Cent. Eur. J. Phys. • 7(4) • 2009 • 711-720 DOI: 10.2478/s11534-009-0041-9

VERS ITA

Central European Journal of Physics

The Yang-Mills fields - from the gauge theory to the mechanical model

Research Article

Radu Constantinescu*, Carmen Ionescu

Dept. of Theoretical Physics, University of Craiova, 13 A. I. CuzaStr., Craiova 200585, Romania

Received 15 November 2008; accepted 24 February 2009

Abstract: The paper presents some mechanical models of gauge theories, i.e. gauge fields transposed in a space

with a finite number of degree of freedom. The main focus is on how a global symmetry as the BRST one could be transferred in this context. The mechanical Yang-Mills model modified by taking the ghost type variables into account will be considered as an example of nonlinear dynamical systems.

PACS (2008): 05.45.-a; 11.10.Ef

Keywords: extended BRST symmetry • Yang-Mills models • nonlinear dynamics

© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

The BRST approach [1, 2] offers an appropriate technique In obtaining a covariant quantum description of the gauge theories, both for abelian and non-abelian models. Many interesting models such as the superparticle, the bosonic string, the Yang-Mills or the Freedmann-Townsend theories have been successfully investigated by using this approach. Its starting point is given by the original action of the considered model, action which in the BRST approach has to be extended with ghost type fields in order to generate correct equations of motion. Customarily, the equations for the real fields only are taken into account. Ghosts are considered as having no physical significance, in the sense that they disappear in the asymptotic states. The idea of this paper is to abandon this attitude and to consider the whole system of equations of motion, both

*E-mail: rconsta@central.ucv.ro

for real and for ghost variables. We find It normal to consider all these equations and to investigate the existence of solutions for this full set of equations. Coming back to the BRST description of the gauge theories, we have to mention that the initial standard approach has been developed towards extended formalisms. Mainly, these extensions were formulated because a nonminimal sector, without a rigorous structure and justification, appears when the standard formalism is developed [3]. By passing to symmetries as sp(2) [4] or sp(3) [5] ones, the non-minimal sector appears in a natural way and the gauge fixing procedure becomes simpler. It had be noticed that, in the most general context, the BRST symmetry can be represented as a sum of anticommuting differential operators [6]:

s = si + S2 +-----h sn; sasb + sbsa = 0. (1)

The standard symmetry is given by si only. The sp(2) formalism generates the first two pieces, si and s2, of

Springer

the generalized sp(n) representation. The sp(3) Is the next step ahead and it offers many possibilities for the gauge fixing procedure. On the other hand, because in the sp(3) case the whole spectrum of the variables is larger, an interesting nonlinear mechanical model is generated when the gauge field theory is transposed in a space with a finite number of degrees of freedom.

In this paper, the sp(3) Hamiltonian description of theYang-Mills theory is presented as a concrete example. The Yang-Mills theory is important by itself, as a toy-model for testing the main properties of the gauge theories, but also because, when the fields are expressed in terms of color factors, it generates interesting examples of nonlinear mechanical models [7]. These nonlinear models are helpful, because the chaotic behavior, which can appear for such systems, seems to play an important role in explaining quark confinement, chiral symmetry breaking and other important elementary particle phenomena [8], [9]. Our aim is to show, as pedagogically as possible, how, starting from the extended Yang-Mills action in the sp(3) frame, one can derive an extended Yang-Mills mechanical model. In this model the global BRST invariance of the fields leads to an external color charge, which allows to express the fields in a suitable form, in terms of some "color" factors. Two different types of such factors are necessary, one type for the real fields and one for the ghosts. Moreover, the ghosts are expressed in a form which takes into account their Grassmann parity and allows transformation of the supersymmetric Yang-Mills theory to a classical system, with bosonic functions only.

The following main results are reported here: (i) the whole set of equations of evolution of the model, both for real and ghost fields are written down for the first time;(ii) by transferring the fields in a mechanical context, an extended Yang-Mills mechanical model is proposed; (iii) a numerical investigation for a particular case of the proposed model is done in order to illustrate how the variables corresponding to the real and the ghost fields are connected.

The paper is structured as follows: after the preliminary considerations, the next two sections make up a brief intraduction to the standard and sp(3) BRST theory, respectively. In the fourth section, we apply these general results to the case of the Yang-Mills theory, seen as a field theory. The next section of the paper makes the transition to the extended mechanical Yang-Mills model in 3 + 1 dimensional Minkowski space, providing the whole set of equations that has to be considered in order to obtain a coherent description. In the sixth section, a particular 4-dimensional case of this model is investigated by computational methods and concrete results concerning the dynamics of the system are presented. Some concluding remarks will end the paper.

2. Basic facts on the standard BRST theory

The standard BRST symmetry designates a global symmetry which, as Becchi, Rouet, Storra and later Tyutin showed, can be attached to any physical system and can embed any other local symmetry the system could have. Denominations like "standard" symmetry have the role of distinguishing between the symmetry as it was pointed out at the beginning and later extensions. In the next section, we will discuss one of these extensions, namely the sp(3) symmetry. It will be in fact the working frame of our approach.

The special importance of the BRST symmetry is connected with the possibility of achieving a coherent quantum description of the singular systems (gauge theories or constrained dynamical systems). The idea issued from papers of Fadeev and Popov [10] who have used for the first time "unphysical" quantities in the Lagrangian description of the Yang-Mills fields. Their approach was further developed by t'Hooft [11, 12], Feynman [13] and de Witt [14]. These quantities were called ghost fields and they do not appear in the asymptotic states of quantum field theories. In this way their dynamics is not usually taken into account. As already mentioned, we shall consider their dynamics, too.

The starting point of the BRST theory is the action So which describe the singular system. It has to be replaced by an extended action, S = S0 + • • •, which will contain the local symmetries of S0 but will generate well-defined path integrals and Green functions. Moreover, S will be invariant with respect to the action of a differential operator, s, called BFV-BRST operator or BFV-BRST symmetry. Any other observable F of the system will meet the same invariance condition:

sS = 0; sF = 0 (2)

The price we have to pay is given by the extension of the space in which the new action is defined. The extended space is generated by the initial physical coordinates but also by new ghost type generators. As in this paper we shall mainly use a Hamiltonian approach, we shall refer to the extended space as extended phase space. An important requirement for s is its nilpotency:

s2 = 0 (3)

This requirement express a feature of differential operator for s. A very useful representation of s, ensuring a

symplectic structure for the extended space and of Implementing the symmetry s by means of a canonical transformation:

s* = [*, Q],

In the previous equation Q is called the BFV-BRST charge and the bracket is a generalized Poisson bracket written in terms of the whole set of canonical variables (real and ghost types). The nilpotency of s combined with the Jacobi identity leads to the master equation, the main equation which allows to effectively obtain Q:

[Q, Q] = 0, (4)

Together with Q, another important observable in the Hamiltonian approach is the Hamiltonian itself. As the action has to be "extended" in order to become a BRST invariant function:

H0 ^ H = H0 + ••• sH = [H, Q] = 0,

The extended Hamiltonian H and the extended action S still contain unphysical degrees of freedom coming from the gauge symmetry H0 and S0. They have to be killed by imposing adequate gauge fixing conditions. As a conclusion, the implementation of the BRST symmetry imposes the following algorithm: (i) the construction of an adequate extended phase space where the ghost type variables are added to the real ones;(ii) the construction of the BFV-BRST charges and the extended Hamiltonian; (iii) the gauge fixing procedure.

3. The sp(3) BRST Hamiltonian theory

We will consider a gauge theory which at Hamiltonian level is represented by a constrained dynamical system described by the set of irreducible constraints {Ga (ql,pt),a = 1, ••• ,m, i = 1, ••• , n} and by the canonical Hamiltonian H0(qi,pi). The Grassmann parities of the constraints and of the Hamiltonian are e(Ga) = ea, e(H0) = 0. The gauge algebra has the form

[Ga,G, ] = fyapGy, [H0,Ga ]= VZGp (5)

where the structure functions fyap and Va can depend in general on the qi and pi.

For this theory we will develop the sp(3) BRST Hamiltonian approach [5]. Hence, we will pass from the original

gauge symmetry to a global symmetry, sp(3) BRST symmetry

ST = S-| + S2 + S3, (6)

So = 5a + da + ■■■ , a = 1,2,3 (7)

sasb + SbSa = 0, a, b = 1,2, 3. (8)

The main steps which need to be followed are: (i) the construction of the extended phase space adequate for implementation of the Sp(3) BRST symmetry (6)-(8);(ii) the construction of the BFV -BRST charges and the extended Hamiltonian; (iii) the gauge fixing procedure. The extended phase space will be generate by the intraduction, for each constraint Ga,a = 1, ■■■ ,m, of the three pairs of canonical conjugate ghost variables {Paa, Qaa, a = 1, 2, 3} with e(Paa) = e(Qaa) = £a +1 and

5aPab = 5abCa (9)

daQab = eadc5dbXac + 1 faayQ^Qy. (10)

In order to secure the crucial property of the Koszul differentials, 5a, a = 1,2, 3 namely the acyclicity of the positive resolution numbers, it is necessary to introduce the new generators, naa with e(naa) = ea and their conjugate Xaa with e(Aaa) = e(naa) = ea so that

5anab = £abcPac (11)

daXab = 5bna + 1 fayAabQyc5ca

+ 112 flfyeFSbCdQacQadQae5a. (12)

The same property, the acyclicity of 5a, imposes the introduction of new generators, na, and their conjugate na with e(na) = e(na) = ea +1 and

5a n a = Sab^ab (13)

dana = 1 f^naQyb5bo

+ 1 (feayfaea + faafly WcQab5ba Qac. (14)

We will denote the whole set of generators of the extended phase space by

Qa = {ql,Qaa,Aaa,na, a = 1, 2,3} (15)

Pa = {Pi,Paa,naa,na, a = 1, 2, 3}.

For two arbitrary functionals F and G defined by the extended phase space, the generalized Poisson brackets with respect to which the conjugation is defined are

[F, G] =

5F 5G 5qa5pA

jqaipa.

The graduation rules of all generators and operators of our theory assume the introduction of the following degree: (i) ghost number (gh) which is positive for ghosts, gh(QA) > 0, negative for ghost momenta, gh(PA) < 0, and zero for real fields, gh(ql) = 0 = gh(pt); (ii) resolution degree (res) which is positive for ghost momenta, res(PA) = -gh(PA), and zero for ghosts, res(QA) = 0 and real fields, res(ql) = 0 = res(pl); (iii) level number (lev) which is positive for ghosts, lev(QA) > 0, negative for ghost momenta, lev(PA) < 0 and zero for real fields, lev(ql) = lev(pl) = 0. For the operators we define the following graduation: gh(5a) = 1, lev(5a) = a — 1, gh(da) = 1, lev(da) = a — 1, gh(sa) = 1, lev(sa) = a — 1. The main quantities of the theory are the three BRST charges and the extended Hamiltonian. The BRST charges represent the canonical transcription of the BRST symmetries:

sa * = [*, 0a], a = 1, 2, 3.

The relation (8) requires that the following equations are fulfilled:

[Qa, Qb] = 0, a, b = 1,2,3. (19)

These equations must be completed by adequate boundary conditions:

5Qa 5Qab

50. 5r/a

= 5abGa,

&abcPac,

Qa=P'a =0

= 5abna

QA=PA =0

where P'A denotes all the ghost momenta except for the one appearing in the right hand side of the same relation. For the extended Hamiltonian the problem is

[H, 0a] = 0, a = 1, 2, 3

P=n=Q=A=0

The solutions to the problems (19), (20) and (21), (22) depend on the type of the theory we deal with. As we will study the Yang-Mills fields, which are bosonic fields and which define a first rank irreducible theory where the structure functions fyap and Vare constants, the BFV-BRST charges have the form [15]:

0a = GaQab5ba + EabcPac*"" + \ f^Q^" + *aan" + \ fty*«^ Q*

+ 1 flfle,ebcnabQaCQ'dQae5m + 2 faYnanyQ?a + ^(C^ + Wa )nyXaC Q°b 5baQ^C.

The extended Hamiltonian [15] will be

H = H0+ VZ (PaaQ^a + naa^" + )■ (23)

In order to avoid the presence of any unphysical degrees of freedom in the theory, we have to apply the gauge fixing procedure. Unfortunately, new problems could be generated by the fact that it is difficult to choose a particular form of the gauge fixing term so that the covariance of the

theory will not affected. We shall propose a general term to overcome such problems and to contain as ghosts the n-momenta of zero order only, the last momenta ensuring the acyclicity of the Koszul differentials. More concretely, the following theorem [5] is valid:

Theorem 3.1.

For any BRST invariant function K a non-constant odd function Y, defined on the extended phase space sp(3) and

with gh(Y) = —3 exists so that 1

and to the first class Hamiltonian

K =-eabc[Qa, [Qb, [Qc, Y]]].

For the phase space generated by (15) and (16) the gauge fixing function Y has the form (in De Witt notation):

Y = fa (q,p)na. The gauge fixed Hamiltonian will have the form

Hk = H + K = H + 1 £abc[Qa, [Ob, [Oc, Y]]

where H is given by (23).

4. The Yang-Mills field theory

In this section, we will discuss the sp(3) BRST Hamiltonian Yang-Mills theory, in 3 + 1 Minkowski space, taking into account an internal symmetry described by a SU(2) Lie algebra. We will start from the action

a = — 1 /'

So[A™] = — -I d4xFJFJ (26)

FJ = d„AJ — dvAJ + geJMA[.

The canonical analysis of the model leads to the irreducible first class constraints

Cj)(x) = pJ(x) k 0

CJ'(x) = —dpJ(x) + g£m„A(x)p'n(x) k 0 (29)

/dd" i\

FmFlJ — - n nJ

lj m 2"m"

+ AJ(—dlpj + germn Alnpir)). (30)

The gauge algebra is given by:

[G|>,c<1>] = 0, g\gv] = o,

[C$, G&] = ge<-mnG?],

[Ho, G^] = C%\ [Ho, GV] = -g^G®. (31) We can use an index A for writing in a condensed form the whole set of the constraints {G^, A = 1,2; m = 1, ■■■ , d}. For the primary constraints A = 1 and for the secondary constraints A = 2.

The generators of the extended phase space have the form

p =< p(A) AA) n(A) n(A)}

'A— lm, ' ma, mab, ma, m i' ,3-,

Qa = {Alm, Q(A)ma, A(A)mab, A(A)ma, n(A)m}

The nontrivial generalized Poisson brackets are

[pm(x),Ajn(y)]x0^0 = —5j5(x — y), (33)

[Pja(x), Q(A>b(y)U=y0 = —5nm5b5AA'5(x — y), [nJkxM«4' )nb(y)]x0=y0 = —5J5ba5AA' 5(x — y), [nJ>(x), niA>(y)]x0=,0 = —5m5AA'5(x — y).

As it is easily to verify, the BFV-BRST charges, solutions of the Eq. (19), have the following expressions [16]:

Da = j d3x (p0mQ{1)mb5ba +(—dlpm + gcJn^p, )Q{2)mb5ba + ¿abc-P^ ^ + ^n^ + I gC (PJC Q^ + nJ^ + nJW* )Q(2)nb5ba + 1 g2£erneJq£dbc

x nJbQ<2"-cQ<2>qdQ<2>nb5ba + 1 gMr^q + e'nqtJrW2]Qt2]nb 5baQt2]qc№rc) (37)

The extended Hamiltonian will be of the form [16]:

H = H0 + J d3x ((Qir)ma + gA0nemnrQ(2)ra)Pt2]a + (Xir)ma + gA0nen^W^ + (nmm + gA0nemnrn(2)r

+ g ef (£abcQ{2)nbQmrcnn2a + (Q(1)na№ra — QWnaA(1)ra)nW) + 6 eeql-e:neabcQ[2)na Q(2)rb Q(1)qc *%>). (38)

For the gauge fixing procedure we will choose the following form of the fermion function [16]:

Y = j d3x (dAf)^ (39)

which leads to the gauge fixed action

Sy = j d4x (—4 FfvFfV + p0m(&Af) + (dPa)(D )nQVna — (d^aW )nX(2)na + (d^w )fni7)n

+ g efr (eabc(dunfa)Q(2)nc (D YeQ{2)eb) + )"eX(2)ea)Q(2)ra — №na(D%Q(2)ea))

+ g2 efeKqeabc (dnW^Q^mgQ^). (40)

On the basis of their equations of motion, the momenta Pfa, nfc and nf and the ghosts Q^)ma, X^')ma and were eliminated from the extended action. For simplicity, from now on we will use simple notations for the remaining generators

P = P( 1) n = n 1) n = n 1)

na na, na na, n n ,

Qna = Q(2)ma, Xna = X(2)ma, =

but we will have in mind that Pna and Qna do not represent canonical conjugate variables. The same assertion will be valid for the pairs nma, Xna and nm, r/m.

5. Towards a mechanical model

As the most important application of the Yang-Mills fields is connected with QCD, let us now consider the Euler-Lagrange equations generated by the gauge fixed action (40) with a SU(2) internal symmetry. The generators of the sp(3) BRST extended phase space are the fields {A^, Qna, Xna, nn, n = 1, 2, 3; a = 1, 2,3} and their conjugate momenta. Direct computations for each of these generators lead to Euler-Lagrange equations of the form:

d,Ffv + gemnrA^Fpvr — gern,n(Qnad,Pra + Xnad,nra + nnd,nr) + t^ ernBeemqeabc (d,nra)QncQqb

+ £ erneeemqdn (XqaQna — XnaQqa) + g3 en„eelqeuvmeabc dn )QqaQnbQvc = 0, (41)

d, (D )f Qna =0 (42)

d, (D,)fXna + g eabcef (d,Qnc)(D, )reQeb = 0 (43)

d, (D,)fnn — gg? eabcenee%Qnb(d„Qqa)(D%Qec + tr efreenqA,q (Xna (d,Qra) + (d,Xna)Qra) = 0 (44)

(D )nmd,Pna — geabcemnr (d^W YeQec + g eabcenreD )"ed,nnaQrc + genmr (d,nn)D YeXeb5ba

+ g efnD )red,nrXnb5ba + ^ eabceneeemw (d,nn)(D" )W,QqbQrc + g2 ea^e^D" )fd,nmQqbQrc = 0 (45)

(D" )nmd,nna — gemnr (d,nn)(D" )rQea = 0 (46)

(D" )ld,nn = 0. (47)

A first remark is that, on the basis of these equations, part of the terms containing only ghost type variables can be eliminated from the gauge fixed action. Although, the terms containing ghost-ghost-gauge fields vertex are still remaining. Another important remark is that, theoretically, by solving the Eq. (42), we obtain the solutions Qna(r, t) which, introduced in (43), lead to the solutions for Xma(r, t). Using the solutions for Qma(r, t) and Xma(r, t) in the Eq. (44), we can obtain the solutions for r/m(r, t). Therefore, it is necessary to solve the Eqs. (42) only. The invariance of the action (40) in respect with both Lorentz and gauge transformations generates, following the Noether prescriptions, invariant quantities. By imposing some restrictions on these quantities, as for example the Gauss law, one obtains equations which give concrete expressions for the real and ghost type fields. In the case of the free Yang-Mills model [17], these requirements, in the Af = 0 gauge, require for fields Af which should be or (i) homogeneous, or (ii) stationary or (iii) irrotational. The case (i) was recorded long ago in [7] and it allows transformation of the Yang-Mills field in a "mechanical" model. In this case, the fields are replaced as unknown variables by a set of color factors fand the system becomes one with a finite number of degrees of freedom. More precisely, by introducing some orthogonal matrices Of, the following hypotheses were considered:

Af = 0, &Af = 0, Af(t)= 1 Off <f>(t),

OLOi = 5mn5lJ.

will choose the following form for the fields Qma(r, t):

Qma(r, t) = h(m)(t)uma(r). (49)

It is important to note that it is not possible to express the ghost fields Qfa(r, t) using the same color factors {f(m), m = 1.....d} as for the real fields Af. In the decomposition (49) we choose e(h(m)) = 0, e(ufa) = 1, because the ghost fields Qfa(r, t) are fermionic in our case. In order that (49) represents a well-defined decomposition, we will choose the fields ufa(r) as a basis, so that

djuma(r) = emnqOnuqa(r). (50)

The relation (50) allows us to write:

djQma(r, t) = emnqh(m)(t)Or'uqa(r). (51)

Similarly, for the ghost momenta we will have:

djPma(r,t) = e^h^^u—M (52)

Using all these relations which express the fields in terms of color factors, the equation of motion (41) gives the following system of "mechanical" equations:

f + fif)({2 — f(f)2) — 2emnrh{n)h{r) = 0; m, n, r = 1, 2, 3.

No summation has to be considered after the color indices put into parenthesis.

What we propose now is the use of the same technique for the extended action S, i.e. for the whole system (41)-(47). Firstly, on the basis of the Noether theorem we establish the conserved quantities of the theory, and also, imposing here some constraints as for the free case, we obtain the equations which leads to the adequate choice of the ghost fields in terms of color factors. In this respect, we will introduce a new set of color factors, h(f), and we

In the last term from the left hand side of (53), the Levi-Civita tensor has been used for convenience, no summation has to be considered between n, r, and for a given value of f the only term which will be considered will be those for which efnr = 1.

In the asymptotic limit, where the ghosts disappear, we have to consider h(n) = 0, (V)n=i,2,3.One obtain the case of the free Yang-Mills mechanical model described by the system:

+ f {m)(f2 — f(f)2) = 0. (54)

It has been Intensively studied as a non-Hnear dynamical system and special periodic orbits, invariance or inte-grability cases have been pointed out [7]. With the same replacements into (42) we have:

h{m\t) - 2h(m)(t) - emnr(f(n)h(r) + f(r)h(n)) = 0, (55)

with similar conventions as in (53): no summation after n, r and the same rule of choosing the terms with emnr = 1 only.

Now, from (53) and (55), we can define an extended mechanical Yang-Mills model which can be written as a system of 2d equations with 2d unknown quantities {f(m), h(mK m =1, ••• ,d}:

f + f(m)(f2 - f(m)2) - 2emnrh(n)h(r) = 0

.. (rn) ■ (56)

h (t) - 2h(m)(t] - emnr(f(n)h(r) + f(r)h(n)) = 0

Let us note, as a first remark, that the system (56) can be seen as a particular case of a more general system of equations written for a vector with 2d components:

of the form:

. = (f (1).....f (d):h(i).....h[d])

w = M(w) • w

The components of the matrix M(w) can depend on the color factors {f(m),h(m)}. It is an interesting result which shows how the evolution of a gauge field system can be reduced to the study of a system with a finite number of differential equations. The fact that the ghost fields were maintained into the equations will allow us to use them in the control and stabilization of the motion. They could have the same role as the Higgs fields have in the abelian gauge model [17] or as the higher order terms have in the chaos control of mechanical systems [18]. We have also to note that the asymptotic limit of the field theory, when all the ghosts disappear, corresponds to the choice h(r) = ■ ■ ■ = h(d) = 0. The system (58) reduces in this case to the Eq. (54) and describe what is known in literature [7] the "mechanical Yang-Mills model". This is way the system (56) was denominated the "extended mechanical Yang-Mills model".

6. A four dimensional dynamics

We will try now to concretely illustrate what is the connection between the two types of variables: the color factors attached to the gauge fields and those attached to

the ghost fields. More precisely, we will be interested in studying what is the concrete form of the variables attached to the ghost fields in order to be compatible with the evolutionary equations for the real (gauge) fields. It is difficult to find out a general solution of the system (58) or to extract interesting information on its behavior. Instead, the particular cases arising from it can be used in order to see what analogy and influences can be established between the dynamics of the gauge and ghost fields. In this section we will restrict ourselves to the study of a particular case. Let us consider d = 2 and use the notation: f(1) = x,f(2) = y,h(1) = u,h(2) = y. By a suitable choose of M(w), the system (58) could come into the form:

x = -x(3 + 2x2 + 2 y3)

y = -y(1 + 1 x2 + 4y2)

u = ux + 2vy

■■ 1

v = —ux + 4vy

Two reasons justify the previous choice. First of all, it seems natural to consider that the dynamic of the real fields is not influenced by the ghost. So, we choose a system in which the first two equations are not coupled with the last two and describe the real dynamics. The last two equations define compatibility conditions for the ghosts' variables u(t) and v(t). Solving these equations one can find out how ghosts are looking in order to be compatible with a given real evolution. A second reason justifying the choice of (59) is a pragmatic one: the first two equations were already studied in [19]. They correspond to a nonintegrable case of the classical Yang-Mills mechanical model. Despite the fact that the solutions x = x(t) and y = y(t) present some wave-like behavior, the trajectory y = y(x) is, generally speaking, chaotic (Fig. 1). Although, some almost periodical trajectories, denoting traces of regularity, were pointed out in [19] (Fig. 2).

What we are doing now is to compare the form of the representations v = v(u) given by the last two equations from (59), for the two cases of orbits y = y(x): chaotic, represented in Fig. 1, or quasi-periodic as in Fig. 2. The computation, made using Mathlab facilities, leads to the following conclusions: in both cases the variables u and v have periodical dependence on t. Although, the aspect v = v(u) is strongly influenced by the regularity of the real trajectory y = y(x). One can notice the differences from the Fig. 3, where v(u) corresponds to the chaotic orbit from Fig. 1, and the same representations from Fig. 4, corresponding to the regular evolution described by Fig. 2.

Figure 1. Chaotic behavior y = y(x).

Figure 2. Regular behavior y = y(x).

7. Conclusions

This paper focused on the time evolution of the variables which are currently used in the description of the Yang-Mills model, starting from the gauge fixed action. We managed to develop the Hamiltonian formalism, applied to the case when a sp(3) BRST symmetry is implemented. The main idea was to take into consideration not only the

Figure 4. Representation of v = v(u) for regular behavior y = y(x).

real non-abelian gauge fields but the ghosts fields too, i.e. to write the evolution equations for the whole set of fields which generated the extended phase space. After the study of the system in the frame of the Quantum Field Theory, we changed the context and transformed the system into a mechanical one. The fields were expressed in terms of two sets of color factors {f(m),h(m), m = 1, • • • , d} and, consequently, the field equations became a system of 2d second order differential equations. A particular choice for the case d = 2 was effectively considered. Our investigation leads to interesting results, as for example: (i) as a gauge field theory in the BRST approach, the Yang-Mills model shows that almost all the terms containing ghosts disappear because of their equations of motion. Although, there are terms generated by ghost-ghost-gauge fields vertex for which Feynman rules have to be considered; (ii) as a nonlinear dynamical system, the extended mechanical Yang-Mills model presents a sensitive dependence on the initial conditions. The mechanical model generated by the free Yang-Mills fields has, as a general rule, a chaotic behavior;however some particular periodic orbits can be found. In our example in d = 2 the equations satisfied by the color factors attached to the real fields are decoupled from those generated by the ghosts. Although, the ghosts evolution is influenced by real fields. Our numerical simulation shows that depending on the behavior of the nonabelian gauge fields the ghosts have different behaviors. One of the main conclusions is that the ghost fields, seen as real fields, follow the regularity of the gauge fields and could be used as a supplementary means in stabilizing the evolution of the whole system. How this influence acts will be presented in a future work.

Figure 3. Representation of v = v(u) for chaotic behavior y = y(x).

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