f A d v a n c e s i n P h y s i c s N 2 N 2 ia For X , µ = 0: µ g 1 g 1 Ib e e Ib e e ~ · (~ + ⇤ ~ )+i ~ · (~ + ⇤ ~ ) I dekl fabl X I fekl fabl X N 2 N 2 1 e ~ · (dE ~ eg ~ Je + 1Ib~ee )e idfe aek r ~ g ~ e ↵J E ~IbJe + d↵I daek r E + + ↵I dJ ekl fabl 2 ( + ✓ ) i · (dIEfekl+fabl ( e +2 ~e✓e)) N N g gg bg e Je e 1 e je Ib ~ ~ ~ ~fbabl· (dIbE ~(⌧e e++↵J E ~jeJe 1 ee ) d dekl fabl+ D · (dE +f↵ e j) +) id i fekl fI fableklD J E (⌧+ +~ I dekl abl j ) + ~ N 2 N2N 2 2N gIb g g 1 g Ib e e Ib e e ~ ⇢·I d(deklE ~feabl ~ Je++✓ )~ee )i+ i⇢I f↵eklI ffekl ~ Ib+· (d ~ Je + 1 ~e) +X ✓E )~ e + ↵J E ablfabl( ↵I dekl fabl + ↵J(E X N N N 2 N 2 g 1 je and µgIb= i: e 1 Ibe je g je 1 jeIb je je g 1 Ib ~ () + ~j ) · (~i e +⇢I f⇤ ~ekle f) abl ( j + ✓j + dekl f+abl ⇤ +fabl X ) + ~ ⇢·I (~ j i + ✓jI fekl I dekl fabl X N 2 N 2 2 j N 2 N g 1 e g g gB ~ b ⇥ (d ~ e + ↵I B ~ Ie ~ b ⇥ (dB ~ e + ↵I B ~ Ie + 1~be ) =d 0Ib e fableD Ib + e ~ e+ id d b ) f f D ekl ekl abl + d f ( + ✓ ) i f f ( + ✓ ) I ekl abl N I ekl abl 2 N N N (92) 2 1 1 g g je je e Ie e e Ie e Ib ~ e Ib e ~ ) + ↵i I B ~ I+ ~ ⇥ )(dB ~ + ↵I B ~ + ~b ) ddaek(⌧ r⇥ ) idf + fabl +(djB fekl~fbabl (⌧ aek +r I dekl+ j 2 2 2N 2N g g g 1 g Ib e e Ib Jee ee Ib e Ib ~ i ⇥ ⇢(dI fB ~ekl f+abl↵J B ~( + ~ ⇥ (dB ~ e + ↵J B ~ Je + 1~be ) + ⇢I dekl fabl ↵(I dekl +f✓abl)X + ✓~b)) + i ↵I fekl fabl X N N N 2 N 2 g 1 je 1 g 1 1 je je je je je Ib Ib @ @ e) ( j (d +E + + ⇢I dekl fabld daek ~✓je ++↵I E ~ Ie ) i e ⇢I fekl fabl ( ~ e + ✓ ~ Ie N 2 j + 2 ~eN) + id faek @t (dEj + ↵jI E + 2 2j ~e ) @t =0 g ~ e + ↵I E ~ Ie + 1 ~ee ) id g fekl fabl b (dE ~ e + ↵I E ~ Ie + 1 ~ee ) + d dekl fabl b (dE (92) N 2 N 2 and µ = and i : µ = i: g 1 g ~ e + ↵J E ~ Je + ~ee ) i ↵I fekl fabl Ib (dE ~ e + ↵J E ~ Je + 1 ~ee ) + ↵I dekl fabl Ib (dE g 1 g 1 b 2 ~N ~ e + ↵I B ~ Ie + ~be ) + id 2fekl fabl D ~Nb ⇥ (dB ~ e + ↵I B ~ Ie + ~be ) d dekl fabl D ⇥ (dB N 2 N 2 g 1 g 1 ije ije ~ Ib1( ije + ✓ije + ~ Ib ( ije + ✓ije + + e I dekl fabl X )e i ) I fekl fabl 1~X Ie e Ie e ~ ~ ~ ~ ~ ~ ~ N 2 N 2 + ddaek r ⇥ (dB + ↵I B + b ) idfaek r ⇥ (dB + ↵I B + b ) 2 e 1 e 2 g Ib ~ 1) + i g g I fekl fabl Ib (~ e + 1 ⇤ ~ e) d f (~ + ⇤ I ekl abl g Ib e Je e Ib ~ ~ ~ ~ ~ ~ e + ↵J B ~ Je + 1~be ) N ⇥ (dB + ↵J B 2+ b ) +Ni ↵I fekl fabl X ⇥ (d 2B ↵I dekl fabl X g N 2 e ~ Ib ( e + ✓e )2 i g ⇢N ~ Ib e ⇢I dekl fabl X I fekl fabl X ( 1 + ✓ ) @ +~ N 1 @ ~ Ie + ~ee ) + id faek N ~ e + ↵I E ~ Ie + ~ee ) d daek (dE e + ↵I E (dE g 1 g @t 2 @t 2 je ~ Ib ( je + ✓ je + ~ Ib ( je + ✓ je + 1 je ) i ⇢I fekl fabl X + ⇢I dekl fabl X j1 j j ) j j g g b NE 2 j ~ e + ↵I E ~ Ie + ~ee ) id 2fekl fabl b (d ~ e + ↵I E ~ Ie + 1 ~ee ) + d dekl fabl N (dE N 2 N 2 =0 g 1 g ~ e + ↵J E ~ Je + ~ee ) i ↵I fekl fabl Ib (dE ~ e + ↵J E ~ Je + 1 ~ee ) (93) + ↵I dekl fabl Ib (dE N 2 N 2 g ~ Ib ije + ✓ije + 1 ije ) i g I fekl f26 ~ Ib ije + ✓ije + 1 ije ) + I dekl fabl X ( abl X ( N 2 N 2 g 1 g 1 Ib Ib ~ e) + i ~ e) (~ e + ⇤ (~ e + ⇤ I dekl fabl I fekl fabl N 2 N 2 g ~ Ib ( e + ✓e ) i g ⇢I fekl fabl X ~ Ib ( e + ✓e ) + ⇢I dekl fabl X N N g 1 je g je je Ib ~ ( ~ Ib ( je + ✓ je + 1 je ) i ⇢I fekl fabl X + ⇢I dekl fabl X j + ✓j + j ) j j N 2 N 2 j 26 =0 (93) ia Multiplying by tk , one gets the corresponding X µ -‐Gauss law: d

27 4944 | P a g e J u n e 2 0 1 7 w w w . c i r w o r l d . c o m


Multiplying by tk , one gets the corresponding Xµia -Gauss law: ~ · (dE ~ e + ↵J E ~ Je + 1 ~ee ) + i↵I faek r ~ · (dE ~ e + ↵J E ~ Je + 1 ~ee ) ↵I daek r 2 2 g g b e Je e b e ~ · (dE ~ + ↵J E ~ + ~e ) i ↵I fekl fabl D ~ · (dE ~ + ↵J E ~ Je + ~ee ) + dekl ↵I fabl D N N 1 1 i Ia k Jb e Jb e e X Jb fe X ~ · ~e d d [IJ] dJb deekl ~e d+abc fcek ) ~ ib·[IJ] (~e ekl ) ~ · ~e [IJ] X+ b [IJ] dabl [IJ] abc cek abl(~ N 2 2 Jb 2 e Ja Jb k e) Kk ~+ ~ ~ ~ eJa ~ k ~ ~ ~ Kk ) fabl fekl X (a(IJ) ~ e + a ⇤ if d X ~ee d+ablad[IJ] b(IJ) (d E + ↵K E + ~ek )(a +(IJ) b(IJ) abl ekl [IJ] ekl⇤ ) (dE + ↵K E N 2 ~ Ja k ~ kJa ~ Jb ~ e) ~abc ~ Kk ) ~ee + b(IJ) ⇤ X · (b ~ek + b(IJ) ) (ddE dcek +[IJ] ib(IJ) dabl fekl⇤ + ↵X K E (b[IJ] N 1~ e 1~ e e d aek~@tJa e ) @et (~ e + ⇤ +E ~+Jb (b ~⇤ ~ k⇤ ~ Kk ) I daek~ + idabc fcek X ) 2 i Ibf[IJ] X (~· (d +)↵K E [IJ] e + b(IJ)2 2 N ije ije~ k Jb kije Kk 1 ije ~ ~ · (d ~ ~ ~ ~ Jb ~ 1Kk )ije ) d )+ i fIekl faek r( ✓ije r( + ✓ + b(IJ) dabl dekl XI aek · (dE + ↵K E ) + ib[IJ] dabl X E++ ↵K+E 2 2 1 ~ b ijee ijeje g e g e b ije ~ +ekl fabl +✓ j ) ) i + ✓ije ) I daek @t (+ + ✓ ) I dD aek(@t (⌧ + Id I fekl fabl D ( 2 2N 2N g e g je 1 b e e e ( @)t (⌧ + i + j I f) abl fekl b ( e ) + i I faek @t ( 2N + ✓ I)fabl dekl i I faek 2N 2 e e e ~ ~ e ⇢eI daek1 r( + ✓ ) +~i⇢I faek e e r(1 ~ e+ ✓ ) ~ ~ + I daek r · (~ + ⇤ ) i I faek r · (~ + ⇤ ) 2~ 1 je 2 1 je je je ~ ⇢I daek r( + ✓j je + + ✓j je + ) g j b ) +e i⇢I faek r( j b e j g ~ ~ 2 D ·~ 2 j I dekl fabl D · ~ + i I fekl fabl g g 2N b ~2N ~ b ( e + ✓e ) + ⇢bI dekl fableD ( eg+ ✓e ) i ⇢b I fekl fable D g e e N N + d f ( + ✓ ) i f f ( + ✓ ) I ekl abl I ekl abl 2N 2N g 3 g je b ~ ~ b ( je + ✓ je + 3 je ) + je ⇢I dekl fabl1D je ( j + ✓j je + jej je ) jei ⇢1I fekljefabl D j j je 4 N 4 j ⇢I daek @t ( j N+ ✓j + j ) + i⇢I faek @t ( j + ✓j + j ) 2) ~ Jb ( ije ) + ia[IJ] fabl dekl X ~ Jb ( ije +ea[IJ]efabl fekl2X e e ⇢I daek @t ( + ✓ ) + i⇢I fJb +✓ ) aek @te( ~ ) ia[IJ] fabl dekl Jb (⇤ ~ e) a[IJ] fabl fekl (⇤ g g b e e b e e + ⇢I dekl fabl 2 ( + ✓ ) i ⇢I fekl fabl ( + ✓ ) N ~ Ja ( ijkN ~ Jb ( ije ) ib(IJ) dabc fcek X ~ Jb ( ije ) + b(IJ) X ) + b(IJ) dabc dcek X g 3 g 3 Nb je je je je b + ⇢I dekl fabl ( j + ✓j jeJb ~ e j je ) i ⇢I fekl fabl ~(e )j + ✓j + 4 j ) b(IJ) dabc dcek (⇤4 ) + ib(IJ)Ndabc fcek Jb (⇤ N + a[IJ] fekl fabl 1Jb (⌧ e + j je )~+Jbia[IJ] dekl fabl iJb (⌧ e + je ) ~ Ib e + u[IJ] fabl fekl X ( e + ✓e ) + u[IJ] fabl dj ekl X ( + ✓e ) 2 2 2 + b(IJ) Ja (⌧1k + j jk ) + b(IJ) dabc dcek Jb (⌧ e +i j je ) ib(IJ) dabc fcek Jb (⌧ e + j je ) N ~ Ib ( je + ✓ je ) ~ Ib ( je + ✓ je ) u[IJ] fabl fekl X u[IJ] fabl dekl X j j j j 2 Jb e 2 1 i + u[IJ] fabl fekl ( + ✓~e )Jb+ ue[IJ] fabljedekl Jb ( Ib + ✓e ) ~ Jb e je + u[IJ] fabl fekl X (2⌧ 2 2 + j ) + iu[IJ] fabl dekl X (2⌧ + j ) 1 i 1 je ~jeJa ~j jeIb+ u[IJ] fabl+ fekl1 vIb(IJ) ( jX +(✓jkje+ ) ✓k ) u+[IJ] fvabl ddabl dJbekl (X ( ✓ej+)✓e ) (IJ) ekl 2 2 N 2 + u[IJ] fekl fabl i Jb (2⌧ e + j je~) Ib + iue[IJ] dekl f 1Jb (2⌧ e~+Ia j jejk ) v(IJ) dabl fekl X ( + ✓e ) abl v(IJ) X ( j + ✓j jk ) 2 N 1 1 i + v(IJ) Ja (1 k + ✓k ) + v(IJ) dabljedekl Ibje( e +i✓e ) v(IJ) dablIbfekl jeIb ( e + ✓e ) je Ib ~ ~ N 2 2 v(IJ) dabl dekl X ( j + ✓j ) + v(IJ) dabl fekl X ( j + ✓j ) 2 jk 2 1 1 i Ia Ib Ib v(IJ) ( 2j + ✓j keJa) k v(IJ) djk ( j je + ✓j jeJb ) + ev(IJ) dabl f ( j je + ✓j je ) abl dekl je ekl ~ ~ N 2 2 + v(IJ) X (⌧ + 2 j ) + v(IJ) dabc dcek X (2⌧ + j ) Nk 2 je Ja Jb e + v(IJ) (2⌧ + j jk ) +X dabc iv(IJ) dabc fcek Jb (2⌧ e + j je ) e dcek je (2⌧ + j ) (IJ) ~vJb iv fcek (2⌧ + j ) (IJ) dabc N =0 =0 (94) (94)



the and the X µ -‐ Aand mpère law: Xµ ia

ia

-Ampère law:

~ ⇥ (dB ~ e + ↵J B ~ Je + 1~be ) + i↵I faek r ~ ⇥ (dB ~ e + ↵J B ~ Je + 1~be ) ↵I daek r 2 2 g g b e Je e b e ~ ~ ~ ~ ~ ~ ~ Je + ~be ) + ↵I dekl fabl D ⇥ (dB + ↵J B + b ) i ↵I fekl fabl D ⇥ (dB + ↵J B N N 1 ~k 1 ~ e i dabl fekl [IJ] ⇥ ~be + a(IJ) fabl fekl X ~ Jb ⇥ ~be + [IJ] ⇥ b + dabl dekl [IJ] ⇥ b N 2 2 ~ Jb ⇥ ~be + 2 b[IJ] X ~ Jb ⇥ ~bk + b[IJ] dabc dcek X ~ Jb ⇥ ~be + ia(IJ) fabl dekl X N 2 ~ Jb ⇥ ~be ~ Ja ⇥ (dB ~ k + ↵J B ~ Jk ) ib[IJ] dabc fcek X b(IJ) X N ~ Jb ⇥ (dB ~ e + ↵K B ~ Ke ) + ib(IJ) dabl fekl X ~ Jb ⇥ (dB ~ e + ↵K B ~ Ke ) b(IJ) dabl dekl X ~ e + ↵J E ~ Je + 1 ~ee ) i↵I faek @t (dE ~ e + ↵J E ~ Je + 1 ~ee ) + ↵I daek @t (dE 2 2 g g b e Je e b e ~ ~ ~ ~ Je + ~ee ) ↵I fabl dekl (dE + ↵J E + ~e ) + i ↵I fabl fekl (dE + ↵J E N N 2 1 i Ja k Jb e + (~e ) + [IJ] dabl dekl Jb (~ee ) (~e ) [IJ] [IJ] dabl fekl N 2 2 2 + a(IJ) fabl dekl Jb (~ee ) + ia(IJ) fabl dekl Jb (~ee ) + b[IJ] Ja (~ek ) N

29



Jbd e d Jbd e f (~edeabc ) fcek ib[IJ] + b[IJ] dabc d+cekb[IJ] (~abc e ) cekibJb (~ e ) cek Jb (~ee ) abc [IJ] 2 k Ja ~Kkk 2 k Ja Jad ~ ~+Kkb + ~edk ) + ~Kk ~ Kk ) ~b(IJ) ~E + ~E b(IJ) dkekl+ Ja + + b(IJ) + (dE + ↵K(d E +↵ ~ekK)E (d ↵K(dE ) ↵K E ablE (IJ) abl dekl N N Ja ~ ~ k ~ Kk ) Jad ~fk ib(IJ) + ib(IJ) dabl+ fekl (dablE ekl+ ↵K(d EEKk )+ ↵K E 1 ~@e (~ e + 1 ⇤ 1 ~@e(~ e + 1 ⇤ e ~ e) faek t) + I daek @t+ (~ e I+daek⇤ i I2f~aek)@t (~i e I+ ⇤t ) 2 2 2 1 ije 1 ije ije1 ijeije ~ ~ije ije1+ ije ije ije ✓ije + ) +i I faek)r( ~ ~+ i ije I d+ aek✓r( + + ✓ ) + I f+ aek✓r( + ) I daek r( 2 2 2 2 g g b ije ije g ~ b ije g ije ~ ije ijefabl D + abl D ~ bI(dekl ~ b f ijef + D + + ✓ije() i+ ✓ I)feklif2N ✓ () + ✓ ) I dekl f2N abl D I( ekl abl 2N 2Ng g b e b e g b I fabl e dekl g( ) + i b I fabl e fekl ( ) I fabl d2N ekl ( ) + i I fabl f2N ekl ( ) 2N 2N e ~ ~ e + ✓e ) e e ⇢ d r( + ✓e )r( faek ~ ~+ i⇢e I+ I+ aek ⇢I daek r( ✓ ) + i⇢I faek ✓er( ) ~je je1+ ✓je je + 1 je )~+ i⇢je ~ je je1+ ✓jeje + 1 je ) r( aek✓r( ~ ⇢jeI d+ j j ) + i⇢I fjaek r( I faek j j) ⇢I daek r( + + ✓ 2 2 j j j j j + 2 j 2 j g g e ~ b eg e ~b e g e fableD e fabl D ~⇢bI(dekl ~⇢bI(fekl + ⇢I dekl+ fablND + ✓ ) ( i +⇢✓I f)ekl fiabl D + ✓e ) ( + ✓ ) N N N g 3 je je ~ bje( je3+ ✓je je + 3g je ) i g ⇢bI fekljefabl D ~ bje g + ~⇢bI dekljefabl D ( j je3+ ✓je + ) j j j j ~ + ⇢I dekl fablND ( j + ✓j + i 4 ⇢I fekl fabl ND ( j + ✓j + 4 j j ) j ) N 4 N 4 ~ Jb ( ije ) + ia[IJ]Jbfablije ~ Jb ( ije ) + a~[IJ] fekl X dekl X Jbfablije ~ + a[IJ] fabl fekl X ( ) + ia[IJ] fabl dekl X ( ) Jb ~ e Jb ~ e a[IJ] (⇤ ) ia[IJ] (⇤ ) abl abledekl Jb f~ e fekl Jbf~ a[IJ] fabl fekl (⇤ ) ia[IJ] fabl dekl (⇤ ) 2 ~ Ja ijk ) + b(IJ)Jb ~ Jb ( ije ) ib(IJ)Jb ~ Jb ije ) 2 +Ja bijk dabc dijecek X dabc fije cek X ( (IJ) X ( ~ ~ ~ + b(IJ) X N ( ) + b(IJ) dabc dcek X ( ) ib(IJ) dabc fcek X ( ) N Jb ~ e Jb ~ e b(IJ) d d ( ⇤ ) + ib d f ( ⇤ ) abc (IJ) abc ~ e ) cek ~ e )cek b(IJ) dabc dcek Jb (⇤ + ib(IJ) dabc fcek Jb (⇤ 1 ~ Jb ( ie + ✓e ) + i u[IJ] fabl dekl X ~ Ib ( e + ✓e ) + u~[IJ] 1 Jbfablefekl X e Ib e ~ 2 2 + u[IJ] fabl fekl X ( + ✓ ) + u[IJ] fabl dekl X ( + ✓e ) 2 2 1 i ~ Ib ( je + ~ Ib ( je + ✓ je ) u[IJ] fabl fjeekl X ✓j je ) u[IJ]Ibfabl djeekl X 1 i j j j je Ib ~ ~ 2 2 u[IJ] fabl fekl X ( j + ✓j ) u[IJ] fabl dekl X ( j + ✓j je ) 2 ~ Jb (2⌧ e +2 je ) + iu[IJ] fabl dekl X ~ Jb (2⌧ e + je ) + u[IJ] f f X j j je Jb abl eekl Jb e ~ ~ + u[IJ] fabl fekl X (2⌧ + j ) + iu[IJ] fabl dekl X (2⌧ + j je ) 1 1 ~ Ja ( 1k + ✓k ) + v(IJ) dabl dekl X ~ Ib ( e + ✓e ) +Ja vk(IJ) X 1 k Ib e ~ ~ N 2 + v(IJ) X ( + ✓ ) + v(IJ) dabl dekl X ( + ✓e ) N 2 Ib e i 1 ~ ( 1+ ✓ e ) ~ Ia ( jk + ✓ jk ) v(IJ)Ibdablefekl X v(IJ) X i j j jk e Ia ~ ~ 2 N v(IJ) dabl fekl X ( + ✓ ) v(IJ) X ( j + ✓j jk ) 2 1 ~ Ib ( Nje + ✓ je ) + i v(IJ) dabl fekl X ~ Ib ( je + ✓ je ) v(IJ) dabl djeekl X j j j 1 i je je je j Ib Ib 2 ~ ~ ( v(IJ) dabl d2ekl X j + ✓j ) + v(IJ) dabl fekl X ( j + ✓j ) 2 2 ~ Ja (⌧ k + 2 jk2) + v(IJ) dabc dcek X ~ Jb (2⌧ e + je ) + v(IJ) X j j 2 jk Ja k Jb e ~ N ~ (2⌧ + je ) + v(IJ) X (⌧ + 2 j ) + v(IJ) dabc dcek X j N ~ Jb (2⌧ e + je ) iv(IJ) dabc fcek X j Jb e ~ (2⌧ + je ) iv(IJ) dabc fcek X j 30 =0 =0 (95) (95) The corresponding vectorial ianchi expressions for the Bianchi identities are The corresponding vectorial expressions for the Bvectorial identities are The corresponding expressions for the Bianchi identities are ~ ⇥ (dE ~ + ↵J E ~ J ) = DJ (dB ~ +D ~ J) D ↵J B ~ ⇥ (dE ~ + ↵J E ~ ~ + ↵J B ~ J) D ) = (d B Dt (96) Dt (96) J ~ ~ ~ D · (dB +~ ↵J B ~) = 0 ~ J D · (dB + ↵J B ) = 0 and 4947 | P a g e and J u n e 2 0 1 7 ~ ⇥ ~e = D ~b w w w . c i r w o r l d . c o m D D~ ~ Dt D ⇥ ~e = b (97) Dt (97) ~ ~ D · b = 0~ ~ D·b=0

and

8

D ⇥ (dE + ↵~J E )vectorial =~ D Dt (dB +the ↵J B (96) The corresponding Bianchi identities are The corresponding expressions forB identities are(96) J) ~B ~the ~ JJ ) (96) ~ ⇥ ~vectorial ~⇥J )expressions ~Jfor ~Dt D (d E +Dt ↵J(d E + ↵Bianchi D (dE + ↵D E += ↵J B )(d JB J JJ ) J= ~ ~ ~ ~ ~ ~ ~ D ~ ~ ~ D ⇥ (d E + ↵ E ) = (d B + ↵ B ) D · (d B + ↵ B ) = 0 Dt J ·~~(dE +↵~↵~JE )== 0Dt J (96) J J (d ~⇥D ~B+ ~BJ= ~+ ~D JB D ~D ~E ~JD )~(d )(d J ~D Dt J J)⇥ J~ + ↵ ~ (96) ~)+= ~JJJ)) (96) D (d E + ↵ (d B + ) (d JB ·⇥ (d B + ↵ B )J(d = D ~ (d ~JD ~J↵B ~B D ~JJ0~E J~E J~ D + ↵ )=↵ = B J↵ J~ JB ~ ~ ~ D ⇥ (d E + E B + ↵ B ) J~B J)(d Dt ⇥ E ↵ E + ↵ ~ ~ ~ I S S N 2 3 4 7 -‐ 3 4 8 7 (96) J ~ ~ ~ ~ ~ ~ ~ ~ J J D · (d B + ↵ B ) = 0 Dt (96) D ·D (dB ↵J+ B ) = 0 J J~Dt J (d ⇥+ (dE ↵ E ) = B + ↵ B ) D ⇥ (d E + ↵ E ) = (d B + ↵ B ) Dt J J ~ ~ (96) J (96) D ·J(dB + ↵J BDt) = 0 Dt (96) V o l u m e 1 3 N u m b e r 6 and D ~ ~ ~ and Dt (96) J (96) ~D ~B ~~JD · ~(d + B )~(d~= 0+ ↵ ~B ~↵·) (d ~)+ ~J J = J0B · (d B + ↵ ~J↵B ·= and D o0 u0r n a l o f A d v a n c e s i n P h y s i c s J ~JJ) )= J~ · (d B + B = 0 D B ↵ B J ~ ~ ~ ~ J and D · (dB +D ↵J· B (dB) = + 0↵J B and D ~) = 0 D ~ and ~ ~ and D b D ⇥ ~ e = D and and and ~b ⇥b~e = Dt ~ ⇥ ~e = D Dt D~ and D (97) (97) ~ and and ~b ⇥ ~e = D ~b ~ ⇥ ~e = Dt D D (97) ~ D ~ ~ ~ ~ D~⇥ Dt (97) · b~e==0 D ·~eb== D 0Dt D b~ ~⇥D (97) D ~~bD ~D ⇥ ~b~ = ~D Dt (97) ~bD D ~eD · = 0 ~ D ~ D ⇥ ~ e = b ~ ~ ⇥ ~ e = b D ⇥ ~ e = b Dt (97) ~b~ = ~ ·D ·⇥~bb~e==0 Dt~b (97) D ⇥ 0~e Dt = D D ~~Dt (97) Dt D · b = 0 (97) (97) ~~· ~b = 0 Dt (97) (97) ~D ~b = 0blocks ~Dt D · bD = 0~bbuilding ~ ~ D · ~ C Lagrangian C Lagrangian building blocks · = 0 D · b = 0 ~ ~ ~ ~ D · b = 0 D · b = 0 C Lagrangian building blocks

C Lagrangian Lagrangian C Lagrangian buildingbuilding blocks blocks C building L =blocks Lblocks LS= LA blocks + LS A +L Lagrangian building CC CLagrangian building C Lagrangian building blocks L = Lbuilding A+L SL = L blocks Lagrangian building blocks C Lagrangian + LzS[µ⌫]z+ C Lagrangian building blocks [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] C Lagrangian building blocks A Lagrangian BuildingLBlocks L +2 zZ L S[µ⌫] Z== L LA[µ⌫] ZA[µ⌫] + L+ Lz = +3 Z L[µ⌫] A = 1 Z[µ⌫] 1+ 2Az[µ⌫] 3 Z[µ⌫] z [µ⌫] S [µ⌫]

(98) (98) LA = 1 Z[µ⌫]L ZL + z z + Z z (98) [µ⌫] [µ⌫] [µ⌫] L = L + L 2 3 [µ⌫] [µ⌫] A S [µ⌫] [µ⌫] [µ⌫] L + LZSz+ Z + z+ zSz[µ⌫] + 3 Z[µ⌫] z[µ⌫] (98) LL = + [µ⌫] [µ⌫] A =[µ⌫] 2A [µ⌫] [µ⌫] [µ⌫] [µ⌫] S [µ⌫] L = L = LL LA L =S = ZLA +LL (98) A S A 1 Z⇠[µ⌫] 2z [µ⌫] Z= ⇠L + ⇠[µ⌫] Z z+ =+L ⇠111A+ ⇠L zZ zL Z Z[µ⌫] + z[µ⌫] + ⇠33ZZ[µ⌫] (98) 1 ZL 2z 3[µ⌫] [µ⌫] S 223 [µ⌫] = +L [µ⌫] [µ⌫] [µ⌫]zz [µ⌫] [µ⌫] [µ⌫] LzL = + A S [µ⌫] [µ⌫] [µ⌫] AZ[µ⌫] S[µ⌫] [µ⌫] [µ⌫] [µ⌫] = ⇠ Z Z + ⇠ z z + ⇠ z [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] L Z Z + z z + Z z (98) S 11 [µ⌫] 2 3 [µ⌫] [µ⌫] A 2 3 [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] LAL= Z Z + z z + Z z (98) L = Z ZzZ[µ⌫] zZ + ⇠333Z =+⇠112Z Z + ⇠⇠3223+ z[µ⌫] + Z[µ⌫] z [µ⌫] (98) (98) [µ⌫] Z [µ⌫] [µ⌫] [µ⌫] [µ⌫] A= [µ⌫] [µ⌫] [µ⌫]z [µ⌫] [µ⌫] Z z2zzZz[µ⌫] ZZ L=A1⇠= = Z z++ (98) z[µ⌫] + [µ⌫]z [µ⌫] AS [µ⌫] [µ⌫] 1 LZ 2Z 3zz [µ⌫] [µ⌫]+ SL 1 Z[µ⌫] 2+ [µ⌫] [µ⌫] L =[µ⌫] ⇠1⇠11Z ⇠2+ ⇠33Z zz[µ⌫] Z + z[µ⌫] zz+ (98) S[µ⌫] [µ⌫] [µ⌫] = Z + zzz[µ⌫] + (98) A[µ⌫] 1 ZL 2Z 3Z [µ⌫] [µ⌫] [µ⌫]+ [µ⌫] A 2 z[µ⌫] [µ⌫] [µ⌫] [µ⌫] z[µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] L = ⇠ Z Z + ⇠ z z + ⇠ Z (1) L = tr(1) Z Z L tr Z Z S 2 [µ⌫] 3[µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫]+[µ⌫] [µ⌫] + [µ⌫] [µ⌫] [µ⌫] LS[µ⌫] = ZZLL + ⇠ z z ⇠ Z z [µ⌫]⇠1 Z1[µ⌫] = ⇠ Z Z + ⇠ z z ⇠ Z z 2 3 [µ⌫] S 1 2 3 [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] [µ⌫] = ⇠ Z Z + ⇠ z z + ⇠ Z z L = ⇠ Z + ⇠ z z + ⇠ Z z (1) L = tr Z[µ⌫] Z [µ⌫] [µ⌫] [µ⌫] S [µ⌫] 1Z [µ⌫] [µ⌫]z+ ⇠3 Z[µ⌫] z L=SS tr = Z⇠[µ⌫] Z= 1⇠ + ⇠33Zz[µ⌫] (1) L Z [µ⌫] L Z[µ⌫]⇠ 2zZ[µ⌫]z + ⇠2+z[µ⌫] [µ⌫] 3 [µ⌫] (1) L = tr Z[µ⌫] (1)ZL[µ⌫] = tr Z1[µ⌫] ZS[µ⌫] 1 I[µ⌫]2 µ[µ⌫]⌫J I 2 [µ⌫] I ⌫ µJ I µ ⌫J [µ⌫] L = 2a a [(@ G )(@ G ) (@ G )(@ G )]⌫ )(@ ⌫ GµJ )] L = 2a a [(@ G )(@ G ) (@ [µ⌫] (1) L = tr Z Z Ktr Z I Z JK µI ⌫ Iµ J ⌫J µ ⌫ µI ⌫ ⌫ µJ µ G [µ⌫] [µ⌫] [µ⌫] (1) L(1) = tr Z ZZLLL (1) = [µ⌫] [µ⌫] = 2a a [(@ G )(@ G ) (@ G )(@ G )] I µ ⌫J (1) = tr Z Z L = tr Z [µ⌫] K I J L[µ⌫] µ = µ IG [µ⌫] ⌫ ⌫J ) [µ⌫] [(@K)µ G⌫I(@ )(@ (@)]µ⌫J GI⌫I )(@ ⌫⌫ GµJ )] I⌫ 2aIµaJ ⌫J ⌫ µIµJ (1) L = tr(1) Z[µ⌫] Z=(3) KG µ ⌫J Z ZL(3) µI K LL(3) =tr2a aJ4iga [(@ )(@ G G )(@ G K µK= µ 2a a [(@ G )(@ G ) (@ G]⌫ )(@ GµJ(99) )] ⌫ ⌫ Lint = I[µ⌫] a (@ G )[G , G ] L(3) 4iga a (@ G )[G , G (99) K I J µ µ µK ⌫J (IJ) K⌫ (IJ) µ I⌫ µI⌫ ⌫J I µ ⌫ µJ int IG (@ µµ G⌫J IG ⌫ µJ I(@ µ] K ⌫J ⌫ (99) µJ µI ⌫ )] ⌫JII = 4iga a )[G , G = 2a a [(@ )(@ G ) )(@ G KL (IJ) K I J µ µ I µ ⌫J ⌫ µJ ⌫ int I µ ⌫J I µJ ⌫ ⌫ (3) LLKL =L 2a a [(@ G )(@ G ) (@ G )(@ G )] = 2a a [(@ G )(@ G ) (@ G )(@ G )] L = 4iga a (@ G )[G , G ] (99) K µI ⌫J (3) I= J2aL I)(@ Jµ µ µ(@ (IJ) KG µI(@ ⌫J⌫)] ⌫ µ ⌫µG (4) ⌫µG µK (4) = 2a G )J)(@ G )(@] ⌫G(99) ⌫J I⌫L ⌫,µK µJ int aµK [(@ G )(@ G 2 IKG J⌫),I)(@ 2J [(@ ⌫J µJ)] II4iga µ µµ K= IK JK µ⌫ Lint =K 4iga aJ(IJ) (@ G )[G ⌫a ⌫ L = aG (@ GIµ]G )[G G ]I⌫⌫⌫L (99) µ (IJ) L = 2a a [(@ G )(@ ) G G )] ⌫JK L g a [G , ][G , ] KG µ(@ (IJ) g a a [G , G ][G , G L = 2a a [(@ G )(@ ) (@ G )(@ G )] (4) ⌫ I µ µ (3) int (KL) (IJ) (KL) 2 I J µK ⌫L ⌫ ⌫ µI ⌫J K I µ µ µ ⌫ int ⌫ int ⌫ (3) (3) (4) K µI ⌫J K ] J µI µI µK⌫J ⌫J ⌫L 2 IK = g4iga a [G , G ][G , G L(4) = a (@ G )[G , G ] (99) (3) (IJ) (KL) (3) 4iga K µ (IJ) µ ⌫ int K µI ⌫J ⌫ int LL = a (@ G )[G , G ] L = 4iga a (@ G )[G , G ] (99) (99) L = g a [G , G ][G , G ] 2 K4iga µK (4) K µ µ4iga (IJ) (IJ) (IJ) (KL) 2 I⌫L (3) = ⌫)[G ⌫I , µ int int L= = aJK (@µI[G GG G ,⌫JG] ⌫L ] µ (99) int ⌫J (3) (@ G )[G G (99) K µI,µK K K= (IJ) L aK [G G ][G ]]]⌫J⌫ ][G ⌫(IJ) int g aL L g4iga a(IJ) G)[G (IJ) (KL) L 4iga aa(IJ) (@ G )[G (99) µa (KL) int Lint = a (@µ,µ,,G ,µK G ] ⌫L (99) µ⌫,⌫⌫L int (4) 2of I µ(IJ) µK int = K⌫⌫J int (4) (4) Equation of motion Equation motion 2 I J 2 I J µK ⌫L L = g a a [G , G ][G , G ] (4) (4) g a(IJ) (IJ) (KL) 2g, µ I ,] G J⌫L µK , G ⌫L ] ⌫ a(KL) 2 I][G J ,G µK int L = a [G ][G Lint = a [G G Equation of motion (IJ) (KL) (4) µ ⌫ int µ ⌫ L = g a a [G , G ][G , G ] 2 I J µK ⌫L (4) L = g a a [G , G ][G , G ] (IJ) (KL) Equation motion (IJ)a (KL) int µ G ⌫ [GµI , G⌫J ][G int = ofgL Lint aint [G ] µK , G⌫L ] = (KL) g2a (IJ) Equation of motion µ, a ⌫ ][G (IJ) Equation of motion [µ⌫] Jb µ [µ⌫] ⌫ Jb [µ⌫] [µ⌫] (KL) Equation of motion tr 1 (4aI @of⌫[µ⌫] ta I @⌫4iga Z (IJ) [t G, t⌫b ]) = 0[t , t ]) (100) trZmotion (4a Z[µ⌫](IJ) taJbG⌫4iga Z [µ⌫] =0 (100) Equation [µ⌫] Equation Equation of of motion trmotion (4aI @of t1a(4a 4iga [ta , taG 0 a b(100) Equation 1motion ⌫ Zmotion b ])Jb= (IJ) G⌫ Z4iga Equation of trmotion (100) [µ⌫] [µ⌫] (IJ) 1 I @⌫ Z [µ⌫] taJb JbZ [µ⌫][ta , tb ]) = 0 Equation of motion ⌫ Equation of tr 1 (4aI @⌫ Z tr t1a(4a4iga [t(IJ) ])⌫ =Z0 [ta , t(100) ta⌫JbZ4iga (100) a , tbG I @⌫ Z(IJ) G b ]) = 0 [µ⌫] [µ⌫] [µ⌫]Jb Jb=[µ⌫] [µ⌫] Taking the above relations’ trace [µ⌫] [µ⌫] Taking the above relations’ trace tr (4a @ Z t 4iga G Z [t , t ]) 0 (100) 1 I ⌫ a a b (IJ) [µ⌫] Jb ⌫ [µ⌫] Jb [µ⌫] tr (4a @ Z t 4iga G Z [t , t ]) = 0 (100) tr (4a @ Z t 4iga G Z [t , t ]) = 0 (100) 1 tI @ I (IJ) ⌫4igat⌫a a G4iga (IJ) Taking the above trace 1 tr Irelations’ ⌫theItr a1[µ⌫] b[t ⌫tZ (4a 0 (100) Jb @above Za[µ⌫] ])[µ⌫] =[t00aa, tbb]) = (100) (IJ) Taking 1 (4a ⌫Z a ⌫Z a⌫,Jb b]) (IJ) Taking the above relations' trace ⌫ Z tr relations’ trelations’ [tGG = (100) tr Z [µ⌫](IJ) ttrace [ta , tb ]) = 0 (100) 1 (4a ⌫ Z trace a I @⌫4iga a , ⌫tbZ 1 (4a a G⌫4iga (IJ) Taking theI @above relations’ trace Taking the above [µ⌫]a Jb [µ⌫]c [µ⌫]a Jb [µ⌫]c 2a @ Z + 2ga f G Z (101) 2a @ Z + 2ga f G Z (101) Taking the above relations’ trace [µ⌫]a Jb [µ⌫]c I ⌫ abc (IJ) I ⌫ abc (IJ) ⌫ ⌫ Taking relations’ trace Taking the above relations’ trace [µ⌫]c (101) 2a @⌫above Zabove +2a 2ga ftrace G⌫2ga Z (IJ) f abc GJb Taking the relations’ Ithe abc Taking the above relations’ trace Z [µ⌫]a + (101) I @⌫(IJ) [µ⌫]a [µ⌫]c [µ⌫]a relations’ Jb [µ⌫]c ⌫JbZ Taking the above relations’ trace Taking above trace 2a @ Z + 2ga f G Z (101) 2athe @ Z + 2ga f G Z (101) I ⌫ abc I ⌫ (IJ) abc ⌫Jb (IJ) ⌫ [µ⌫]a [µ⌫]c [µ⌫]a Jb [µ⌫]c [µ⌫]c (101) Multiplying by t and taking the trace: [µ⌫]a Jb [µ⌫]c Multiplying by t and taking the trace: 2a @ Z + 2ga f G Z k k t Multiplying b y a nd t aking t he t race: I ⌫ abc [µ⌫]a Jb (IJ) ⌫ 2a @ Z + 2ga f G Z (101) [µ⌫]a Jb [µ⌫]c Z 2ga ZG (101)(101) k I(IJ) ⌫ fabc G abcG ⌫ Z (IJ) Multiplying by tk2aand taking the trace: I @⌫2a 2a @ Z 2ga fabc (101) [µ⌫]a Jb @⌫Z Zt+ + 2ga f⌫abc Z[µ⌫]c I+ ⌫taking Multiplying and the trace: [µ⌫]a ⌫Jb [µ⌫]c I by (IJ)+ ⌫(IJ) 2a 2ga G Z (101) I@ ⌫ tk abc2ga (IJ) f 2a @ Z + f G Z (101) ⌫ (IJ) Multiplying by and taking the trace: I ⌫ abc Multiplying by t and taking the trace: ⌫ k k [µ⌫] [µ⌫]e Jb [µ⌫]e [µ⌫]e Jb [µ⌫]e [µ⌫] (2)Multiplying L = atrI(2) z[µ⌫] zMultiplying (d if )@ Z + ga f (d if )G Z (102) L = tr z z a (d if )@ Z + ga f (d if )G Z (102) [µ⌫] by t and taking the trace: [µ⌫]e Jb [µ⌫]e aek aek ⌫ abc ekc ekc (IJ) aek aek ⌫ f abc)G ekcZ ekc (102) k I)@z[µ⌫] ⌫ tkga and taking the(IJ) trace: [µ⌫]e (2) Lif Multiplying by tk= and taking the trace: azI (dzaek Z zbyby (d+ekc if ⌫Jb Z [µ⌫]e [µ⌫] [µ⌫] Multiplying and taking the trace: aek abc (IJ) Multiplying by tk⌫aek and taking the trace: ⌫ekc [µ⌫]e kaek atr ift+ )@ ga fekc ifekc )G (102) I t(d ⌫Z abc (dJb (IJ) [µ⌫]e (2) L = trMultiplying ⌫Jb Z [µ⌫]e [µ⌫]e by and taking the trace: [µ⌫] k Multiplying by t and taking the trace: a (d if )@ Z + ga f (d if )G (102) k a (d if )@ Z + ga f (d if )G Z [µ⌫] aekI ⌫(4) aek [µ⌫] aek (IJ) ⌫ abc ⌫ekc ekc (102) (IJ) ekc ⌫ I aek abc ekc [µ⌫] [µ⌫]e (2)LL==trtr z(d[µ⌫] I J µKI ⌫L (2) = tr)@ z[µ⌫] z= (4) JJb]Jb[µ⌫]e µK[µ⌫]e ⌫L (102) [µ⌫]e Jb [µ⌫]e [µ⌫]e (2) zI[µ⌫] z z L if [µ⌫]e (4) a Z + ga f (d if )G Z L b b [G , G ][G , G I J µK ⌫L [µ⌫]e Jb aek aek ⌫ abc ekc ekc (IJ) L = b b [G , G ][G , G ] (IJ) (KL) a (d if )@ Z + ga f (d if )G Z (102) ⌫ Jbif [µ⌫]e [µ⌫]e µ + ⌫ga if int (KL) I⌫ Z aek aek ⌫f abc ekc ekc)G aI (daek ifaeka )@ + ga (d Z (102) (IJ) ⌫⌫ ][G int ⌫ Z L)@ =abc bJ(IJ) bµK [G ,)G G(d ,ZG ] ⌫Jb ekc ekc (IJ) Z fµµabc (102) (4) [µ⌫]e [µ⌫]e (IJ) (KL) (daek )@⌫Z Zif + (d,(IJ) if )GJb (102) Iif aek aek ⌫Iga ekc ekc ⌫L int [µ⌫]e [µ⌫]e aek abc ekc ekc (IJ) ⌫ Z aaII(d if(d )@ + ga fIfabc (d if )G (102) [G , G ][G G ] aek Linta aek ⌫ b(KL) ekc ekc J µK ⌫L (IJ) (dbaek if )@ Z + ga f (d if )G Z (102) (IJ) ⌫ I J µK ⌫L µ ⌫ I= aek ⌫ abc ekc ekc (IJ) ⌫ (4) I 2b + 2bb(IJ)(4) c(KL) [G G ]{G }µKµK ,, G I, G I⌫L J⌫J ]{G ⌫L⌫L } (4) = b + c [G I= J(IJ) µK ⌫L µJ,][G ⌫µK (KL) L [G , G , G ] µ + 2b c [G , G ]{G G } L b b [G , G ][G , G ] (IJ) (KL) (IJ) (KL) Lintint b(KL)int [GIµ ,µG(IJ) ][G ⌫ (KL) J µK, G ⌫L µµ]} ⌫ ⌫ (IJ)c ⌫I ]{G +=2bb(IJ) , G, IG J µK ⌫L (KL)b[Gµ , G JJ] µK ⌫L IG⌫µJG J⌫b[G µK ⌫L I⌫L µK ,µK + 2 IG [IJ] (KL) + 2 G [G I µK [IJ] (KL) +2b2b c [G , G ]{G , G }J⌫ ]{G + 2 b G G ,G G⌫L µµ + 2b c [G , ⌫}G , G]] ⌫L } (103) (IJ) (KL) [IJ] (KL) ⌫ [G µ ⌫ + c [G , G ]{G , G I J µK ⌫L (IJ) (KL) (IJ) (KL) µ µ ⌫ 31 (103) 31 (103) + 2 [IJ] b G G [G , G ] I J µK ⌫L (KL) µ ⌫ I J µK ⌫L 31 + II,,G ⌫L IcJµJ, G µK ⌫L + c[IJ] c[KL] {G }{G GJ⌫J}{G }µKµK , ⌫L 31 c {G G } I µK ⌫L ⌫ (103) [IJ] [KL] + c c {G + 2 b G G [G , G ] µ [IJ] (KL) bµK Gµ]µ G⌫⌫[G , G ] µ2 ⌫[IJ][KL] 2 [IJ] b(KL) [G , 31 G I + J⌫[IJ] (KL) 31G µG (103) ++c[IJ] c[KL] {G ,µK G⌫L }⌫L I J ⌫L (103) µ, G ⌫ }{G (103) IIG JJ } µK ⌫L µKµK ⌫L⌫L IG J ⌫c{G µK + 2 c G ,IG 31 [IJ] [KL] J 31 + 2 G G {G , G } I J µK ⌫L µ + 2 c G [IJ] [KL] + c c {G , G }{G , } 31 µ ⌫ [IJ] [KL] [IJ] [KL] 31 µ, G ⌫⌫ }{G c c {G , G } µJ ⌫ µK + c c {G , G }{G , G } I µ+ ⌫L 31 [IJ] [KL] [IJ] [KL] µ ⌫ + 2 [IJ] c[KL] Gµ G31 GGI⌫L } I{GJ ,µK I⌫L JJ µK µK ⌫L I⌫ J J ⌫ µK µK31 G G G [IJ] [KL] µ[IJ] I+ ⌫L G G{G G , G⌫L } IG} µK + G +2 2+[IJ] c G G {G , G [KL] [IJ] [KL] µ ⌫⌫J µ [IJ] [KL] µ ⌫ + c G G {G , G } + c G G J 2⌫ µK ⌫L [KL]I µ [IJ] [KL] µ ⌫ + [IJ] [KL] Gµ GI⌫ GJ µK G ⌫L I G J G µK G ⌫L I J µK ⌫L [IJ] [KL] Equation of motion: µ ⌫G ⌫[IJ] G ++of GG + [IJ] [KL] Equation of motion: Equation motion: [KL] Gµ G⌫ G G µG Equation of motion: Equation of motion: [µ⌫] Equation motion: [µ⌫] Jb [µ⌫]2 [IJ] GJb z [µ⌫] [ta , tJb [µ⌫] tr 2 of 4bof(IJ) GJb [tofaG ,G tJb ] z+ 4c[t[t GJb+ z [µ⌫] {t G , tJb + ] [µ⌫] = 0[ta , tb ] = 0 Equation motion: bJb b} [IJ] Equation motion: tr 4b(IJ) z[µ⌫] 4c[IJ] ⌫22z4b tr 4c z {tJb a,,ttbb]⌫] + (IJ) [IJ]aG a a , t[µ⌫] b }⌫+ 2 [IJ] G⌫b z ⌫ ⌫ ⌫ ⌫ Jb [µ⌫] Jb [µ⌫] tr 2 4b(IJ) G⌫ zJb [µ⌫] [ta , tb ] + 4c[IJ] G⌫ zJb [µ⌫] {ta , tb } + 2 [IJ] G⌫ zJb [µ⌫] [ta , tb ] (104) = 0 (104) Jb [µ⌫] Jb [µ⌫] Jb [µ⌫] 4b G z [t , t ] + 4c G z {t , t } + 2 G z [t , t ] =0z0[µ⌫] [t , t ] = 0 Jb [µ⌫] Jb [µ⌫] Jb a b a b a b 2 4b (IJ) [IJ] [IJ] ⌫ ⌫ ⌫ trtr 2Taking G z [t , t ] + 4c G z {t , t } + 2 G z [t , t ] = trace above equation afrom b G aequation b G⌫ z [IJ]{ta⌫, tb } +(104) a [IJ] b G (IJ)the [IJ] tr 4b z [t , t ] + 4c 2 ⌫ ⌫ a b a b 2 (IJ) [IJ] Taking the trace from above ⌫ ⌫ Taking the trace from above equation Taking the trace from above equation (104) (104) Taking the trace from above equation (104) Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c [µ⌫]c Jb [µ⌫]c Takingthe the trace from above equation Jb [µ⌫]c Jb Jb [µ⌫]c 2ib f G z + 2c d G z + d G z (105) Taking trace from above equation 2ib G⌫ [IJ] z abc+ + 2c 2c[IJ]equation G⌫[IJ] +⌫ [IJ] dabc G⌫ zz (105) abc 2ib (IJ) Taking ⌫ (IJ) ⌫ abcG abc [IJ]d thefftrace above zfrom ddabcG zz Jbabc[µ⌫]c + (105) G ⌫ [µ⌫]c(IJ) abc ⌫ Jb [µ⌫]c [IJ] abc ⌫ 2ib(IJ) fabc GJb z + 2c d G z + d G z (105) abc abc [IJ] [IJ] ⌫ Jb [µ⌫]c ⌫ Jb [µ⌫]c ⌫ Jb [µ⌫]c tk (IJ) Multiplying Multiplying by and ftabc aking the Jb [µ⌫]c Jb trace Jb [µ⌫]c 2ib G zttrace 2c dabc G z ++ dabc G z (105) Jb [µ⌫]c [µ⌫]c [µ⌫]c [IJ] by and taking the ⌫+ 2ib G ++ G z [µ⌫]c dJbabc G (105) Multiplying by and taking the trace (IJ) fabc [IJ] [IJ] ⌫ ⌫z2ibk(IJ) ⌫ taking ⌫ ⌫z+ [IJ] dabc f2c G[IJ] zand 2c[IJ]the dabc G z GJb (105) Multiplying by t⌫dtkkabc trace abc ⌫ ⌫ z Multiplying by tk and taking the trace 2Jb [µ⌫]e Ja 22[µ⌫]k Ja [µ⌫]k Multiplyingbybytktkand andtaking takingthe thetrace Jbtrace [µ⌫]e Multiplying b(IJ) fabc (idbbeck feck G+ Multiplying by t(id and trace Jb [µ⌫]e feck+ )Gthe z[IJ] + c[IJ] GJa z[µ⌫]k k)G abc eck (IJ) f ⌫ ztaking ⌫ z c ⌫ c (id f )G 2 eck eck (IJ) fabcJb N ⌫ zJa [µ⌫]k N [IJ] G⌫⌫ z N b(IJ) fabc (ideck feck )G⌫ zJb[µ⌫]e + c G z [µ⌫]e 2 2[IJ] 1⌫Jb Jb [µ⌫]e JaJa [µ⌫]k Jb + [µ⌫]e Ja [µ⌫]k N[µ⌫]e 2Ja z11[µ⌫]k b(IJ) fabc (id fc[IJ] )G z)G c+ G z [µ⌫]k eck(d eck [IJ] (106) ⌫z(d ⌫[µ⌫]e (106) Jb [µ⌫]e Ja [µ⌫]k b(IJ) f+abc (id f )G + c G z c[IJ] dabc + if z G+ + d + if )G z GzJa z [µ⌫]k Jb eck eck [IJ] eck eck [IJ] abc eck eck ⌫ ⌫ ⌫ ⌫ c[IJ][IJ] ⌫ N b f (id f )G z + G (106) + c d (d + if )G z + G abc eck eck (IJ) ⌫ ⌫ 1 abc eck eck [IJ] [IJ] N ⌫⌫ z 4948 | P a g e N N ⌫ Ja [µ⌫]k N N Jb [µ⌫]e (106) + c d (d + if )G z + G z 1 abc eck eck [IJ] [IJ] ⌫ ⌫ Jb [µ⌫]e Ja [µ⌫]k 1 1 1 J u n e 2 0 1 7 + c d (d + if )G Jb [µ⌫]e JbG [µ⌫]e [µ⌫]k [µ⌫]e [IJ] (106) N zeckJb+ abc eck eck [IJ]abc 1 if (106) ⌫(d ⌫z z 1 + dif ifeck )G zJa (deck +(d z )G + dabc +)G z+ JbJb [µ⌫]e Ja [µ⌫]k abc [IJ] eck [IJ] eck [IJ] ⌫G ⌫eck ⌫ [µ⌫]e ⌫+ www.cirworld.c+ o1 mc[IJ] + d (106) N c d (d + if )G z + G + d (d if )G z abc eck eck [IJ] [IJ] 2 2 eck ⌫ ⌫ [IJ] abcJb eck ⌫ z [µ⌫]e N N 2 + d (d + if )G z 1[IJ] abc eck eck ⌫ JbJb[µ⌫]e [µ⌫]e 1 2 [IJ] dabc )G⌫z z eck1 + eck [IJ] ++ dabc (d(d ifif )G Jb [µ⌫]e eck eck ⌫ [µ⌫] +[µ⌫]+z[µ⌫] = tr Z [IJ] dabc (deck + ifeck )G⌫ z 2Z2[µ⌫]Lz [µ⌫] (3) L = tr (3) 2 (3) L = tr Z z [µ⌫] [µ⌫]

Takingthe thetrace tracefrom from above equation Taking above equation (104) the trace from above equation Taking theTaking trace from above equation Taking the trace from above equation [µ⌫]c [µ⌫]c [µ⌫]c JbJb[µ⌫]c JbJb[µ⌫]c JbJb[µ⌫]c Jb [µ⌫]c 2ib fabc zJb 2c d[µ⌫]c dzabc zJb 2ib fabc GG +f+2c dabc GG +d+abc dJbabc GG abc (IJ) [IJ] S(105) S(105) z 2 [µ⌫]c 3 4 7 -‐ 3 (105) 4 8 7 ⌫z z [µ⌫]c Jb[IJ] [µ⌫]c [µ⌫]c (IJ) [IJ] [IJ] ⌫ ⌫zG ⌫+ ⌫ ⌫z+ 2ib G[IJ] z 2c G dIabc GNJb abc (IJ) [IJ] ⌫ ⌫ ⌫(105) 2ib(IJ) fabc z + 2c d G z + d G z abc abc [IJ] [IJ] ⌫ ⌫ ⌫ 3 [µ⌫]c N u m b e r 6 Jb [µ⌫]c Jb [µ⌫]c V o l u m e 1Jb 2ib(IJ) fabc G⌫ z + 2c[IJ]Jdoabc G z + v adnabcc G z (105) u r n⌫ a l o f A d[IJ] e s⌫ i n P h y s i c s MultiplyingbyMultiplying bytk tkand andtaking taking the trace Multiplying the trace by t and taking the trace k Multiplying by tk and taking the trace Multiplying by tk and taking the trace Jb[µ⌫]e [µ⌫]e 2 2 [µ⌫]k Jb JaJa[µ⌫]k 2 [µ⌫]e Ja [µ⌫]k b(IJ) fabc (ideckb feck feck )G zJb [µ⌫]e cJb G zJa 2[IJ] b(IJ) fabc (id )G + c[IJ] ⌫zeck [µ⌫]k ⌫ )G ⌫ ⌫z (id f+ )G zG + eck (IJ) fabc ⌫ N b(IJ) fabceck (ideck feck z + c G z N 2 c[IJ] G⌫Jaz [µ⌫]k [IJ] ⌫ ⌫ N Jb [µ⌫]e b(IJ) fabc (ideckJb [µ⌫]e feck )G c[IJ] G⌫ z 1⌫ z Ja+ 1N Ja[µ⌫]k [µ⌫]k Jb [µ⌫]e 1 N (106) Jb [µ⌫]e Ja [µ⌫]k c[IJ] dabc + if )G z + G z 1 (106) ++c[IJ] dabc (d(d + if )G z + G z eck eck [IJ] ⌫ ⌫ eck+ c[IJ] eck [µ⌫]e (106) ⌫ eck Jb ⌫ dif (d )G +Ja z1[µ⌫]k abceck eck [IJ] G⌫ z (106) ⌫ z [IJ] + c[IJ] dabc (deck + )G⌫+zif + [IJ] G NN ⌫ N Jb [µ⌫]e Ja [µ⌫]k N (106) + c d (d + if )G z + G z abc eck eck [IJ] ⌫ ⌫ 11 [µ⌫]e JbJb[µ⌫]e 1+[IJ] N Jb [µ⌫]e dabc + if )G z 1 dabc ++ [IJ] (d(d if )G z eck eck [IJ] ⌫ eck eck [µ⌫]e ⌫ eck Jb +(d1eck[IJ]+ dif (d abceck eck )G⌫ z )G⌫+zif 2 2+ [IJ] dabc [µ⌫]e 2 + 2 [IJ] dabc (deck + ifeck )GJb ⌫ z 2 [µ⌫] [µ⌫] (3)LL==trtrZZ [µ⌫] (3) z zL = [µ⌫] (3) L = [µ⌫] tr(3) Z[µ⌫] z [µ⌫]tr Z[µ⌫] z [µ⌫] (3) L = tr Z [µ⌫] z ⌫K (3)(3) I I [G µJµJ , G =2a2abI(JK) b(JK) G ] I µJ , G⌫K ] LL @(3) GµJ⌫K@]µ G µ= ⌫2aII b,(JK) µ@G ⌫ [G intint=(3) L int Lint =I 2a b @ G [G , G⌫K⌫ [G ] µJ ⌫K I (JK) ⌫µJ ⌫K (3) I Iµ µJ ⌫K I (107) L = 2a b @ G [GµJ ,,G G⌫K }] (107) c[JK] @G {G , G } I ++2a2a @µint {G , G } I µ (JK) µG ⌫ ⌫ I cI[JK] µJ@µ G ⌫K{G (107) c[JK] (107) ⌫ } + 2aI c[JK]+ @⌫ µI2a GI⌫IµJ {G , G I µJ ⌫K ⌫K I 2a µJ c G ⌫K Equations+of motion: (107) + @ G {G , G } +2a2a @ G G I µJ ⌫K @µ G µG⌫ G G I[JK] µ+ [JK] ⌫2aIIG[JK] µJ @⌫K ⌫G +I 2a I [JK] @µ G⌫I G[JK]G µ ⌫I µJ ⌫K ✓ ◆ +I2aI JG[JK] G⌫⌫L G, t ]G [µ⌫] Jb @ [µ⌫] Jb [µ⌫] t 2iga z + 2b G Z [t , t ] µKµ [t (4)(4) 2aI @⌫ z I J µK ⌫L a a b a b (IJ) (IJ) ⌫ ⌫ (4) = iga iga(IJ) b(KL) [G ,G ][G , GI ,]G]⌫L J µK LL ,µG G Equations of tr motion: =0 ⌫b (KL) J , [G µK 3int=(4) ⌫I ][G int Jbµ [µ⌫] Lbint =[G iga , [µ⌫] G⌫L ] ,t } (IJ) µ,2G[IJ] ⌫ ][G Lint = (IJ) iga [G ,{t Ga(KL) ][G ]GJb +2c Z , t Z {t (IJ) (KL) b} + a b [IJ]bG µ ⌫ ⌫ ⌫ (4) I J µK ⌫L I J µK ⌫L I ,iga J ]{G µK , G ✓ ◆ iga c[KL] [G L = [G}µ },J⌫L G G(108) ] I⌫L µK , ⌫L iga c[KL] [G ,µGG ,[G G (IJ) ⌫(IJ) intJb J b(KL) µK [µ⌫] [µ⌫] Jb⌫ ][G [µ⌫] Equations of motion: µ ⌫I ]{G (108) (109) iga cb[KL] ,[tG }](108) of(IJ) motion: (IJ) Equations (108) µ ,, G ⌫ ⌫]{G 2aofI @motion: z t 2iga G z [t , t ] + 2b G Z , t iga c [G , G ]{G G } ⌫Equations a a a b (IJ) (IJ) (IJ) [KL] ⌫ µ ⌫ Equations of motion: motion: I J J µK µK ⌫L⌫L I J µK ⌫L of motion: Iiga tr Equations = 0 3 Equations of (108) Jb [µ⌫] Jb [µ⌫] iga [G , G ]G G c [G , G ]{G , G } I J µK ⌫L iga [G , G ]G G ✓ ◆ (IJ)t [KL] (IJ) [KL] µ+ ⌫ a ,Jb [KL] ✓ ◆ J G⌫[G µK Z {t tbJb }taking ZµµG, ⌫L tb }G[µ⌫] ◆Jb [µ⌫] µ ⌫I2⌫ [IJ] ✓ Equations of+2c motion: iga GJb{t Multiplying by the trace a , [µ⌫] [IJ] G⌫ (IJ) [µ⌫] [µ⌫] (IJ) [KL] kJband ⌫ ]G [µ⌫] Jb] + [µ⌫] iga [G , G ]G [µ⌫] [µ⌫] Equations of motion: (IJ) [KL] ✓ ◆ 2a @ z t 2iga G z [t , t 2b G Z [t , t ] µ ⌫ ✓ ◆ I ⌫ a a b a b (IJ) (IJ) 2a @ z t 2iga G z [t , t ] + 2b G Z [t , t ] ⌫ ⌫µK[t[µ⌫] 2a @ z t 2iga G z [t , t ] + 2b , t ] I Ga⌫ JZb Jb ⌫L ✓ ◆=0 I ⌫ a a b (IJ) (IJ) ⌫ ⌫ I ⌫ a a b a b (IJ) (IJ) [µ⌫] Jb [µ⌫] Jb [µ⌫] Equations of motion: ⌫ [µ⌫] Jb [µ⌫] iga [G Ga[µ⌫] G (IJ) Jb [µ⌫] [µ⌫] 3 ✓ 2aItr 2a @2iga z[µ⌫] taa{t 2iga [t , tt]G ]Jb +Z 2b G Z◆ [t0aa ,, ttbb ]] ◆ Equations (IJ) [KL] [µ⌫] Jb tr o3f trmotion: = 0 =[t @⌫ z3+2c t2a GaJb zb[µ⌫] [t2aG ,G t[IJ] ] zz+ 2b G [tJbaG ,(109) tJb ]Z µ ,[t ⌫ ⌫Jb b{t (IJ) Jb [µ⌫] Jb [µ⌫] aJbIIG bJb ⌫ ⌫b[µ⌫]c (IJ) (IJ) @ z t , ] 2b [µ⌫] [µ⌫] ⌫2iga ⌫+ Z , t } + G Z , t } ⌫ a b ✓ (IJ) (IJ) [µ⌫] [µ⌫] [µ⌫]a Jb [µ⌫]c ⌫ ⌫ a b [IJ] +2c G Z {t , t } + 2 G Z {t , t } tr = ⌫ ⌫ G Z {t , t } + 2 G Z {t , t } tr✓ 3 2a+2c = 0 a b a b [IJ] [IJ] 3 @⌫ z3[IJ] ta2a zZ [t2aG ,G⌫tbJb 2b Z [t⌫aG ,Z tJbb ]Z ⌫Jb ⌫Jb a+ G b ⌫(IJ) aG [IJ] [µ⌫] [µ⌫] [µ⌫] = 00 z2iga ga f[µ⌫] + f,Jb G[µ⌫] ⌫]z+[µ⌫] Jb+2c [µ⌫] Jb [µ⌫] Itr a⌫I @G (IJ) (IJ) ◆ abc abc (IJ) ⌫b+ Jb G {t }Jb +[t 2a[µ⌫] G Z {t , tt[µ⌫] }=[t0a , tb ](110) t[IJ] ,ib t[IJ] ][µ⌫] 2b Z t } + Z t } I⌫@⌫⌫ z+2c a{t[µ⌫] b{t a⌫,, ttzbG b} a(109) b} (IJ) (IJ) [IJ] [IJ] tr 32a @ zby Multiplying and taking the trace ⌫ [µ⌫]tk+2c Jb Jb a ,2iga b a b ⌫ ⌫ [IJ] [IJ] G Z {t + 2 G Z {t , ⌫ a a b Jb [µ⌫] ⌫ (109) =0 2iga(IJ) zJba ,⌫Jb [tb }a ,+ tb ]2+[IJ] 2bG G [ttba}, t[µ⌫] ] 3a I ⌫ tr t+2c b(109) ZG⌫G {t Z2Jb ⌫ Z{t [µ⌫]c a , Jb [IJ] ⌫ d ⌫+ +2c G⌫t[µ⌫]c Z [µ⌫] , tdb(IJ) }[µ⌫] {ta(109) , tb0} (109) tr 3 = 32 +JbcG + {t[IJ] G aJb [IJ] [IJ] G⌫ Z abc32 abc [IJ] [µ⌫] (109) ⌫ }Z+ ⌫ Z 32 +2c G Z {t , t 2 G Z {t , t } aand a[µ⌫]c b [IJ] [IJ]the⌫ trace [µ⌫]a 32taking (109) Multiplying tk and taking the trace ⌫ by tG Multiplying tkbyand taking trace (109) k Jb aI @⌫by zMultiplying zb[µ⌫]c + ib GJb abcthe (IJ) f (IJ) ⌫ Z Multiplying by and taking the trace 32fabc Multiplying by+tkgaand taking the trace (109) Multiplying by ttkk ⌫and taking the trace (110) [µ⌫]c Jb the trace taking the trace Multiplying Multiplying by tk a+ nd taking t[µ⌫]a he G kJband Jb [µ⌫]c Jb [µ⌫]c by and trace [µ⌫]a Jb [µ⌫]c Jb [µ⌫]c k [IJ] dby Z@ + dabc ZG[µ⌫]c [µ⌫]a Jb Jb+ abc @Multiplying ga ftthe ztaking ib fabc Z [µ⌫]c ⌫+ a(IJ) +zG[µ⌫]c ga fib z[µ⌫]c ib fabc GJb Iby ⌫z abc (IJ) ⌫ ⌫ (IJ) aI @⌫acz[IJ] +and ga⌫+ G +G f(IJ) G ZG[µ⌫]c Itaking ⌫fz abc (IJ) [µ⌫]a [µ⌫]c ⌫Jb ⌫JbZ [µ⌫]c abc abc (IJ) Multiplying t trace [µ⌫]a Jb [µ⌫]c Jb ⌫ ⌫ k [µ⌫]a Jb a @ z + ga f G z + ib f G Z [µ⌫]c (110) aI @⌫ z[µ⌫]a + ga f G z + ib f G Z I ⌫ abc abc (110) (IJ) (IJ) ⌫ ⌫(110) abc abc (IJ) (IJ) a @ z + ga f G z + ib f G Z ⌫ ⌫ Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c I ⌫ abc abc (IJ) (IJ) ⌫ ⌫ Jb z[µ⌫]c Jb Z [µ⌫]c [µ⌫]a Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c Jb [µ⌫]c 1 g (110) + c d G Z + d G Z a @ z + ga f G + ib f G (110) + c d G Z + d G Z abc abc I ⌫ abc abc [IJ] [IJ] (IJ) (IJ) [µ⌫]e Jb [µ⌫]e a @ z + ga f G z + ib f G Z (110) ⌫ ⌫ ⌫ ⌫ + c d G Z + d G Z abc abc [IJ] [IJ] I ⌫ abc abc (IJ) (IJ) Jb [µ⌫]c Jb [µ⌫]c ⌫[µ⌫]c ⌫(d abct ⌫and abc [IJ][µ⌫]a ⌫ )G z Jb [µ⌫]c Jb⌫ a[µ⌫]c ⌫ + Jb [µ⌫]c aIG (d if[IJ] @[IJ] fJbG [µ⌫]c ifekc Multiplying taking the trace aek aek ⌫Zz[µ⌫]c abc (IJ) +Jb cZ GzJb GJb Zekc ⌫(110) c[IJ] + d+abc G+ Z kdabc abc abc [IJ] (110) aI @ ⌫ + z by + ga f[IJ] ib+ fabc G ⌫ ⌫ + ddG Z ddabc Z ⌫c[IJ] ⌫2 abc (IJ) (IJ) abc [µ⌫]c Jb [µ⌫]c [IJ] ⌫G ⌫ Z 2 ⌫ ⌫ Jb [µ⌫]c Jb [µ⌫]c + c[IJ] dabc G G +⌫ cZ G⌫ [IJ] Z dabc +⌫ Z [IJ] dabc+ [IJ] dabc G⌫ Z (110) Jb1 [µ⌫]c Jb [µ⌫]c 1 Jb trace [µ⌫]e Ja [µ⌫]k Multiplying by t and taking the trace + c d G Z + d G Z 1 g Multiplying by t and taking the k abc [IJ] Multiplying[IJ] by abc tk and the trace k ⌫ taking ⌫ + b f (id f )G Z + c G [µ⌫]e Jb [µ⌫]e abc eck taking eck (IJ)by [IJ] ⌫ Z ⌫ trace Multiplying t and the (d if @ z + a f (d if )G z by t and taking the trace k I aek aek ⌫ abc ekc ekc (IJ) Multiplying Multiplying by ta a nd t aking t he t race k Multiplying by t and taking the trace ⌫ 2 N (111) 2k1 Multiplying by1tk 1and taking the trace the Multiplying by tkk2and g taking 1 [µ⌫]e gJb trace 1 g [µ⌫]e Jb [µ⌫]e [µ⌫]e Ja [µ⌫]k [µ⌫]e Jb [µ⌫]e [µ⌫]e Jb 1 1 if @⌫ zd[µ⌫]e a[µ⌫]e fabc if )G zif Multiplying byaekif t 1and the trace 1+ (daek if )G(d Z + Gekc [µ⌫]k a@I⌫aek (d @Z + fJa (d )GJb z [µ⌫]e I (d ekc ekc 1 aaek g⌫+ ⌫[IJ] abcif eck eck [IJ] aI (d zctaking + a+Jb fz(IJ) (d ifG )G z Jb aek abc ekc (IJ) g2g⌫ca ⌫ Z [µ⌫]e abc ekc ekc (IJ) [µ⌫]e ⌫N fabckaek (id feck )G + Z [µ⌫]e aeck (daek if +N(d a[IJ] f⌫if (d ifekc )G⌫⌫Jb z [µ⌫]e ⌫ @ 2(d @aek + fabc )G 1if gaekc Iaek aek abcekc ekcJb ekc)G g2@⌫⌫azz(IJ) (IJ)f 2 + 122ab(IJ) 2 a if + (d z ⌫ I (daek 2 ⌫ z [µ⌫]e ⌫ zif [µ⌫]e Jb [µ⌫]e I aek abc ekc (IJ) [µ⌫]e 2 2 2 2 a (d if @ z + a f (d if )G @aek + a(IJ) fabc (d if )G (111) 1 (IJ) 2Jbekc Iaek aek Jb ⌫ abc ekc aek 2if1 ⌫z ekc 1 [µ⌫]e ⌫ z 1 ⌫ z ekc 1 [µ⌫]e Ja [µ⌫]k 1 1 21+aI1(d g [µ⌫]e Ja [µ⌫]k 1 Jb [µ⌫]e Ja [µ⌫]k Jb [µ⌫]e 2 2 2 +fZabc faek feck )G +c ⌫Jb c[IJ] G Z11z[µ⌫]k + (d if Jb Ja + beck fd+ (id f+eck )G + c[IJ] GJa Z [µ⌫]k abc 1 bf(IJ) 1Zif abc eck [IJ] ⌫ [µ⌫]k +a+I (d b(IJ) (id fifeck )G Z G Z if @(id (dekc )G abc eck (IJ) 12⌫21z+ [µ⌫]e abc eck aek ekc (IJ) [µ⌫]e Ja ⌫a ⌫ ⌫N d Z⌫Jb[µ⌫]e + G Jb abcf(d eck eck N + fabc (id feck )G Z + c[IJ] G⌫⌫Ja Z [µ⌫]k ⌫)G ⌫ GZ 1bbeck (id f)G Z [µ⌫]e +[IJ] c[µ⌫]e Z1[µ⌫]k abc eck eck 12 b(IJ) 1[IJ] (IJ)f [IJ]G N 2 2 2+c[IJ] 2 + (id f )G Z + c Z ⌫ ⌫ abc eck [IJ] Jb [µ⌫]e Ja(111) [µ⌫]k (111) ⌫ ⌫ eck (IJ) Jb Ja N (111) ⌫ 2 N + b f (id f )G Z + c G 2 N feck )G + Jb eck eck (IJ) abc [IJ] ⌫ Z 1 b(IJ) fabc (id 212 eck N ⌫1 c[IJ] GJa 1 ⌫ Z[µ⌫]e ⌫ Z (111) 111+ 1 1 Jb [µ⌫]k (111) [µ⌫]e Ja [µ⌫]k (111) N Jb [µ⌫]e Ja [µ⌫]k Jb [µ⌫]e Ja [µ⌫]k 21dcf[IJ] N d(id (d+ + )G Zif +c[IJ] G+Ja 1eck Jb [µ⌫]e +eck c+if dif (d abc eck [IJ]Z 1ZG (111) ⌫ Z + (d )G ZZZ⌫Jb+ ++ ++ c+ b[IJ] fif )G G abc eck )G [IJ] [IJ] GJa 1Z 111 [µ⌫]k Jb [µ⌫]e [µ⌫]k (111) ⌫[IJ] ⌫JaZ [µ⌫]k abc eck eck abc eck (IJ) (µ⌫) [µ⌫]e ⌫⌫ ⌫eck ⌫⌫ Z+ d (d )G Jb [µ⌫]e abc eck eck [IJ] 2 N 1 + c d (d + if )G Z G Z (4) L = tr Z 2 N + c d (d + if )G Z + G Z 1 1 abc eck eck [IJ]dabceck [IJ]G⌫⌫ 222 N N + eck c(µ⌫) (d )G⌫⌫JbZ [µ⌫]e JaZ [µ⌫]k [IJ] abc [IJ] + ⌫+ if ⌫ N[µ⌫]k eck [IJ] Jb [µ⌫]e Ja c[IJ] (deck Z abc eck⌫ + eck )G [IJ] [IJ] G⌫ (111) NZ [IJ] G+ +dif Zif + ⌫Jb Z 12 c[IJ] dabc+(d2122eck eck )G ⌫ N 11+ 1 Jb [µ⌫]e [µ⌫]e N Jb [µ⌫]e Ja [µ⌫]k 2 N + d (d + if )G Z 1 + d (d + if )G Z abc eck eck [IJ] 1 + dabc (d )G Z I⌫Jb + c[IJ] +G eck⌫J [IJ] [IJ] 1+ µ I ⌫ µJ [µ⌫]e ⌫Jb G eck= eck abc [µ⌫]e ⌫ eck ⌫ Z Jb L 2if [(@ G )(@ )+ (@[µ⌫]e +(d dJif (d +Zif if )G Z 21eck + 12 Z[IJ] dabc µ G⌫ )(@ G )] K I+d µ)G (µ⌫) abc eck⌫Jb eck [IJ] JbZ [µ⌫]e N)G + (d eck ⌫+ abc eck eck [IJ] [µ⌫]e ⌫⌫ (4) L = tr22Z + d (d + if )G Z (µ⌫) 2 abc eck eck [IJ] (112) 2 ⌫ + d (d + if )G Z eckµ ⌫I ↵I ⌫ ↵J µ⌫ ↵J 1 2 [IJ] abc +22eck JbG [µ⌫]e 8⇢ g (@ )(@ G ) + 4⇢ ⇢ g g (@ G )(@ G )] J µ⌫ ↵ I K µ⌫ ↵ ↵ += tr[IJ] d(µ⌫) )G Z abcZ(d eck + ifeck(µ⌫) (4)=L Ltr ZZ(µ⌫) ⌫J ⌫ I ⌫ µJ L =(µ⌫) trGZI )(@ (4) L Z2 (4) (µ⌫)µZG (µ⌫)) + (@µ G⌫ )(@ G )] 2(4) [(@ (µ⌫) (µ⌫) I L JZ µtr ⌫ = Z Z (µ⌫) (4) LK==(µ⌫) tr(4) Z (µ⌫) L = tr Z Z (µ⌫) (µ⌫) (4) LZ=(µ⌫) tr Z Z ↵J (112) (4) L =+tr8⇢ Z(µ⌫) µ ⌫I(µ⌫) µ⌫ µJ ↵I ↵J Iµ ↵⌫J ⌫J I+ ⌫J I )(@ ⌫ ↵G µJ )] (@J [(@ G )(@ Gµ[(@ )G + 4⇢(@ ⇢µG gI⌫µ⌫ g+ (@↵GG I= µJ⌫ (@ J2g(µ⌫) µ⌫ IIµ K L = G )(@ G ) G )(@ G )] L 2 )(@ ) )(@ G )] K I µ ⌫ ⌫ G⌫= )(@2I IG Jµ[(@ )+ µ I⌫IµI ⌫⌫ ⌫JµJ K µ (@ (4) L L =Ktr=Z2(µ⌫)I ZJ [(@L ⌫J µJ ⌫Iµ G⌫µµ)(@⌫J µ(3) ⌫J IG ⌫ )] µJ K I⌫+ µG ⌫J G )(@ + (@ G (112)↵J LK = 2 I LL [(@= G⌫J )I⌫⌫I )(@ G )]⌫I⌫ )(@ K I IJJµ µ (112) 224i(g [(@ G )(@ ))))@ + (@ g(@ , ↵I G⌫ G ] )] Jint µG µµ K µ ⌫ I)(@ = [(@ G )(@ G +µ⌫G (@µ⌫ G )(@ Gµ⌫µJ )] J K J µG IGK I⌫I I⌫↵J µ ↵J ↵J(112) ⌫µJ µµ ↵I µ[G K I)(@ J↵J ⌫ ⌫ µ ⌫I µ⌫ ↵I ↵J (112) L = 2 [(@ G )(@ G ) + (@ G )(@ G )] + 8⇢ g (@ G ) + 4⇢ ⇢ g g (@ G )(@ G )] (112) 8⇢ g (@ G )(@ ) + 4⇢ ⇢ g g (@ G )(@ G )] µ I µ⌫ JG +µ ↵ I K µ⌫ ↵ ↵ ⌫ ⌫ J µ⌫ ↵ I K µ⌫ ↵ ↵ + K8⇢J gµ⌫J(@ )(@ G ) + 4⇢ ⇢ g g (@ G )(@ G )] (112) µ ⌫I ↵J µ⌫ ↵I ↵J µ⌫ ↵⌫J ↵ µ⌫ µ 8⇢ ⌫I µ⌫ ↵I ↵J ↵I µ↵J ⌫I I)(@ ↵J ↵J ↵K µ(112) µ↵ ⌫J (@ IK↵ ⌫ g µJ ↵J µ⌫ G ↵I ↵J J ggµ⌫ G + 4⇢ (@I⌫+ G 4⇢ Gµ⌫ )(@ )]@ )(@ µ⌫ II⇢ K µ⌫ ↵ 8⇢ gJ)(@ )(@ )))µIµ⌫ + 4⇢ G G[G )], G↵K ] 8ig gG ,g)] G ] ↵K gµ⌫ G↵↵↵⌫I J gµ⌫ ↵⇢ IG KG ↵I ⇢↵ LK = 2+I 8⇢J [(@ )G +µ↵J (@)⌫I@µ+ GG )(@ +µ)(@ 8⇢ (@ G )(@ G⇢⇢[G + 4⇢ JJ K µ⌫ ↵ K (@ J↵J ⌫I µ⌫ I(@ ↵I g µ G+ JG µ⌫I(@ ↵G Kg8ig µ⌫ ↵ ↵ ⌫↵ + 8⇢ g (@ G )(@ G ) + 4⇢ g g (@ G )(@ G )] J µ⌫ ↵ I K µ⌫ ↵ ↵ (3) (112) K ⌫J I ↵I Lint+=8⇢4i(g gJ JG [G↵µI , ↵K Gµ⌫ ] , G↵J I J µ K ⌫I 8ig I ↵J K))@ gµµ⌫G g⌫µ⌫ ] ↵ G↵J )] + 4⇢ (@ )(@ J gµ⌫ (@ G )(@ ↵ ⇢I ⇢K I ⇢@ K gG µ⌫ g [G ↵↵ ↵K µI µ ⌫I J ↵K K µI ⌫J µI @ G ⌫J [G↵ , G ⇢I (3) ,KGµg⌫J ] 8ig ⇢] [G ] (3) (3) 8ig (113) µI ⌫J J IL KJgµ⌫ ↵ GgJ [G IK J gµ⌫ = I4i(g )@ G ,K G =Kg@4i(g )@ , G⌫J ] I µ KG (3) ⌫ I[G K JK K µG Lint L =int 4i(g , G ] µI ⌫J (3) ⌫K int (3) JL K J4i(g I I KJ)@ µI ⌫J (3) = K µI ⌫ [G K µI ⌫J g )@ G L(3) = 4i(g g )@ G [G , G ] I J K J I K µ µ⌫ ↵K I ↵J L = 4i(g g )@ G [G , G ] L = 4i(g g )@ G ⌫ I int J Kg @ IJ I K µG JJK II ] K IGJ↵K J K K ⌫⌫J µI µIµµ ⌫J⌫⌫⌫J µ ⌫I J µ ↵K⌫I int ↵K ↵K 8ig ⌫J J⇢ [G ↵K J ⇢IJ⇢Iint K Lint8ig =J 8ig 4i(g )@µI [G ,µI G, G ]⌫J ↵ ⇢gµ⌫ g↵8ig [G ] [G 8ig GI⌫I ] [GJJ↵J , G↵K g,[G @µ,,↵G ]⇢J@gµ8ig gµ⌫ ] J IµI K@K µ⌫ ↵↵JG KJ⌫J µ⌫ ⌫I ⌫⌫J ↵ ,@G Jg I⇢ KK µ⌫ G@↵K [G G ]G 8ig gIµ⌫ G@⌫IKµ[G ,JG ]µµG ↵K ⌫I I ⇢K gIµ⌫ KµI ↵K µI J ↵K ↵K µI ,I ⇢ ⌫J ↵K ] ↵K ↵ (3) int K µI ⌫J 8ig ⇢ g @ G [G G ] 8ig ⇢ g @ G [G , G @IJJµ⌫ ,↵↵G 8ig ⇢J gµ⌫ @K G↵K ] ↵↵ K µ⌫ K J[Gµ⌫ µ⌫ g[G @µ⌫ [G I ⇢K gg8ig µ⌫ ↵G K I ↵J Kµ µ⌫ K Lint = 4i(g8ig )@ G [G G ], G III⇢ K µ⌫ ↵ ↵ ,(113) ↵K µI µ GII⌫IJ J ↵K I ⌫J,]↵J I JJ K J J K ↵K I ⌫ µ⌫ ↵K I ↵J [G ,µ⌫G ]↵J↵K ⇢↵JJ ]gµ⌫I @ GJ [G ⇢Kgg8ig , ↵]GG ] 8igI↵K (4) ⇢G@ ⇢↵KG g↵K gG GI ↵J ↵ µ⌫ µK ↵ , G ⌫L ] I2(g µ⌫ ↵@ 8igJ 8ig ⇢I ⇢JK⇢gIµ⌫ @g@↵Jµ⌫ G [G ,[G ↵K ↵K = gJ[G gL I↵⇢,,,IG ,G µIJ J ↵K 8ig ⇢KG g⌫Jµ⌫II]I↵@@,L↵G G [G GK )[G ] µ µ ,⌫IG⌫ ][G int ⇢Ig⇢µ⌫K@L g↵8ig J I⇢ K µ⌫ ↵G ⇢⇢I@ gg↵K JK µ⌫ ↵ Jµ⌫ 8igJ8ig Gg↵K [G ,gµ⌫ Gg[G 8ig ] ] ↵J ]K[G I⇢ I↵ J gµ⌫ @ G [G↵ , G (113) 8igJ ⇢I ⇢K gµ⌫ g @↵ G [G↵ , G ] µI ⌫J K ↵L (113)(113) (113) (114) (113) 8g µ⌫ ↵KJ gLI I K ↵Jgµ⌫ [G , G ][G↵ , G ] 8ig [G↵ , G ] I J µK ⌫L (113) (4)J ⇢I ⇢K gµ⌫ g @↵ G µ⌫ G I][G ↵J , GK ] ↵L Lint = 2(gJ gK I L 4ggJJggLL⇢ I I⇢ K K g )[G 4949 | P a g e g µ , [G ⌫ , G ][G , G ] (113) µ⌫ ↵

↵

J u n e 2 0 1 7 (4) (4) 8g g (4) g [G µI , G⌫J ][GK , GI↵L ] J µK (114) I, G⌫L ⌫L I)[G J , G µK ⌫L =JJ2(g gKJIgLIK )[G ] µK =I µ⌫2(g g, G , GJJ⌫J ][G ] J IIg(4) KK L IL K↵ (4) µK , G⌫L JLg J g⌫IµL][G IJ K ,)[G ⌫ ][G = 2(g gKL L g g G w w w . c i r w o r l d . cLointmL int (4) int L J µK I µ L = 2(g g g g )[GµI ,]G⌫L ][GµK , G⌫L ]

= 2(g 2(g g33 L(4) = 2(g )[G ,G ][G ] ggKK ggJK G⌫L ,G ] L JKg IJ K)[G µ,,G IgL ↵J ↵L JLgint K = Lg µ⌫ JµI IL JJ[G I↵L ⌫ ]K ⌫⌫ ][G int 4g µK ⌫JKK µI ⌫J K µ, G↵L gLJ⇢L ⇢int gIIµ⌫ ,L,IIG ][G ,Iµµ,L,LG µIg[G ⌫J ↵L J8g Ig K (114) Lint = 2(g g g )[G , G ][G ] ↵ ↵ g g G ][G G ] J K L J I K 8g g [G G ][G , G ] L I K µ⌫ ⌫ (114) J L I K µ⌫ ↵ 8g g g [G , G ][G , G ] µI ⌫J K ↵L ↵ J L I K µ⌫ 8gJ gL µI ⌫Jµ⌫ µIJ, G↵L ⌫JµK K⌫L ↵L ] (4) ↵][G I[GK g ][G , G I ,K K)[G (114) ↵ 8g gµ⌫ [G G , G ] g [G , G J L I K µ⌫ ↵ J L I µ⌫ Lint = 2(gJ8g gKJ ggLI LI K8g g , G ][G , G ] ↵ ⌫KI ↵L ↵L I ⌫J ↵J J Lµ⌫ I K µ⌫K ↵J ↵ K ↵L I⇢µI ↵J ↵L (114) G G ]↵L ⇢ ⇢ Kg4g gµ⌫µ⌫ ,gG ][G gg[G ⇢, K gµK µ⌫ µ⌫ II,, G ↵J K ↵L] J[G Lµ⌫ IG µ⌫][G ↵ 4gJ g4g ,[G ,↵[G G, K ]G µ⌫ ↵J]][GK K , G↵L ↵ L ⇢JI ⇢LK gIµ⌫K µ⌫ I↵ ↵J I↵ g][G ↵J ↵ ↵ µI ⌫J K ↵L J L I K µ⌫ 4g g ⇢ ⇢ g [G , G ][G , G ] (114) 33 ↵ ↵ [G ↵,, G ] ↵ 4gJ gL ⇢I ⇢K4g gµ⌫ I⇢K K,gG µ⌫g ][G JJgLL⇢I[G µ⌫

(114) (114)

L = 4i(gI J↵KK µIgJ ⌫J )@µ G⌫ [G , G µ] ⌫I J ↵K 8igJ int [G , G I ]K 8ig ] I ⇢K gµ⌫ @↵ G K I ⇢J gµ⌫ @ G [G↵ , G ↵K 8igJµ⌫ I ⇢K g↵K [GµI , G⌫J ] 8igK I ⇢J gµ⌫ @ µ G⌫I [GJ↵ , G↵K ] µ⌫ @↵ IG ↵J 8igJ ⇢I ⇢K gµ⌫ g @↵ G [G↵ , G ] I S S N 2 3 4 7 -‐ 3 4 8 7 8igJ ⇢I ⇢K gµ⌫ g µ⌫ @↵ G↵K [GI↵ , G↵J ] V o l u m e 1 (113) 3 N u m b e r 6 J o u r n a l o f A d v a n c e s i n P h y s(113) i c s (4)

LEquation gJ gL I K )[GIµ , GJ⌫ ][GµK , G⌫L ] K I L int = 2(gJ g of (4) motion ⌫L Lint0 = 2(gJ gKµI I L⌫J gJKgL I↵LK )[GIµ , GJ⌫ ][GµK , G(114) ] 1 8g g g [G , G ][G , G ] J L I K µ⌫ ↵ Equation of motion (µ⌫) µ (⇢ ) ta[GµI4⇢ @ K Z ↵L ta + Equation of motion (114) 8gJ4gLIµ⌫@⌫I ZK ,KGIJbg⌫J⇢↵L ][G I gµ⌫↵J 0 1] ↵ , G ] A (µ⌫) 4g g ⇢ ⇢ g g [G , G ][G , G @ J L I K µ⌫ Equation of motion (µ⌫) µ (⇢ ) ↵ ↵ 0 1 tr ⇠ = 0 (115) +4i(g g )G Z [t , t ]+ Equation of motion 1 I J J I a b ⌫ 4 @ Z t 4⇢ g @ Z t + µ⌫ I ↵J K ↵L Equation of of motion motion (µ⌫) I ⇢g µg (⇢ Equation 4gtaaJ gL ⇢4⇢ [G)↵taG G ,G ] 1 4 II @⌫⌫ Z0 g⇢ (µ⌫) @ ⇢Z )g µJb ][G I⇢ µ⌫ IK a, + ↵) [t Jb of 0 motion 1 +4i(g ⇢ g Zµ(⇢ Equation oEquation f motion @ A Equation of motion (µ⌫) I J J I ⇢ 0 1(⇢ tr ⇠10 = 0 )at,at+b ] (115) +4i(g g )G Z [t ,)4⇢ t(⇢ (µ⌫) µ (⇢ 1 Jb (µ⌫) I J J I a b ]+ 4 @ Z t g @ Z ⌫ @ A I ⌫ a I ⇢ (µ⌫) µ ) 4 I @4⌫of tagJ ttaI4⇢ g⌫4⇢ tab ]+ +) tta+ tr ⇠1 Equation =0 +4i(g )GI33 ZI@gg⇢Z(⇢ , (⇢ motion ⇢µJb IZ @J0 Zg(µ⌫) @[tµµa)Z Z[t + 0 1 1(115) ⌫Z 4⇢ (µ⌫) IIJ@ ⌫@ a ⇢ Jb I(µ⌫) ⇢Z @ a (µ⌫) (⇢ +4i(g4tr ⇢4It)g G tg)btZ ]aA (µ⌫) µ [t(⇢ )tb ]+ A = 0 µJb (⇢ IZ )4⇢ I ⇢⇠ J+4i(g a ,Jb (115) g )G , @ 4 @ Z 4⇢ g @ + 1 I J J a Jb (µ⌫) @ Z t @ Z t + ⌫ 0 I ⌫ a I ⇢ tr ⇠1tr = 0 (115) )G Z [t , t ]+ +4i(g ⇢ g ⇢ )g G Z [t , t ] @ A I ⌫ a I ⇢ a Jb (µ⌫) I J ⇢I⌫)G a 33 baa,, ttbb]+ I Jtrace Ig A(⇢)= the trTaking =) t00 + 1 (115) +4i(g )G Zt (µ⌫) [t ]+ (µ⌫) µ (⇢ I JJ J 4g ⇠⇠11 @ (115) +4i(g [t ⌫ Z µJb I J I a b ⌫ Jb @ Z 4⇢ g @ Z Jb (µ⌫) @ +4i(g Aa ,= µJb gµJb ⌫ a⇢I))g ⇢]+Z [t @ ⇢IG)G G [ttba]+ tr ⇠1+4i(g 0a, tb ] A =(115) gJ⇢I⇢II)g ZgJ(⇢ [ta⇢a,)),⌫It[t tbb]Z JJ tr 0 (115) +4i(g )G (⇢ I gJJ+4i(g ⇢gIg)g [t J ZI(⇢ µJb ⌫GZ I ⇢⇠J1I⇢ +4i(g ⇢ Jb +4i(g G [t ,, tt(µ⌫) Taking the trace ⌫)aZ A I@ JJ +4i(g JJ⇢ II)g ⇢⇢µ aaµJb bb]] (⇢ (µ⌫)a Jb (µ⌫)c µJb (⇢ ) ) tr ⇠ = 0 (115) g )G Z [t , t ]+ 1 I J J I a b ⌫[taI , tJbZ Taking the trace +4i(g gJ2⇢ ⇢I ⇢I)g@J⇢ ZG(⌫ ]+ gJ[t aI,)f gJ ⇢Z G t babc ] G⌫ Z I @⌫ Z I ⇢J +4i(g I )g ⇢2(g µJb (⇢ ) Taking the µtrace (116) +4i(g gJ ⇢I )g⌫)c⇢ G ZJb [t ⌫)a I ⇢J (µ⌫)a a , tb ] Taking the the trace Taking trace2(g ⌫)a Z 2⇢II⇢@JµZ 2(g gJ I )fabc G⌫JbZ (µ⌫)c (µ⌫)a (µ⌫)c I µJb J + trace (⌫gJ + ⇢ )f G Z Taking the Taking trace II@@⌫⌫the I abc Z 2⇢ @ Z 2(g + g )f G Z (⌫ I I J J I abc ⌫ Taking Taking the trace the trace (⌫ ⌫)a (116) (µ⌫)a µ Jb (µ⌫)c ⌫)a ⌫)a ⌫)cI @ Z (µ⌫)a(µ⌫)a I @⌫ Z µ µJb (µ⌫)c 2⇢ 2(g +ZgJb )f (116) I G J Jb J I (µ⌫)c abc G Jb (µ⌫)c ⌫ Z Taking the trace (⌫+ gJ 2⇢ @ Z 2(g )f (µ⌫)a µµ ZZ ⌫)a ⌫)c 2(g ⇢ + g ⇢ )f G I @⌫IZ I I J I abc µJb @ Z 2⇢ @ 2(g + g )f G Z J J I abc ⌫ (⌫ I(µ⌫)a I ⌫)a J + gJJ II)fabc abcG⌫Jb (⌫ 2⇢ @ Z 2(g Z ⌫ ⌫)a (116) (⌫ ⌫⌫Z I I J µ Jb 2(gIII⇢@@JMultiplying + g ⇢ )f G Z (µ⌫)a µ (µ⌫)c (⌫(⌫2⇢ taking J I @ abc by the trace Z ZµJb +G gJ⌫ ZI )fabc(116) G⌫ (116) Z (µ⌫)c @+tZ g2(g k (⌫and I ⌫2⇢ I @ 2(g J abc (116) I ⌫Z I µJb J +⌫)c J II )f (⌫I ⌫)a ⌫)c 2(g ⇢ g ⇢ )f G Z ⌫)c (µ⌫)a µ Jb (µ⌫)c I J J I abc µJb ⌫)c (⌫ 2(g 2(gI2(g ⇢J I+ µJb (116) (116) ZGabc 2⇢ @ ZµJb Z J Ig)f 2(g +⇢gand ⇢@abctaking )f GZ Z(⌫ I ⌫)c I J + gJ I )fabc G⌫⌫)e (⌫ Z abcG ⇢⇢JJgt+ ⌫)c (⌫ JJI⇢II⌫)f Multiplying the trace µJb (µ⌫)e µ (⌫ k g 2(g ⇢ + g ⇢ )f G Z 2(gIIby ⇢J t+ ⇢ )f G Z (116) I J J I abc Multiplying by and taking the trace (if d )@ Z + ⇢ (if d )@ Z abc k J I aek aek aek aek (⌫ ⌫ µJb (⌫ ⌫)c I (⌫ 2(g ⇢ + g ⇢ )f G Z Multiplying by t and taking the trace ⌫)e I J J I abc k (µ⌫)e µ (⌫d Multiplying tby and taking the trace Jb (µ⌫)e Multiplying by tk aInd tby aking race kthe Multiplying and taking the trace (if )@ Z ⇢⇢II)f (if )@ µZ aekby aek Multiplying tdttkkaek and taking trace (117) + (g⌫⌫Iby g+ (ifthe d)@aek )G (⌫ ⌫⌫)eZ Jthe abc aekdaek )@ ZJ(µ⌫)e + Z Multiplying t+ taking trace I (if aekby tdkaek I (if aek aek Multiplying and taking the trace k and (⌫ ⌫)e (µ⌫)e µ Jb+ (µ⌫)e (µ⌫)e(µ⌫)e µ aekµ⌫)e daek )@ Z )Gthe ⇢)@ (if µ)e Z(117) ⌫)e I (if aek aek )@ µJbd⌫)e (⌫ Multiplying by tZ and taking trace + (g + )f dd⇢⇢⌫aek (if + ⇢II+ (if dZ Z µZ (µ⌫)e k (µ⌫)e I(if Jaek Iaek abc aek I+ aek ⌫Z aek (g ⇢J⌫⌫(if + gaek )f (if )G Z )@ Z (if dIdaek )@ ⌫Jb (⌫ I)@ J + abc aek aek ⌫)e Iaek aek aek dd)@ (if d )@ Z (µ⌫)e (g +dgaek gJJ+ )f (if )G Z ⌫)e (µ µ (117) (⌫ II(if aek I aek (µ⌫)e µ I Jaek I abc aek ⌫ (⌫ (if d )@ Z + ⇢ (if d )@ Z Jb (µ⌫)e I aek aek ⌫ I aek aek (if d )@ Z + ⇢ (if d )@ Z (⌫ ⌫)e I aek aek ⌫J + gJ I )fabc I (if aek aek µ)e µJb (117) Jb (µ⌫)e (⌫ + (g d )G Z I aek aek Jb (µ⌫)e (µ⌫)e ⌫ (µ⌫)e µ)e (g ⇢⇢JJI+ +Jggg+ )f (if )G (117) ++ )f dddaek Z Jb µJb J⇢ abc aek J abc (117) +III(g (g + )f(if (if )G Z d )@ Zdd)G +Z )@ µ Z (µ Iabc abcaek aek aek⌫)G (117) aek aek ⌫aek I (if aek Jb daek (µ⌫)e + )f (if Z +(g (g ⇢IggII(if )f (if )G ⌫Z⇢ J J+ I abc aek aek (⌫ Jb (µ⌫)e JI + IJJ aek aek (µ⌫) ⌫ (117) (gII )fJ abc +(if gJ I )fabcd(if )G⌫µJbZ µ)e aek ⌫ (µ (5) +L(g =I trJ z+(µ⌫) (117) )G Zddaek Jz aek µJb +g)f (gI ⇢(if ⇢Id)fabc)G (if )G Jb Z(µ⌫)e µ)e J + gJaek aek aek µJbµ)e µ)e (µ + (g+ ⇢(g g ⇢ Z µJb J I+ J I abc aek aek +I(µ⌫) (g ⇢ + g ⇢ )f (if d )G Z (117) µ)e + (g + g )f (if d )G Z (µ J J I abc aek aek µJb gJ(g ⇢I )f daek )G Jabc (ifJaekI abc aekµJbZ(µ aekµ)e ⌫ (µ I ⇢J ++ +(if gJ aek ⇢I )fabc (if daek )G Z(µ (5) zz (µ⌫) II ⇢ Jabc aek + (g ⇢ + g ⇢ )f d )G Z I J J aek (5) L L= = tr trzz(µ⌫) (µ (µ⌫) µ)e (gzI(µ⌫) ⇢(4) )GµJb⌫LZ(µ (5)z (µ⌫) L= tr z+ J + gJ ⇢I )fabc (ifaek I J daek µK (µ⌫) (µ⌫) (5) (5) L = tr z (µ⌫) L = b b [G , G ][G , G ] (µ⌫) (5) L L= = tr tr zz(µ⌫) z (µ⌫) [IJ] [KL] (µ⌫) (µ⌫)z µ ⌫ int (µ⌫) L z(4) = tr z(µ⌫) (5) L = tr(5) z(µ⌫) (µ⌫)z I J µK ⌫L µI ↵L (4) = b[IJ] b[KL] [G , G[IJ] +µI2b u[KL] [G]] , G⌫J ][GK µK,gG ⌫L ⌫J][G int ↵ ,G ] (5) LL tr zb[IJ]zb(µ⌫) L= , µ⌫ G [KL] int = (µ⌫) (4)[Gµ , G⌫ ][G I J µK ⌫L ⌫J[G ⌫L K ↵L ⌫J, G K] µI ↵L (4) (4) b[G bv,[KL] G ][G I 2b J µI µK I[IJ] J(KL) µK ⌫L µ ,[G ⌫G int + uuL gg= G ][G ]] ]{G g][G ,↵L GµK (4) µI ⌫J K µ⌫ [IJ] [KL] I[IJ] J LintL b2b b[IJ] [G+ ,G ][G ,µK G ],,⌫L µ⌫ ↵ ,G } (4) [IJ] [KL] L= = b b [G , G ][G G ] II ,,↵ J ⌫L µ ⌫ + 2b [G , G G (4) [KL] = b b [G , G ][G G ] J µK ⌫L µ ⌫ int µ⌫ [IJ] [KL] (4) ↵ ⌫⌫L [KL] µK, G Lbint = b2b b[KL] [G ][G ] µI ,µI ⌫J ][G µI ⌫J, G K [IJ] ⌫J gµ K⌫µI [IJ] [KL] Lint = b[IJ] [G G G ],,↵L ⌫J µK ↵L⌫L µ int +µ⌫ u [G G ][G⌫L ]} [IJ] [KL] µIµI ⌫J ↵L µ[IJ] ⌫µI int 2b µ⌫,K [KL] ⌫J K ↵L ↵ ,G v g [G , G ]{G , G } + 2b c g [G G ]{G ,G ⌫J KG ↵L (4) [IJ] (KL) µI ⌫J K ↵L ++ 2b u g [G , G ][G , ] I J µK µ⌫ [IJ] (KL) ↵ µ⌫ [IJ] [KL] + 2b u g [G , G ][G , G ] µI ⌫J K ↵L ↵ ++ 2b[IJ] vL(KL) g [G , G ]{G , G } µ⌫ [IJ] [KL] µI ⌫J K ↵L 2b[IJ] u g [G , G ][G , G ] = b b [G , G ][G , G ] ↵ µ⌫ ↵ µ⌫ [KL] µI ⌫J K ↵L [IJ] [KL] + u ggµ⌫ [G ][G ]] µ ][G ⌫↵ int µI,,IG [IJ] [KL] + 2b 2b u [G ][G↵↵K,,K,G G ↵L µ⌫ [IJ] [KL] µI µK ⌫L⌫J↵J + 2b[IJ] u[KL] g[G [G ,⌫J G µ⌫ ↵L 2b g⌫J [G ,G G }] µ⌫ µI ⌫J K ↵L ↵,K µ⌫ [IJ] (KL) ↵↵ ,G µI ⌫J ↵L 2b cc(KL) g+ ,v[KL] G ]{G G }}]]{G + u g g [G , G ][G G µI ⌫J µK ⌫L µ⌫u [IJ] (KL) µI K ↵L ++ v g [G , G ]{G , G } µ⌫ [IJ] µI ⌫J K ↵L ↵ µ⌫ [IJ] µI ⌫J K ↵L + 2b v g [G , G ]{G , G } ↵ +2b 2b g [G , G ]{G , G µ⌫ [IJ] (KL) µI ⌫J K ↵L + [IJ] 2b[IJ] v(KL) g [G , G ]{G , G } + u g [G , G ][G , G ] ↵ µ⌫ (KL) µ⌫ µI ⌫J K ↵L (118) 2b v ]{G , G } µ⌫ [IJ] [KL] ↵ ⌫J ]{G µK K ⌫L}↵L ↵↵ [IJ] + 2b ggµ⌫ [G ,,,⌫L G µ⌫ [IJ] (KL) ↵J KµI + 2bu[IJ] v(KL) gµ⌫ [GvvcII(KL) ,G ]{G G } ]{G µ⌫ I↵L ↵J ↵ , ,G +2u 2b [G G ]{G G, G } } µ⌫ µI ⌫J µK ↵ µ⌫ [IJ] (KL) µI ⌫J µK ⌫L u g g [G , G ][G , G ] + g g [G , G µ⌫ ↵J K ↵L µ⌫ [IJ] [KL] µI ⌫J µK ⌫L ++ 2b c g [G , G ]{G , G } µ⌫ [IJ] (KL) µI ⌫J K ↵L ↵ ↵ ↵ ↵ µI, ,G ⌫J ⌫L µ⌫ (KL) 2bu[IJ] [Gcv↵(KL) Gg⌫J ]{G G ++ u+[IJ] g+ ggµ⌫ , ,,GG ][G µ⌫[G (KL) ⌫J]⌫L µK ⌫L 2b cc(KL) ]{G ,, G }} µK [G G ]{G ,,,GG }(118) µ⌫2b [IJ] [KL] [G ,IG ]{G }} µI µK µ⌫ (KL) µ⌫↵µI ↵J ↵L µ⌫ ↵K [IJ] + 2b c(KL) gµ⌫ [G ]{G G µ⌫ [IJ] µ⌫[G I , ↵J ↵J + 2b[IJ] c(KL) gg[IJ] G ]{G ,G G↵L }][G I K,↵G ↵J µK ⌫L (118) + u u g g , G ] µ⌫ [IJ] µ⌫ Ic[KL] K [G ↵L ↵ µ⌫ I ↵J K ↵L + 2u v g [G , G ]{G , } + 2u g [G , G ]{G , G } µ⌫ I ↵J K ↵L µ⌫ [IJ] (KL) µ⌫ I ↵J K ↵L ++u+ u g g [G , G ][G , G ] µ⌫ [IJ] (KL) µI ⌫J µK ⌫L ↵ ↵ µ⌫ I ↵J K ↵L ↵ µ⌫ [IJ] [KL] + u u g g [G , G ][G , G ] (118) ↵ ↵ 2uu[IJ] v g g [G , G ]{G , G } µ⌫ µ⌫ I ↵J K ↵L [IJ] [KL] u g g [G , G ][G ] ↵ ↵ µ⌫ + 2b c g [G , G ]{G } (KL) u u g g [G , G ][G , G ] ↵ ↵ µ⌫ I ↵J K ↵L µ⌫ [IJ] [KL] µ⌫g µ⌫ [IJ]u[KL] (KL) ↵g,µ⌫ ↵↵⌫L [IJ] IG ↵J KG (118) ↵L ↵ + u2u [G , ][G , ] (118) µ⌫ [IJ] [KL] I ↵J µK + u u g g [G G ][G , G ] ↵ ↵ µ⌫ I ↵J µK ⌫L + v g g [G , G ]{G , G } µ⌫ (118) [IJ] [KL] µ⌫ I ↵J K ↵L (118) ↵ ↵ µ⌫ [IJ] (KL) ↵ ↵ ,↵L ↵J{G K ↵L 2u cc(KL) ggµ⌫ [G G ,↵,G }↵J + vg(IJ) g↵J ,G }{G G↵L } Iv,µ⌫ µK (118) µ⌫ [IJ] (KL) ↵L ++ g+ [G ,↵J GggII↵]{G ,IG }↵J µ⌫ ↵ ↵⌫L +[IJ] 2uv[IJ] [G G ]{G G ↵[G +2u 2u [G ,µ⌫ G G µ⌫ ↵J KG ↵L µ⌫ [IJ] (KL) + 2u vv(KL) gg[IJ] g↵guµ⌫ ,,µ⌫ G ]{G G }}K ↵↵J µ⌫ [IJ] (KL) u g]{G [G ,IIK G ][G ,K ] }}(118) 2u v(KL) gµ⌫ gg↵Jµ⌫ ,,,,}G ]{G ,,⌫L G I]{G K ↵L µ⌫ µ⌫ [KL] [IJ] (KL) I[G µK ↵ ↵ ↵ ↵ ↵ + 2u v g [G G ]{G G µ⌫ [IJ] (KL) µ⌫ I ↵J µK ⌫L + 2u v g g [G , G ]{G , G } ↵ ↵ I , ⌫L ↵J]{G µK ⌫L + g↵Jµ⌫}{G [GµK ,G }} µ⌫ [IJ] (KL) I{G µK (118) ↵g ↵G [IJ] I(KL) ↵J ⌫L vv+ gg+ g2u ,, G 2v ccI(KL) {G ,G G }{G ,G µ⌫ I↵J ⌫L µ⌫ (KL) ↵J µ⌫]{G ++ [G ,G ]{G ,II↵µK G },G⌫L (IJ) µK ⌫L µ⌫ ↵J ↵L ↵ ↵µK µ⌫ [IJ] (KL) 2uvvc[IJ] [G G ,II,G G }}}]{G ↵[G +2u g+ ggµ⌫ {G G }{G ,↵J ↵J µK , K ⌫L µ⌫ [IJ] (KL) +(IJ) 2u cc(KL) ]{G , } ↵ µ⌫2u c g [G , G ]{G G } (IJ) (KL) v g [G G , G } ↵,,G I ↵J µK ⌫L µ⌫ [IJ] (KL) µ⌫ (KL) µ⌫ ↵J µK ⌫L ↵ ↵ ↵ ↵ + 2u c g [G , G ]{G , G } µ⌫g µK [IJ] ↵ + 2u[IJ] c(KL) g{G [G ,↵J G , µK G }G ⌫L J µK ⌫L + vµ⌫ v,(KL) {G ,⌫L G }{G ,G } µ⌫ I(KL) ↵J µKG ⌫L µ⌫I]{G ↵ (IJ) Ig ⌫L cc(KL) ggµ⌫ G }{G },}↵J cI↵µ⌫ ,↵J GµK }µK I(KL) ↵J µ⌫ I{G µ⌫ I,I↵µK ++ v(IJ) g+ ggcv2u {G ,G }{G G }, ⌫L I⌫ ,}{G ↵J µK ⌫L ⌫L µg↵J µ⌫ (IJ) (KL) +(IJ) g(IJ) g {G , G }{G G } ↵ +v2v 2v {G , G }{G , G µ⌫ ↵J µK µ⌫ (KL) v g {G , G }{G , G } + vv(IJ) vv(KL) g {G , G }{G , G } ↵ µ⌫ (IJ) (KL) + c g [G , G ]{G , G ↵ µ⌫ I ↵J µK ⌫L (IJ) (KL) µ⌫ µ⌫g {G [IJ] (KL) µK ↵g,µ⌫ + , ↵J G }{G G}⌫L ↵I ↵ µ⌫ (IJ) (KL) + v(IJ) v(KL) gIJJv}{G G }{G G}{G } , G, ⌫L ⌫L + IgvI2v c{G g↵J {G , ↵G }} µ⌫ ↵JµK µK ⌫L , ⌫L µ⌫ (IJ) (KL) ↵ ↵ I µK + c c {G , G , G } µK ⌫L (IJ) (KL) I ↵J µK ⌫L ↵J µK ++2v g+µ⌫µ2v ,cGI↵,, G }{G G }⌫L µ⌫ ,I} I ,↵J ↵J } µKµK (IJ) (KL) 2vccmotion {G G }{G G c+(IJ) {G G↵⌫{G ,IgG µ⌫ (IJ) (KL) {G ,, G }{G , G ⌫L}⌫L + 2v cc(KL) gg,µ⌫ }{G (KL) vµ{G {G G }{G µ⌫ (IJ) Equation ⌫v}{G I gg ↵J µK ⌫L} (KL) J↵ µK ↵ + c(KL) gµ⌫ {G G }{G ↵,, G ⌫L }, G, G} } µ⌫ (IJ) (KL) ↵}{G + of 2v(IJ) {G , G }{G G } +I2v cg(IJ) c {G , G , G µ⌫ (IJ) c(KL) J µK ⌫L (IJ) (KL) ↵ µ ⌫ J µK ⌫L ↵J II , G ⌫L II⌫,}{G µK +✓ c(IJ) c(IJ) {G ,(IJ) GJb G ◆ µK} ⌫L (KL) + cc(IJ) {G G }{G µK ⌫L µ{G (KL) + c2v {G ,,{G G }{G ,,Jb G cc(KL) G ,G+ G}J⌫J⌫I↵⌫L }} µK ⌫ }{G (µ⌫) (µ⌫) cJ⌫(KL) g,,µ⌫ ,G }{G ,G }t } Equation of motion of motion + Equation (KL) J µK ⌫L µ (IJ) µIµ,,c + c c {G }{G G } (IJ) (KL) +4b G z [t t ] 4c G z {t , µ + c c {G G }{G , G } Equation of motion a b a b [IJ] (IJ) ⌫ ⌫ (IJ) (KL) µ ⌫ tr ⇠2 =0 ✓ J µK µJb µJb (⇢ ) Equation of (µ⌫) motion Jb (µ⌫) ✓ of motion + c c G }{G ,}G⌫Lg}⇢ ◆ +4u G g z [t{G , tIµJb + G z◆(⇢ ) {ta , tb } (IJ) (KL) ⇢ aG b,]z [IJ] (IJ]) Equation ⌫ 4u +4b G z [t , t ] + 4c {t , t Equation of motion Jb motion (µ⌫) a b Jb (µ⌫) a b [IJ] (IJ) Equation of ⌫ ⌫ Equation of motion ✓ ◆ +4b G⌫ zmotion [t , t ] + 4c(IJ) G⌫µJbz {t(⇢a ,)tb }Jb (µ⌫) = 0 Equation tr ⇠⇠2✓ [IJ] motion µJb of (⇢ ) a b Jb (µ⌫) (119) ◆{ta◆ trEquation = ✓ of ◆ z4u [taG ,JbtµJb + , tb0} +4u ggJb [t[ta[IJ] ,,ttbG ]] Jb + gg⇢4c zz(IJ) {t ✓ ◆ 2 µJb (⇢+4b )(µ⌫) (⇢ )G b ](µ⌫) ✓ ⇢ zz(µ⌫) b} [IJ] G (IJ]) ⌫+4c ⌫a ,,ztt(µ⌫) Jb Jb (µ⌫) ✓ ◆=0 (µ⌫) Jb +4u G 4u G {t } +4b z motion [t , t ] + G z {t , t } Jb+4b (µ⌫) Jb (µ⌫) tr+4b ⇠[IJ] ⇢G a b ⇢ a b [IJ] (IJ]) ✓ ◆ a b a b (IJ) +4b G z [t , t ] + 4c G z {t , t } Equation of 2 G G z [t , t ] + 4c G z {t , t } Jb (µ⌫) Jb (µ⌫) ⌫ ⌫ µJb (⇢ ) µJb (⇢ ) a b a b [IJ] (IJ) a , tG b ] ⌫Jb a(119) b} , t } (IJ) z) (µ⌫) [taGg, ⌫⌫⇢tb ]z+ 4c[t(IJ) {t ,⌫tbz} g⇢ {t ⌫ +4b ⌫ z4u aG (µ⌫) G tr ⇠2tr ⌫Jb +4u G[IJ] z(⇢= trace a µJb a , t0{t b=a0 (IJ) tr ⇠⇠the 0 b ◆= bG+ [IJ] (IJ]) tr ⇠⇠+4b µJb[IJ] (⇢ (⇢ ) (⇢ 2 z[t tb4u ] (⇢ +(IJ]) z4c {t 2 µJbG (⇢ [IJ] ) t[t]a ,+ µJb )t, bt}} Taking ttaking he trace a ,a⌫)µJb )4c(IJ) )= (119) ⌫ z ⌫µJb trG = 00 g[IJ] ztrace G g z {t µJb (⇢ (⇢ taking the 2✓ ⇢ a ,)µJb bag ⇢ b +4u G g [t , t ] + 4u G g z {t , t } [IJ] +4u G z [t , t ] + 4u G g z {t µJb (⇢ ) µJb (⇢ ) ⇢ b ⇢ a b [IJ] (IJ]) tr ⇠22+4u+4u = ⇢ a b ⇢ a ,,0ttb } [IJ] (IJ]) 34 G g z [t , t ] + 4u G g z {t , t } Jb (µ⌫)[t(IJ]) Jb (µ⌫) ⇢ [IJ] a b ⇢ a b [IJ] µJb (⇢ ) µJb (⇢ ) +4u G g z , t ] + 4u G g z {t } (119) ⇢ a b ⇢ a b (IJ]) [t(IJ]) ] +(µ⌫)c 4cg(IJ) z a , tb{t +4u G gJb z(µ⌫)c [taG, ⌫tb ]z+ 4u G z G {t }(119) a , tbJb a , tb } [IJ] ⇢ +4b (119) (119) tr [IJ] ⇠[IJ] =0 2ib +g 2cz(IJ) d) [tabc Gt ⌫] + z 4u⇢ G⌫Jb 2 fabc G⌫ z µJb (⇢ (µ⌫)c µJb (µ⌫)c (⇢ ) (119) Jb (119) 34 +4u2ib G , g z {t (119) dabc ⌫ z ⇢ (120) a + b 2c(IJ)(IJ]) a , tb } [IJ][IJ] f abc⇢G⌫ z 34 ⌫)c ⌫)c µJb (120) (119) + 2iu[IJ] fabc GµJb34 z(⌫ + 2uµJb d⌫)c z(⌫ ⌫)c abc G (IJ)34 µJb 34 + 2iu[IJ]34 fabc G z34 + 2u d G z abc (IJ) (⌫ (⌫ 34 34 Multiplying by tk and taking trace taking the trace 34 Multiplying by ttkhe and Multiplying by tk and taking the trace 2 (µ⌫)k b[IJ] fabc (idekc + fekc )GJb z (µ⌫)e + Jb c(IJ) GJa (µ⌫)e (µ⌫)k ⌫ z 2 b[IJ] fabc (id⌫ekc + fekc )G z + c(IJ) GJa N⌫ ⌫ z N 4950 | P a g e ekc )G Jb (µ⌫)e Jb (µ⌫)e +c(IJ) dabc (dekc if ⌫ z +c d (d if )G abc ekc ekc J u n e 2 0 1 7 (IJ) ⌫ z (121) 2 (121) ⌫)e µJb µJa 2 ⌫)k w w w . c i r w o r l d . c+u o m f (id z(⌫ + µJb v(IJ)⌫)e G z(⌫ ⌫)k ekc + fekc )G [IJ] abc µJa +u f (id + f )G z + v G z ekc ekc N [IJ] abc (IJ) (⌫ (⌫ N ⌫)e µJb +v(IJ) dabc (dekc ifekc )G z(⌫ ⌫)e +v(IJ) dabc (dekc ifekc )GµJb z(⌫

(120) Jb (µ⌫)cµJb (120) ⌫)c dµJbGJb⌫)c ⌫)c µJb µJb 2ib[IJ] G + z2c+(IJ) zd(µ⌫)c ⌫)c JbG(µ⌫)c Jb G (µ⌫)c abcf abc ⌫ fz+ ⌫ G +f[IJ] 2iu + 2u z abc abc [IJ] (IJ) 2ib z 2c d z 2iu f G z + 2u d G z Jb (µ⌫)c Jb (µ⌫)c (⌫ (⌫ abc G abc (IJ) abc z abcz [IJ] G (IJ) ⌫ ⌫ (⌫ (⌫ (120) 2ib f + 2c d G [IJ] abc (IJ) ⌫)c ⌫ ⌫)c ⌫ µJb abc (120) ⌫)c2u(IJ) dabc G ⌫)c + 2iu GµJb zµJb + zµJb ISSN 4 8 7 abcf (⌫ (⌫ ⌫)c ⌫)c 2 3 4 7 -‐ 3(120) + [IJ] 2iuf[IJ] G z + 2u d G z(⌫d GµJb µJb abc2iu (IJ) abc (⌫ + f G z + 2u z V o l u m e 1 3 N u m b e r 6 abc abc [IJ] (IJ) Multiplying by t and taking the trace (⌫ (⌫ k Multiplying by tk and taking the trace J o u r n a l o f A d v a n c e s i n P h y s i c s Multiplying by tby taking the the trace k and 2 (µ⌫)e Ja2 (µ⌫)k Ja (µ⌫)k Multiplying tk and taking trace the Jbtaking by t and k bMultiplying f (id + f )G z (µ⌫)e +Jb c(IJ) G [IJ] abc bekc (idekc⌫+ fekc )G ztrace +⌫ z c(IJ) G⌫ z [IJ] fabcekc 2 ⌫N N Jb (µ⌫)e Ja 2Jb G b[IJ]bfabcf(idekc + f+ + +cJb z (µ⌫)k Ja2 (µ⌫)e ekc )G⌫ z Jb (µ⌫)e (IJ) ⌫ G Jb(µ⌫)k (µ⌫)e Ja (µ⌫)k (µ⌫)e f )G z c +c d (d if )G z ekc ekc [IJ] abc (id (IJ) N abc ekc ekc (IJ) ⌫ +df ⌫ ⌫z c +cekc (d)G )G z(IJ) ⌫z ifekc+ b[IJ] fabc (id G⌫ z abc ekc⌫N (IJ) ekc (121) N Jb (µ⌫)e (121) 2 +c(IJ) d (d if )G z Jb (µ⌫)e ⌫)e ⌫)k abc ekc ekc µJb ⌫)GµJbz ⌫)e µJa ⌫)k Jb2 z (µ⌫)e µJa +cekc d (d if +u[IJ] fabc+u (id + f )G z + v G abc ekc ekc (IJ)f ⌫ ekc (IJ) +cekc dabc (d)G )G z (IJ) G (121) f(⌫ekc zif z(⌫ ekc (IJ)+ [IJ] abc (id (⌫ekc +⌫ v(⌫ 2 N (121) ⌫)e µJb µJa N⌫)k (121) 2 ⌫)e ⌫)k +u[IJ] f (id + f )G z + v G z µJb µJa 2 ⌫)e µJb abc ekc ekc (IJ) ⌫)e µJb ⌫)k (⌫ (⌫ ⌫)e µJb µJa +u[IJ] fabc+u (id + f )G z + v G z ekc +vekc d (d if )G z (IJ) N f (id + f )G z + v G z abc+vekc ekc ekc (IJ) abc (⌫ (dekc Nif(⌫(⌫ekc )G z(⌫ ekc [IJ] (IJ) (IJ) dabc (⌫ N (⌫ ⌫)e µJb ⌫)e +v(IJ) d (d if )G z µJb abc ekc ekc +v(IJ) dabc+v (dekc if (dekc )G (⌫ if z(⌫ekc )GµJb z ⌫)e (IJ) dabcekc (µ⌫) (⌫ (µ⌫) (6) L = tr Z z (6) (µ⌫) L = tr Z(µ⌫) z (µ⌫) (6) L = tr Z(µ⌫) z (µ⌫) (6) L = tr Z(µ⌫) z tr Z(µ⌫) z (µ⌫) (6) L= (3) I µJ I ] µJ ⌫K Lint = 2 ILb(3) , G⌫K [JK] =@µ2GI⌫b[G ] [JK] @µ G⌫ [G , G int (3) I µJ ↵I ⌫K µJ ⌫K ↵I ] µJ ⌫K (3)2+I b Lint = @µ G [GI↵bG, G ]@ µJ[G ⌫K [JK] g⌫µ⌫ I, G ⌫K [JK] +@ g, µ⌫ ,G Lint = 22⇢ bb(3) [G G ] µJ[G II⌫ b [JK] IIL µ2⇢ [JK] = 2G@↵I ] ] µ G↵ [JK] @ ⌫ [G , G int µJ ⌫K µ[G ⌫I , G J µ ]↵K ⌫I] µJ J ↵K + 2⇢+I b2[JK] g[JK] @g↵µ⌫ ↵I µJ µ⌫+ G ↵I 2G@@I↵Ibu gµ⌫ G⌫K + 2⇢IIbu[JK] gµ⌫ G [G G ][G↵ ,,G µ⌫↵ [JK] + 2⇢ g[G @@,,↵G G [G G⌫K]] [JK] J ↵I↵K J µ⌫[G ↵K µ⌫ ↵IJ] J↵K ↵K + 2 +I u2⇢ gµ⌫+@gµµ⌫ G⌫I ,gGJ [G µ u[JK] @⌫I G[G µ]↵↵K (122) ↵↵G uG g, G [G , G] ] (122) + 2 [JK] [G µ⌫ ↵]G [JK] +gµ⌫ 22⇢@gII u gµ⌫ G@,⌫I IIu[JK] ↵@ [JK] ↵ , G↵ µ⌫ µ ↵I J ↵K ⌫I J ↵K ⌫I]↵K + 2⇢+I u2[JK] gµ⌫+ g 2µ⌫@@↵µ⌫vGG [G ,G J ↵K↵K ,µJG }↵I J↵ , G ↵[G (JK) g↵I G {G } (122) + 2⇢IIuv[JK] @↵ Gg{G ,G (122) µ⌫ (JK) +ggµ⌫ 2⇢gIIu[JK] g↵@µ⌫ ] (122) µ⌫ ↵ G ][G ↵@ ↵, G µ ⌫I J ↵K µ⌫ ↵I J ↵K µ⌫ ↵I J ↵K + 2 +I v2⇢ g @ G {G , G } J{G µ⌫+g2⇢ v(JK) @⌫I G ,⌫I G}{GJ{G } ↵K ↵↵ g, µGG @↵K , G} } ↵ + 2(JK) {G µ⌫↵@ ↵G (JK) +gµ⌫ 2µ⌫@gIIµvG gµ⌫ IIv(JK) (JK) ↵ , G↵ µ⌫ ↵I JµJ ↵K ↵I ⌫K ↵I µJ + 2⇢+I v2⇢ g g @ G {G , G } µ⌫ ↵I J ↵K J ⌫K↵K µ⌫+gµ⌫ (JK) c(JK) G GG }↵I }{G, G 2⇢g@II↵v↵c(JK) gµ⌫ {G } } + 2⇢ @↵ G{G µ⌫↵{G ↵,G (JK) +gµ⌫ g@µ⌫ IIv(JK) ↵@,↵G ↵, G ↵I µJ ⌫K + 2⇢I c(JK) gµ⌫ @↵ G {G }⌫K µJ ⌫K ↵I , G + 2⇢I c(JK)+gµ⌫ G↵I } ,G } 2⇢@I↵cG gµ⌫µJ @↵, G {G (JK){G (4)

I J µK (4) Lint = i(gL GJ⌫⌫L][G ] µK , G⌫L ] I int J b= I b[KL] )[G [KL] i(gI gJJb[KL] gJ µI,bG )[GIµ ,, G ⌫ ][G [KL] (4) I J µK ⌫L I )[G (4) Lint = ggJµ⌫⇢I[G b[KL] , GµK GµK I⌫ ][G ↵J⌫L ],µK (4)i(g 2ig I ] ⌫L J J bJ[KL] ⇢JIbb= , Gg↵Jµ⌫)[G ][G ,[KL] G µ [KL] 2ig ][G , GJ⌫ ⌫L L i(g g[G )[G ][G Lint = Ii(g ] ] µK , G⌫L ] [KL] IJ gJJI b↵ J ↵ [KL] I int I b[KL] [KL] µ, G µI,bG ⌫ ][G I ↵J µK ⌫L ⌫J ,µK K ]↵J ↵L µK µI ⌫J ↵L 2igJ2ig ⇢ b I ug[KL] [G ,[G GI µI ,][G ⌫L ↵J µ⌫ ][G ] K ↵⇢ µ⌫ 2ig [GI↵G G⌫L 2ig ggµ⌫ [G ,G ][G ↵ 2igI JJ[KL] ⇢I b[KL] ggµ⌫ [G , GG ][G ,, G ]][G µ⌫ [KL] JJ IIbu ↵ , G ]] ↵[KL] µI µ⌫ ⌫J I K↵J ↵LK ↵L K K ↵L ↵L ↵J 2igJ2ig u ⇢I u[KL] gµ⌫ [G µI µI[G ][G ⌫J K gI,uG G↵][G ][G ,⌫J G][G ]↵ , ↵G, G] ] 2ig g,µ⌫G [G↵I↵],↵L G 2ig [G ↵g,µ⌫ 2igI JJ [KL] ggµ⌫ ,G ] ][G J ⇢[G Jµ⌫ [KL] I u[KL] ↵ ,G (123) (123) µ⌫ IµI ↵J⌫J K K ↵L↵L µI K K ↵L↵L 2igJ2ig ⇢ u I v(JK) gµ⌫ ggµ⌫⇢[G ,G ][G ,↵G µ⌫ I G⌫J] ↵J µ⌫ IG ↵J K ↵L [G , ]{G , } ↵[KL] ↵ 2ig v g [G , G ]{G , G } 2ig u g g [G , G ][G , G ] 2igI JJ⇢[KL] u g g [G , G ][G ] J I µ⌫ (JK) J I µ⌫ ↵ ↵ I [KL] µ⌫ (123) ↵ ↵↵ (123) µI µ⌫ ⌫J I K ↵LK (123) ↵J ↵L K K ↵L ↵L µI µ⌫ I }⌫J ↵J 2igJ2ig gµ⌫ [G ,vG , G µI[G ]{G ⌫J K, G ↵L Iv (JK) ⇢ v g g , G ]{G , G } ↵ J I µ⌫ (JK) 2ig g [G ]{G , G } 2ig ⇢ g [G , G ]{G , G } ↵ µ⌫]{G↵ , ↵ ↵ 2igJ I v(JK) gJµ⌫ ,G G } ↵ ↵ I (JK) J [G I K ↵L µ⌫↵I ,, G IKJ ]{G KJ⌫L }↵L I ,G 2ig ⇢I vI (JK) gµ⌫ g µ⌫⇢g[G , GI↵J }↵J µ⌫ ↵LµK +J2ig i(g c+ G 2ig g G]{G g↵J [G , G)[G ]{G G µK } , G⌫L } I c(KL) (KL) i(g c↵(KL) g)[G Equation of motion µ (JK) gµ⌫ [G }, G↵⌫,]{G IJ g JIJv J ]{G I c(KL) ↵ J ⇢J I v(JK) ↵ , µ⌫ ↵ ⌫, G µ I J µK ⌫L Equation of motion I )[G ↵J , G I J⌫L +Equation i(gI 2ig c ⇢of ggJIµ⌫⇢IJ[G c(KL) G IIµK µK ⌫L µK J ⌫L µ ⌫,c]{G c+ G },µK motion Equation of motion i(g g[G )[G ,G ]{G (JK) 2ig gµ⌫]{G G, ↵J ]{G , }G } , G⌫L } J ↵ ↵I ,cG + i(gJ IJ(KL) )[G J gJ Icc(KL) (JK) JIc(KL) (KL) µI (KL) ⌫ ]{G µ , G⌫ } ↵J µK ⌫L ↵J 2igJ ⇢I c(JK) gµ⌫ [GJI↵⇢,I G ]{G G } ]{GµK , G⌫L } 1 ↵J [G,IµK 2ig cI↵(JK) ,G Equation of m0 otion (µ⌫)J ⇢I c(JK) µ [G (⇢G)gµ⌫]{G Jb} (µ⌫) Jb (µ⌫) 2ig g , , GG⌫L ↵ µ⌫ 2 I @⌫ z 0 ta 2⇢I @(µ⌫) g⇢ z35 ta +µ2b35 Z [ta , tb ]Jb+ 2c {ta , tbJb }+ (µ⌫) [IJ](⇢ (IJ) G⌫ Z ⌫ ) (µ⌫) 0 2 µI @⌫ z(µ⌫) 0 @ ta (⇢ 2⇢)I[t@µ,gt⇢Jb]z+ 2bµJb GJb Z , tb ] + 2c(IJ) G1⌫JbZ (µ⌫) {t (⇢ (µ⌫) ) Jb[ta(µ⌫) [IJ] (⇢(µ⌫) ) ta +G ⌫ Z (⇢ g tr ⇠3 2 I @⌫ z (µ⌫) ta 2⇢ +2u 2I @ [IJ] zzµJb @ aµJb g35 2b [t +a(⇢,2c ⇢ ⌫Z a g⇢⌫G +2⇢ 2bI[IJ] G +gG 2c Z,at,bt}+ t)b(IJ) }+ G⌫ Z A {taa 35t) ta⇢a Z I@ ⇢b⌫ zZ 2v a[t b ]{t a, t b ][IJ] (IJ) G{t ⌫ µJb (⇢t(IJ) )+ @ tr ⇠3 @ +2u G , t g]⇢+Z2i(g [t , t ] (⇢ +⇢2v GgµJb gz⇢(⇢ Z) [t(⇢ ,)t{t]a , tA 35 Jb µJb b }+ [IJ] [t (IJ) (⇢ µJb µJb (⇢ ) z (µ⌫) ) )G )G tr+2i(g ⇠+2u +2u Z(µ⌫) {t }+ I JG gg J⇢ IZ bg⇢Jb J 2v I{t(IJ) ⇢ g⇢ Z aµJb b a , tb(⇢ tr ⇠3 @ , tbG] aµJb + 2v G )I[t⇢agaJ,⇢tbbZ] g+ ⌫ [t 3 [IJ] a a , tG b }+ [IJ] (IJ) +2i(g gJ I )GJb z(µ⌫) [ta , tb ] + µJb 2i(gI ⇢J(⇢ ) gJ ⇢I )GµJb g⇢ z (⇢ ))[ta , tb ] I J Jb (µ⌫) ⌫ z +2i(g g )G [t , t ] + 2i(g ⇢ g ⇢ )G g z [ta , tb ] +2i(g g )G z [t , t ] + 2i(g ⇢ g ⇢ )G g z [t , t ] I J a bJ I a J bI Ja I b ⇢ =0 I J J I ⇢I J ⌫ I J ⌫ = 0 (124) =0 =0 (124) (124) (124) taking the trace Taking the trace taking the trace trace taking the trace taking the(µ⌫)a (µ⌫)c (gI J (µ⌫)agJ I )fabc GJb I @⌫ z ⌫ z Jb (µ⌫)c (µ⌫)a @⌫zz (gJb gJ I⌫)c )fabcG GJb z(µ⌫)c (µ⌫)a I@ I J(µ⌫)c ⌫ z ⌫)a (g g )f µ µJb @ z (g g )f G z I ⌫ I J J I abc ⌫ I ⌫ ⇢I @ z I J(gI ⇢JJ I gJabc ⌫abc G ⇢ )f z I (⌫ (⌫ ⌫)a ⌫)c µJb ⌫)a ⌫)c ⌫)a ⇢II(µ⌫)c @µµgzz(⌫ (gµJb ⇢JJzJb ⌫)c g(µ⌫)c ⇢II )f )fabc GµJb z(⌫(125) I⇢ J⇢ abcG ⇢ @ (g g z Jb ⇢I @+µ zib(⌫[IJ] fabc(g ⇢ ⇢ )f G I J J J(⌫ abc dabc G (⌫ Z (⌫ GI⌫ Z +I c(IJ) (125) (125) Jb (µ⌫)c (µ⌫)c ⌫ Jb(125) (µ⌫)c Jb Jb (µ⌫)c + ib f G Z + c d G Z Jb (µ⌫)c Jb (µ⌫)c abc abc [IJ] (IJ) ⌫ ⌫ ⌫)c ⌫)c + ib f G Z + c d G Z µJb µJb abc abc [IJ] (IJ) + ib[IJ] f G Z + c d G Z ⌫ abc ⌫ abc G + iu Z(⌫ (IJ)+abc v⌫(IJ)⌫ dabc⌫)c G Z(⌫ [IJ] f ⌫)c µJb µJb GµJb ⌫)c + µJb ⌫)c ⌫)c + iu f G Z v d Z(⌫ ⌫)c µJb + µJb abcG abcG [IJ] (IJ)dabc Z (⌫ Z + v(IJ) abc + iu[IJ] fabc G Z(⌫ iu[IJ] + vf(IJ) dabc GZ(⌫ (⌫ (⌫ Multiplying by tk and taking the trace Multiplying by tk and taking the trace Multiplying by by ttktrace and taking taking the the trace trace k and Multiplying Multiplying 1 by tk and taking the 1 ⌫)e µ d1aek )@⌫ z (µ⌫)e + ⇢ I (if 4951 | P a g e aek d1 aek )@ z (⌫ I (ifaek ⌫)e 1 (if (µ⌫)e ⌫)eaek J u n e 2 0 1 71 2 I(µ⌫)e aek 1 d d aek 2)@ ⌫zz(µ⌫)e )@ +)@1µ⇢⇢zII(if (if daek )@µµzz(⌫ ⌫)e aek)@ (if + d I aek aek ⌫ aek (if d )@ z + ⇢ (if d (⌫ I aek aek 2 I aek aek 2 (⌫ 2⌫ www.cirwo 2 r l d .+c o1 m 2 aek daek )GJb (gI J + gJ 1I )fabc (if z (µ⌫)e ⌫ 1 Jb (µ⌫)e (µ⌫)e 2 1 Jb Jb (µ⌫)e + (g (gII JJ + + gdgJJ II)G )fabc daek abcz(if aek aek )G⌫⌫ z + aek + (gI 1J + gJ I )f (if abc aek aek ⌫ 2 µ)e 2 + (gI ⇢J + gJ ⇢2I )fabc (ifaek daek )GµJb z (µ 11 2 µ)e 1 µJb µ)e

I I J J I abc (⌫ (⌫ (125) (µ⌫)c (µ⌫)c (125) + ib[IJ] fabc GJb +Jbc(IJ) dabc GJb ⌫ Z ⌫ Z (µ⌫)c Jb (µ⌫)c + ib[IJ] fabc G Z + c d G Z abc (IJ) ⌫ ⌫ ⌫)c ⌫)c I S S ⌫)c N 2 3 4 7 -‐ 3 4 8 7 + iu[IJ] fabc GµJb Z(⌫ + v(IJ)⌫)c dabc GµJb Z µJb + iu[IJ] fabc GµJb Z(⌫ + v(IJ) d(⌫abc V oGl u mZe(⌫ 1 3 N u m b e r 6

J o u r n a l o f A d v a n c e s i n P h y s i c s

Multiplying by tk and taking the trace Multiplying by tk and taking the trace 1 1 ⌫)e (µ⌫)e µ + ⇢(µ⌫)e I (ifaek 1 daek )@⌫ z I (ifaek 1 daek )@ z(⌫ ⌫)e 2 daek )@2⌫ z + ⇢I (ifaek daek )@ µ z(⌫ I (ifaek 2 2 1 + (gI J + g1J I )fabc (ifaek daek )GJb z (µ⌫)e (µ⌫)e 2 + (gI J + gJ I )fabc (ifaek⌫ daek )GJb ⌫ z 2 1 µ)e + (gI ⇢J + g1J ⇢I )fabc (ifaek daek )GµJb z(µ µ)e + (gI ⇢J + gJ ⇢I )fabc (ifaek daek )GµJb z(µ 2 2 1 1 Jb (µ⌫)k (µ⌫)e (126) 1 ekc + fekc )GJb + b[IJ] fabc (id +Jb c(µ⌫)e Z (IJ) G⌫ 1 ⌫ Z (µ⌫)k + b[IJ] fabc (idekc + fekc )G⌫ N Z + c(IJ) GJb 2 ⌫ Z 2 N 1 (µ⌫)e 1 ekc ifekc )GJb + c(IJ) dabc (d Z (µ⌫)e + c(IJ) dabc (dekc ⌫ ifekc )GJb 2 ⌫ Z 2 1 1 ⌫)k µJb 1 ekc + fekc )GµJb Z(⌫ ⌫)e + + u[IJ] fabc (id ⌫)e G 1 Z(⌫ ⌫)k µJb v(IJ) + u[IJ] fabc (idekc + fekc )G NZ(⌫ + v(IJ) GµJb Z(⌫ 2 2 N 1 1 ekc ifekc )GµJb Z(⌫ ⌫)e µJb ⌫)e + v(IJ) dabc (d + v(IJ) dabc (dekc ifekc )G Z(⌫ 2 2

(126)

References Acknowledgments References This work was partially supported by the Brazilian funding agency CNPq.

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