Non-abelian whole gauge symmetry

The wholeness principle is analysed for non-abelian gauge symmetry. This principle states that nature acts through grouping. It says that physical laws should be derived from fields associations. At this work, we consider on the possibility of introducting a non-abelian fields set { aI A  } under a common gauge parameter. A Yang-Mills extension is studied. Taking the )

A further step is on how such non-abelian whole symmetry is implemented at ) (N SU gauge group.For this, it is studied on the algebra closure and Jacobi identities, Bianchi identities, Noether theorem, gauge fixing, BRST symmetry, conservation laws, covariance, charges algebra.As result, one notices that it is installed at ) (N SU symmetry independently on the number of involved fields.Given this consistency, Yang-Mills should not more be considered as the unique Lagrangian performed from ) (N SU Introducting the BRST symmetry an invariant eff L is stablished.The BRST charge associated to the N -potential fields system is calculated and its nilpotency property obtained.Others conservations laws involving ghost scale, global charges are evalued showing that this whole symmetry extension preserve the original Yang-Mills algebra.Also the ghost number is conserved.These results imply that Yang-Mills should be understood as a pattern and not as a specific Lagrangian.
Concluding, an extended Lagrangian can be constructed.It is possible to implement a non-abelian whole gauge symmetry based on a fields set { a I

Introduction
Gauge symmetry is guiding physics [1].The physical laws search is being determined by symmetry groups.They carry the lemma where the numbers of gauge fields should be given by the number of generators of a given group.Under this principle Yang-Mills theories have been developed [2,3].Nevertheless it is possible to move beyond to this situation by including an undefined number of potential fields rotating under a same symmetry group [4].Different origins based on Kaluza-Klein [5], supersymmetry [6], fibre bundle [?],  -model [7] have already been studied to consider an initial set of fields transforming under a common gauge group as , .= ' .The index a is an internal indice and run according to the group's choice.
Eq. ( 1) indicates the existence of a non-abelian gauge symmetry involving different potential fields.It introduces the meaning of wholeness through gauge symmetry.Consequently emerges a new concept for physical laws be understood.Its whole symmetry deploys an ab initio for describing a systemic nature.For this, it constructs a fields association under a same gauge parameter.Considering that such fields satisfy the Borscherâ€™s theorem [8], one can redefine them.To get a better transparency on symmetry, one should write the model in terms of the Thus the origin of the potential fields can be treated back to the vielbein, spin-connection and potential fields of higher-dimensional gravity-matter coupled theory spontaneously compactified for an internal space with torsion [5].
Nevertheless by definition, the physical fields are that ones which physical masses are the poles of two-point Green functions.For this, one has to diagonalize the transverse sector by introducting a matrix  [9].The } , basis is not the physical basis.It yields an operation guaranteed by the Borscher's theorem saying that physics must be nondependent under fields reparametrizations [8].Thus, the physical basis } { I G  is obtained rotating as and so, given the  matrix invertible condition The outline of the paper is organized as follows.The methodology is first to expose the new aspect originated from ) (N SU symmetry, and then, understand how through the gauge parameter and group generators the non-abelian symmetry is implemented.So at section 2, the non-abelian fields set symmetry { a I A  } is proposed through an extended Lagrangian with respect to Yang-Mills.In section 3, from internal mechanisms one studies that this extended model is consistent with the symmetry skeleton which antecedes the Lagrangian.In the next two sections, Bianchi and Noether identities are derived.At section 6, the Lagrangian is divided in different pieces according to the scalars produced from generators decomposition.In section 7 one extracts on the classical equations showing about granular and collective space-time evolutions, covariance and with relationships beyond Lie algebra.Conserved currents are explored at section 8.The energy-momentum tensor is expressed at section 9. On symmetries as BRST, ghost scale and global gauge transformations, corresponding charges algebras and ghost number conservation are left for section 10.The corresponding Slavnov-Taylor Identity is written at section 11.Concluding remarks are posted at section 12, saying on the possibility of a systemic physics be described based on the whole symmetry principle.

Non-abelian whole Lagrangian
A non-abelian gauge symmetry association is defined through equations ( 1), ( 3), ( 5), (8).They are showing that the ) (N SU symmetry can be worked out through different fields basis as { is called the constructor basis due to the fact that, under this field-referential, the gauge invariance origin for the Lagrangian terms become more immediate.This is because the field  D works as the usual gauge field and the fields i X  transform covariantly.
The candidate for non-abelian whole Lagrangean will contain granular and collective contributions coming from antisymmetric, symmetric and semi-topological sectors [10].Working out the Lagrangian in constructor basis, one gets ) where  Z is the most general covariant field strength with granular dependence on fields, and  z is associated to collective fields.
Similarly for the collective field strength, Notice that  Z and  z are not necessarity Lie algebra valued as it is  F in the usual Yang-Mills theory.However in order to explore the abundance of gauge scalars that such extended model offers one should also consider all possible group-valued structures in the non-irreducible sector contribution.
Besides that, one can yet to express the gauge fixing term so that the Lagrangian Eq. ( 9) become The transverse diagonalized gauge invariant Lagrangian, which means the physical Lagrangian, is given by where the corresponding field strengths one written in terms of physical fields.Rewriting Eq. ( 11 For Eq. ( 14), , = ) ( ) ( Notice that eqs.( 21), ( 23), ( 26), (28) transform covariantly.

Symmetry skeleton
We consider five preliminary types of fundamental mechanism analysis on the conditions for including more fields.They are based on counting the number of degrees of freedom, geometry, supersymmetry, symmetry and dynamics.This section intends to explore the most subtle, the fourth hability.It is based on the following instructions: algebra closure, BRST algorithm, Bianchi identities, local Noether theorem, covariance.The function of these symmetry topics will be to study whether the

) (N SU
gauge group accomodates the presence of N potential fields rotating under the same group parameters.These arguments purely based on symmetry will work as basis for including an extended Lagrangian to SU(N) Symmetry Group.Gauge fields, being Lie algebra-valued carry group properties.Therefore, a first call of command from symmetry is to verify whether the set of gauge transformations implemented by such general gauge theory is able to build up one algebra.Considering the physical sector, one gets the field transformations and taking two successive gauge transformations, one gets that the algebra of infinitesimal transformations closes: The Jacobi identity of the Lie algebra imposes a next relationship.It is necessary to show that these infinitesimal transformations generate the whole invariance groups.Verify that the Jacobi identity acting on field Eq. (32) and Eq.(33) apply to any tensorial combination.Concluding this first consistency test, one can state that the local properties for the N -potential fields of the classical transformations are summarized by Eq. (30), Eq. (32) and Eq.(33).
The next text includes quantum aspects.It is the BRST algorithm.BRST transformations [11] have been considered a very useful technique to probe the internal structure of a gauge theory.By taking supplementary fields with unphysical statistics it was noticed, initially, as a method to originate the Ward identities and also to compensate the effects due the quantum propagation of zero modes which are contained in a potential field.However it was later understood that the BRST framework also reveals more intrinsic aspects of the theory.Besides solving the gauge dependence of the gauge-fixing term, it brings a perspective where it anticipates the notion of Lagrangian.This means that BRST signature appears at the level of first principle for detecting a full Lagrangian.In this way, as the ghosts and the auxiliary fields are unphysical quanta, one could say that the BRST method works like the X-ray technique for detecting a possible physical illness embedded in the body of the theory.For instance, by computing the cohomology of the BRST charge, one is able to infer about the stability and absence of anomalies in the theory [12].
Considering that the BRST and anti-BRST symmetries [13] penetrate in the symmetry instructions for organizing the most general gauge invariant Lagrangian, our proposal is to use it for testing how for gauge theories will be able to absorb the presence of more potential fields.We are going to follow the Baulieu & Thierry-Mieg prescription [14].There the ghost technical device takes from the very beginning, for predicting the Lagrangian, a set of basic fields ; and in our case it should be complemented by the presence of i X  , 1  N massive vector fields.S e p t e m b e r 2 1 , 2 0 1 5 The inclusion of the auxiliary field b , interpreted as a Lagrangian multiplier for the gauge-fixing condition, promotes the BRST and anti-BRST as fundamental symmetries of gauge theories.The symmetry generators s and s of fields into ghosts become independent of the notion of Lagrangian in the sense that transformations do not depend any more on the gauge-fixing term of the Lagrangian.Writing in terms of the fields, Information contained in Eq. ( 37) can be checked by Eq. (34).They completely determine the properties of an extended gauge symmetry.
The algebraic method will be scheduled in terms of Lorentz invariance, dimension analysis, ghost number, BRST and anti-BRST invariance, hermiticity and global invariance.It builds up the following expansion .However, according to the prescriptions, we discard them because they break the s -invariance of the Lagrangian.However a difference here is that these symmetric tensors can appear conveniently in the theory through the

D
-gauge field but also a function of i X  , vector fields.Then, as BRST invariance is equivalent to the classical gauge invariance, the most general possibility is given by where  Z and  z are written in the previous section.One can indeed check, after some algebra, that really which is the last requirement to establish our final Lagrangian.We have therefore shown that the most general non-abelian Lagrangian satisfying Baulieu and Thierry-Mieg programme is really the effective Lagrangian we have been using from the departure: The b field can be eliminated by using its equation of motion and in the limit of Landau gauge one gets the generating functional for Green functions: The third instruction dictated by the symmetry relies exclusively on algebraic identities, as for instance, the Bianchi identities.Mathematical considerations yield two relationships to be analyzed and explored by each particular theory.These are: Identities Eq. (54) and Eq.(55) will take different forms relative to the structure of the particular theory under consideration, as it becomes evident if we apply them to the cases of general relativity or Yang-Mills.Nevertheless, their implementation in physics is not immediate.In order to transform them into a type of constraint equation, they must first obey a kind of physical closure.This means that Eq. (54) and Eq. ( 55) must be consistent with dimensionality and covariance considerations.Thus a good candidate that gauge theories provide to surpass such a convenience is the covariant derivative.So, from Eq. (54), one gets the following identity the operational identity will be The significant physical question for the Bianchi identities of the extended theory concerns the possible covariant derivatives that can be built up.Since this model provides two basis  36).Now, taking these covariant derivatives in Eq. (56) or Eq.(59), one gets different kinds of Bianchi identities.While the second Jacobi identity is more useful for effective theories, Eq. (54) serves our interest of exploring about the physical fields.Thus taking the physical set, the corresponding covariant derivative is . From eq (54) one gets the most general identity which contains the basic conditions for being proposed as a physical equation.It has the covariant property and correct dimensionality.Then, splitting up the corresponding field strength in symmetric and antisymmetric piece, one gets the following identity:  The attempt in this section is being to identify the existence of instructions in gauge theories for assuming a number of potential fields different from the number of group generators.So as a final aspect for analyzing a possible origin for this extended model is by means of invariance of the action.It leads to Euler-Lagrange equations which will be studied in the following sections and contributions from surface terms.The effort here will be just of introducing more fields at the minimal action principle.It gives, S e p t e m b e r 2 1 , 2 0 1 5 .Eq. (64) shows that while the conservation laws are to be manifested for all the system, the equation of motion appear individualized for each field, separately.Therefore an emphasis from this result is that the different identities which the Noether theorem and total angular momentum gives rise to are conservation laws for all system containing N fields.
The local Noether theorem for a non-abelian gauge involving N potential fields in the same group is understood by the three following equations: Thus, from the analysis of the global and local instructions given by Eq. ( 65) and Eq. ( 66), one gets that there is an explicit information on how symmetry moves a room to accommodate the ia X  fields.It is calculated through Eq. ( 68).On the other hand, Eq. (67) only informs that L is totally antisymmetric.However, implicitly, from dimensional analysis and gauge invariance, it is also possible to guess that there can be In the physical set the local Norther theorem is transposed as It then appears clearly that Coulomb's law does not contain a necessary compromise with a non-dynamical origin.It was just a coincidence for the case involving one field.Observe also that even its strongest condition, Eq. ( 73) does not require for the fields I G  being not dynamical.

Another conservation law concerns to the total angular momentum
and the spin-current To conclude this section, we would note that the so called four types of internal mechanisms work not only to detect the presence of N fields but also to isolate the identity carried by each of them.The first instruction shows, formally, the possibility of more than one field to be transformed under the same group parameter ) (x a  ; from the Baulieu & Thierry-Mieg procedure one gets a method to assume an extended Lagrangian; the existence of different equations associated to each field spots be developed through the Bianchi identities; the minimal action principle brings a conjunction between the whole system involving N fields and the individualization of each quanta through the variational principle.
There the identity of each field is obtained through its correspondent covariant equation of motion, while the system identity is organized through conservation laws.This means that the conservation of energy-momentum, angular momentum and internal charges are instructions only for the system as a whole.
Consequently the symmetry skeleton is able to support more "flash": the presence of more potential fields besides the usual gauge field.The principle that the number of potential fields must be equal to the number of group generators is enlarged.The , one also derives the following expressions

Noether identities
The local Noether theorem provides three relationships

Lagrangian scalars
The potential fields Lagrangian plays with different quanta.From group theory arguments one knows that a quadrivector carries information about different spin states.Neverthless as gauge invariance acts differently one the vector and scalar sectors, one expects that it will work as a source for rendering explicit a different dynamics for each one of those parts.So we should now split the Lagrangean in antisymmetric and simmetric parts rewrite Eq. ( 9) as A new aspect in this whole gauge model is that fields strength are not just Lie algebra valued.They can be decomposed through groups terms

Field Equations
The on-shell informations also will be depending on this generators expansions.It gives for Taking the trace in the above equation, one gets Multiplying the equation of motion by k t and taking again the corresponding trace, we have The corresponding equations of motion at physical basis are  Eq.( 101) contains three features.First, it is covariant which proves that the introduction of this extended symmetry is consistent.Notice that it not only show on covariance but also on the presence of a conserved current when Eq. (7.6) is not considered The charge associated to this current as the same symmetry boundary condition as in the usual QCD [15].

Directive and Circumstantial Symmetries
The whole physics introduces the meaning of an integral organization including two kinds of symmetries.They are the directive and the circumstantial symmetry.Their qualitative difference is that while the director symmetry appears as a natural instruction from the gauge parameter, the circumstantial symmetry will be depending on relationships between the so-called free coefficients studied at Apendice B.
From these two types of symmetries one derives currents conservations.Associated to the gauge parameter one gets the Slavnov-Taylor identity (off-shell) and the Noether identity (on-shell) which yield one conserved current with N-fields contributions Eq. ( 9.3) provides the conservation law

Charges Algebra
Although the gauge fixing term breaks the gauge it is possible to show that there is a symmetry that is preserved in the Lagrangian which is the BRST symmetry.Considering the group parameter as   a a gc = where a  is a bose quantity, a c a fermi quantity and  some anticommuting global quantity, we will derive the BRST invariance.For convenience it will be studied at constructor basis.where the corresponding terms GI L is defined at Eq. ( 9), the covariants derivatives with to this extended model are The fundamental object in a gauge theory is not the Lagrangian but the functional generator ot the Green's functions.It is given by S e p t e m b e r 2 1 , 2 0 1 5

Considering the general Lagrangian
For simplicity, we are going to separete in antisymmetric and symmetric parts and the relationships working as the "ghost-scale" generators of the fields operators transformations Concluding, we obtain that the charges algebra is the same as in QCD: Eq. ( 155) is showing that the charges algebra depends only on the symmetry involved.It does not depend on the number of potential fields being considered at the fields set.
where the total Lagrangian in terms of fields and sources is and by substituting the BRST transformations, one gets in which it is easy to show that the Jacobian of those transformations is unity.Those expression is written in terms of derivatives of the generator functional in relation to sources We convert this differential equation into an expression in terms of the one particle irreducible 1PI generating functional  , then we use the Legendre transformation by using W and  as function of the sources ) , , , , ( Hence the expression (11.6) becomes For simplify the form of this expression, the generator functional Z has the following dependence in terms of The equation ( 169) is Slavnov-Taylor identity for extended ) (N SU .It will give us the important relations between Green functions of the massless, massive gluons and Faddeev-Popov ghosts that imply into the renormalizability of the model.The question of full renormalizability will not demonstrated here cause it is necessary a detailed analysis on Slavnov-Taylor identities and redefinitions of the parameters into the Lagrangian.It will be dedicated in a next paper.

Conclusion
The effort in this work is to implement the whole gauge principle at non-abelian level.Gauge symmetry depends on two variables which are the gauge parameters and group generators.They define Lie algebra valued fields transforming under gauge symmetry.The purpose is to show that these two variables, a  and a t , also work to accommodate the gauge symmetry for a fields set transformation as Eq. ( 1).Consider on the possibility of an antireductionist physics where N -non-abelian fields act together.Given the ) (N SU symmetry drive new association features which go further than Yang-Mills understanding.This means to preserve the symmetry pattern and introduce a new Lagrangian.
Eight aspects attached to group generators and gauge parameters were analysed in order to express the consistency of introduction of this extended gauge model.From group generators: algebra closure and Jacobi identities, Bianchi identities; from gauge parameters: Noether theorem, gauge fixing, BRST symmetry, global transformations (BRST, ghost scale, gauge global); charges algebra, covariant equations of motion plus PoincarÃ© lemma from both symmetry variables.And so, they are showing that ) (N SU gauge group acts as an operator where it does not matter the number of fields involved on its transformations.Consequently, given a certain

) (N SU
gauge group it is possible to derive a Lagrangian where the number of potential fields is not necessarily equal to the number of group generators as ruled by Yang-Mills theory. Eq. ( 1) introduces that symmetry should be treated as an environment.A fields association physics appears.In a further work we will analyse on more details other classical aspects, renormalizability, unitarity.For instance, study on its consequences on the Slavnov-Taylor identity.And so, understand on possibilities for a systemic physical process be described through this non-abelian whole gauge principle.Complexity should be an achievement related through a gauge totality principle.

Group relationships
Gauge theory considers fields as Lie algebra valued.So one should express under the corresponding group generators properties where Eq. ( 172) does not belong to the algebra.And with the following traces properties where where only the first three meshes in Eq. (94) contributes to , ia T  .They are  , ) ( From mesh 5 to 8 there is no more propagating terms.Eq. (177) will receive just sources.Thus from we get calculations which will not contribute to the scope of this work.For instance, mesh (5) gives Consequently, the above Eq.( 187) shows how others sources from Eq. (186) will not interfere on the covariant property of i X  equations of motions due to the fact that they depend only on i X  -fields whose transform covariantly.

Conserved currents
Classically, in order to avoid undesired degrees of freedom we should relate them to conserved currents.For this every field in this whole model must be associated to a corresponding conserved current.Noether and Slavnov-Taylor identities already inform on the existence of only one natural conservation law.In this apendice one explores the conserved currents through the circumstantial symmetry.

. 4 B
Thus the field  D works as the usual gauge field and the fields i X  as a kind of vctor-matter fields transforming in the adjoint representation.Geometrically, the potential fields i X  can be originated from the torsion tensor of the higher-dimensional manifold that spontaneously compactify to is the Minkowski space-time and k B some k -dimensional internal space.

.
The presence of different coupling constants means on the possibility for coupling with different currents.

g
the metric tensor of Minkowski space and ISSN 2347-3487 2837 | P a g e S e p t e m b e r 2 1 , 2 0 1 5

3 are 2 K 3 ,
polynomial functions on all fields satisfying the above conditions.The operators s and s obey the following nilpotency relations to the closure of the classical algebra and to the Jacoby identity.Now, the next step is to prove that this full extended BRST invariance Eq. (34) leads to the most general non-abelian gauge independent physics.From the fact that Q L has dimension four and zero ghost number, one immediately extends the result for Yang-Mills theory by one explores a combination of four independent monomials, three present in the discussion of Yang-Mills theory , c b (42) S e p t e m b e r 2 eliminate Eq. (43) -Eq.(45) from the game since they are of the type 2 K s .So the result here is the same as in ordinary Yang-Mills theories to emphasize here that for most of the classical Lie groups, but not these cases one must still consider others candidates to ext K

3 , 3 . 3 K 1 L
one proceeds in the same way as for ext K The result is again as in Yang-Mills theory , Fadeev-Popov and the gauge-fixing terms are reproduced in the extended Yang-Mills approach.Nevertheless a main difference lies on the fact that is not only a function of the  the external source associated to  D field and i J to the i X  vector field and

G
 one should take them both as a laboratory to grow the covariant derivatives.From the first set, one gets two types of covariant derivatives: where 62) means that this extended model contains N Bianchi identities, where each one is associated to a corresponding physical field.A similar result one gets from Eq. (59) for effective cases.

iXF
 fields.Eq. (67) contains indications for their presence through a coupling with the genuine gauge field  D .It can be made through mixed propagators and interacting terms.For instance, Eq. (11) plus Eq. (17) satisfy Eq. (67).The inclusion of more potential fields should rather be characterized as an extension of the usual case.Therefore our preference in writing the Noether equations in terms of the set is easy to get the boundary conditions by turning off the i X  fields.From this basis, we will analyze three pieces of information from Noether theorem.First it is to reobtain the old result where symmetry current derived from inhomogeneous  D field will play a dual role.Its expression obtained from Noether theorem coincides with the relationship which will be performed for the corresponding  D -equation.Another consistency test is from Eq. (66), or taking its divergence.Then, the proposed Lagrangian must verify the equality between the left-hand side and right-hand side.The third information that Noether theorem provides should not be understood as a conservation law but as a constraint of the theory.Substituting the weaken condition Eq. (67) in Eq. -symmetric tensor depending on  D and the i X  fields. J in our case is essentially made of the "matter" of i X  fields we have put in the game of the extended model.Thus the axiomatic approach to defining gauge theories as the theories where the equation , should be obtained as a symmetry constraint is enlarged ( a  is the QCD field strength).Eq. (69) reexamines this reflex S e p t e m b e r 2 1 , 2 0 1 5 between symmetry and Coulomb's law.

.
It contains the orbital angular momentum plus the spin contribution.It gives group allows to introduce different fields rotating under the same symmetry and associated with different symmetry weights 1   I , and coupling constants I g .However it is still necessary to ascertain a fifth consistency of the above skeleton for assuming more fields.It is to study on the covariance properties of the equations of motion.It will be considered in the subsequent sections 4 Bianchi identities Considering the covariant derivatives Eq. (35), Eq. (36) and the collective expression one gets an expansion where each term transforms covariantly.It yields a Lagrangian whole expansion which englobes the usual Yang-Mills sector and the whole extension.Defining the field strength one expands the symmetric field strength 88)considering the antisymmetric sector, for the symmetric sector, one obtains a 8 meshes decomposition.Given where abc f are the anti-symmetric structure constants and abc d are the components of the completely symmetric invariant 3  rank tensor of the group.
2347-3487 2847 | P a g e S e p t e m b e r 2 1 , 2 0 1 5

5
100) S e p t e m b e r 2 1 , 2 0 1 In order to understand more specifically the model one should also express the equations of motion through Eq. (88) sectors.Considering first the  D -field equation of motion, one gets that Eq. (7.7) depends only on ie X fields, one gets that through this model

DX 5
Second, deriving the Noether theorem expression Eq. (68), one gets -field equation of motion without the right hand side a D S j  , .Third, due to the PoincarÃ© lemma, one derives the expression , is conserved covariantly.Its relationship with Noether current is . fields, we get the following covariant equations of motion which corresponding expressions are in Appendix B.Considering that the main proposal at this section is to show that the introduction of a fields set in the ) (N SU gauge symmetry preserves covariance it will be not necessary to calculate the physical fields equations of motions.S e p t e m b e r 2 1 , 2 0 1Given that the minimal action expressions between two generic fields reference system that its corresponding equations can be related to the 111) So eqs.(111) generate that the explicit covariance obtained through eqs.(7.4) and (7.11) are preserved at


current is explicitly derived at Appendix C. Considering that Eq. (8.3) coincides with Noether identity, ) (D J a  conservation is a directive.It takes obligatory one degree of freedom from a D  field.Similarly for ia X  fields, one gets at Appendix C. Consequently the classical decoupling of the longitudinal sector gauge fixing term at Eq. (19) can be rewritten in terms of the scalar auxiliary field is invariant under BRST transformation.Considering that the part involving eff L was already proved, we have now to demonstrate on the invariance of the measure that it is a constant that does not depend on fields and that can be absorved by a functional constant.In fact by introducing the fields set model contains the BRST symmetry the Noether theorem leads to the conserved current BRST J  .
following expression for the BRST charge

.
is with respect to the scale global symmetry for ghosts be studied corresponds to the global gauge transformation.The corresponding infinitesimal transformations are . (150),one gets that they are separatedely invariants.Thus one derives the following Noether conserved current

Finally, in order
to close this section we are going to calculate the ghost number operator.It is defined as Another ingredient on this non-abelian extension is to consider the Slavnov-Taylor identity.Now we perform those BRST transformation on generator functional to obtain the Slavnov-Taylor identities for the extended symmetry ) (N SU .It is convenient to define the generator functional in terms of sources for fermions and bosons The three last terms of Eq. (160) have been introduced of a way that the total Lagrangian remains invariant by BRST transformations in accord with the nilpotent relations.The others sources anticommuting too.Now the invariance of the generator functional under BRST symmetry implies that get the functional differential equation equations of motionEq.(7.11) corresponds to ( N -1) covariant equations of motions related to i X  -fields.At this Appendix it will be expressed the correspondent currents the Lagrangian antisymmetric sector, eq (89).It gives

J
 are calculed from the 8 meshes that build up S L according to Eq. (94).It gives 190) S e p t e m b e r 2 1 , 2 0 1 5 Eq.(C.3) can be rewritten as the following expansion 191) Notice that eqs.(3.24) and (3.27) coincide.Similarly for  ia X fields equations of motion, eqs.(7.2) and (8.4), are the following  ia J currents expression: free coefficients expressions one can decouple the longitudinal sector.Given the model symmetry circumstance, one getsThe volume of circumstances measure the number of invariant terms in the Lagrangian.It is an interesting property that fields association physics can offer.It relates the free coefficients associated to scalar terms as 2 .Physically these free coefficients can take any value without violating gauge symmetry.As an example, we are going to the case possible to rewrite some of these structures in more elementary terms : this, we have, in general, that the total volume of circumstance of the Lagrangian is