Z. Phys. C 67, 695-700 (1995) ZEITSCHRIFT FOR PHYSIK C �9 Springer-Verlag 1995

On the BFT-BFV quantization of gauge invariant systems with linear second class constraints

Ricardo Amorim

lnstituto de Fisica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528 21945-970 Rio de Janeiro, RJ, Brazil (e-mail: ift01001@ufrj)

Received: 1 December 1994

Abstract. We study the Hamiltonian path integral formu- lation for generic systems with first class and linear second class constraints.

1 Introduction

The quantization of Hamiltonian systems with both first and second class constraints [1] can be achieved in several ways. Second class constraints are conveniently treated at classical level with the use of Dirac brackets (DB's) [1] [2], which can be in principle replaced by (anti) commuta- tors in the process of canonical quantization. The physical states which describe the relevant solutions of the theory are selected among all the other possible states, by impos- ing that they are to be annihilated by the first ctass constraints, written as operators at the quantum level. The second class constraints are supposed to value as oper- atorial identities. Even if there are no ordering problems, which could introduce central terms in the first class algebra, in general it is not an easy task to represent the fundamental variables of the theory as operators, due to the usual complexity of DB structures [2].

This difficult is essentially kept when other methods of quantization are taken in account, as the BRST oper- atorial quantization procedure [3] or its Hamiltonian functional counterpart, the Batalin, Fradkin and Vil- kovisky (BFV) method [41, later adapted to consider also second class constraints by Fradkin and Fradkina (FF) [5].

In an attempt to unify the methods of quantization for systems with both classes of constraints (and at the same time define extended algebras in terms of Poisson brackets (PB's) in place of the original ones constructed with the aid of DB's), Batalin, Fradkin and Tyutin (BFT) [61 have develop a systematic algorithm which implements the Abelian conversion of the second class constraints of the theory under consideration.

In a recent work by the author and Das [7], it has been shown that when the second class constraints are linear in the canonical variables, the BFT algorithm is

implemented by a proper shift in those variables, permit- ting the Abelian conversion to be done in a generic way. We observe that a great amount of interesting works has recently been done concerning the application of the BFT formalism to Hamiltonian systems where linear second class constraints are present [8]. The obtainment of general results which have to be satisfied by all of such systems becomes then a relevant point. Actually it can be shown [9] that the models treated in ref. [8] are BFT-extended just by the shift procedure described in [7] and improved in the present work. At the same time those systems [8] can be quantized along the functional methods to be introduced in section 4. Incidentally, we observe that the collective field formalism, due to Alfaro and Damgaard [10], very powerful in deriving the BV [11] formalism from the BRST one, as well as in the obtainment of general Ward identities, can be implemented in the scope of Abelian converted linear second class constraints ori- ginally associated to first order Lagrangians [7].

Our paper is organized in the following manner: In section 2 we give a review of Hamiltonian systems with both first and second class constraints and establish the notation to be used through out the paper. Following this we present some of the results due to Fradkin and Frad- kina [5] concerning the functional quantization of such systems. In section 3 we extend the results of [7], by showing the general correspondence between the con- straint algebras constructed with the original phase space variables and those obtained with the aid of the BFT variables. By using the results of section 3, the functional quantization of gauge invariant systems with Abetian con- verted linear second class constraints is studied in section 4. We also show that in the unitary gauge the results of FF are reobtained. Section 5 is devoted to some final com- ments and conclusions.

2 Hamiltonian systems with first and second class constraints

Consider a Hamiltonian system evolving in a phase space P described by the canonical set of variables

696

yU(g = 1, . . . , 2 N ) , satisfying the fundamental P B ' s ~

{yU, y~} = f ~ ' , (1)

where fu~ is an antissymmetric and invertible matrix which gives the unconstrained symplectic structure of P. From the above expression, the PB between two any functions A(y) and B(y) defined on P is given by

0A u~ c?B {A, B} = (2)

Generically, the evolution of the system is governed by some first class Hamiltonian H = H(y). The system may also be restricted to some constraint surface in P given by a set of first class constraints 7~ = 7,(Y) (a = 1, . . . , fi) as well as by a set of second class constraints 7~ = )/~(Y) (~ = 1, . . . , 2n) , w i t h N > n + r~.

Accordingly to the classification given above [1] [2], it follows the PB structure

M :

= C~b 7c +

{vo, = CabT~ + Tabz~Za,

= V~?b + IT~)~Z~,

H} = + (3)

which defines the first order structure functions of the theory (In general they depend on the canonical variables y~). Since )G are supposed to be second class, the con- straint matrix A,~ is regular. Quadractical terms in )/~ may be present since the square of second class constraints is a first class one, and by using Jacobi identity it can be seen that the PB's between two first class quantities is necessar- ily first class.

In principle the process of canonical quantization of a system like the one appearing in (3) can be implemented with the use of DB's. Also the BRST operatorial quantiz- ation [3], as well as the corresponding Hamiltonian func- tional quantization developed by BFV [4], when applied to gauge invariant theories which also contain second class constraints, are more conveniently settled with the use of DB's than in terms of PB's [5].

The DB between two phase space functions A(y) and B(y) is defined as El]

{A, B}, = {A, B} - {A, x,}A~{Z~, B}, (4)

where A ~p is the inverse of A~ appearing in the first of equations (3). Definition (4) guarantees that the DB be- tween any quantity and a second class constraint vanishes identically. From (4) we see that fundamental PB's of the theory are deformed to

{y,, y,}, = f u ~ _ {y**, )G}A~{Z/~ ' y~} - - / Z V

= f , , (5)

For simplicity we a s sume discrete bosonic coordinates . The exten- sion to the c o n t i n u u m as well to the case where the dynamica l sys tem presents fermionic degrees of f reedom can be done in a s t ra ight forward way

where now f,u~ is in general a singular matrix. It can be shown [2] that on the second class constraint surface, f~ ' induces a two-form which actually is regular. This corresponds, however, to a reduced phase space treatment which can bring explicitly breaking of covariance as well as the appearance of non-linearities.

By using expression (5),'(4) can be written as

c~A ~,,o OB {A, B}, = ~yVf, ~Ty~. (6)

The introduction of DB's permits to write relations (3) in an alternative way as

M , = 0,

= 0,

H } , = 0, (V)

where the structure functions appearing above in general are not the same as those appearing in (3).

The BFV [4] quantization of a system like the one appearing in (7) can be done along the lines described by FF [5]. The vacuum functional is defined by

I 1 Z,v = ~ [dy ~j Idetf[ ~[dPa] [dt/a]6[Z~] IdetA ]2

exp{i I dt[�89 y~'F,~,9 ~ + P~O ~ - H~]}, (8)

where in the measure appears the determinant of the second class constraint matrix [12] and the determinant of the symplectic matrix given by (1). This is so to make the measure invariant by canonical transformations on P [2]. The argument of the exponential is the BFV gauge fixed action. There, Fu~ is some matrix which permits to write the usual first order (in the velocities) kinetic term in terms of the y" coordinates. It is the inverse o f f ~ in case of being independent of the canonical variables, which can always be done with a proper choice of the coordinates. P~ and 17 ~ represent the usual ghost degrees of freedom necessary to construct the BRST generator [4] [5]

~2 = 7~/~ + ~' P, , ..-P,~ ~2;1[~ ~.'~ b,.-- r/b . . . . r] b~+ ~. (9)

The ghosts have odd Grassmanian parity (in our formula- tion ghosts are fermionic quantities) and satisfy the usual PB relations

{po, = - a o. (10)

Also

H~ = H, + {T, ~2}, (11)

is the BRST invariant gauge-fixed Hamiltonian, which depends explicitly on the gauge fermion T. The proper form of f2 and //1 depends on the rank of the specific theory considered and can be determined [4] [5] when one imposes the nilpotency of the BRST generator,

{O, O}, -- 0, (12)

as well as the involution relation

{H1, *2}, = 0, (13)

697

where the DB's appearing in (11-13) are the same ones defined on (6) for functions of the original canonical coor- dinates y, but are trivially extended to the ghost sector in order to accommodate the fundamental brackets (10).

For rank one theories, for instance, it can be shown that (9 13) lead to

(2 = yarl a - �89 tlbrlcC~bPa (14)

and

H~ = H + PaV~7 b, (15)

where the above structure functions are the same as those appearing in (7). It can also be proved [5] the indepen- dence of the functional integral by changes in the fermion function 7* and that there exists general Ward identities associated to the original gauge invariance of the theory.

As a final comment we observe that in practical ap- plications of the above formalism [2] [5], the phase space coordinates y~ are splited in two sets. The first one is spanned by the original dynamical quantities and the second one is formed by Lagrange multipliers 2" and their conjugated momenta =a. As 2" and ~ are auxiliary fields, they have no dynamics and consequently the ='s are actually first class primary constraints. Although in gen- eral H1 does not depend on such variables, this does not occur with the gauge function 7*. It is just this fact that permits, after integrating in the pairs 2 ~, ~a, to achieve delta function, Gaussian or other usual gauge fixing for the original first class constraints y~ belonging to the true dynamical sector of the theory.

3 The BFT method for linear second class constraints

The general description given in the last section can for- mally solve the problem of quantization of systems with mixed constraints. As we have already commented, the canonical quantization procedure is implemented by let- ting the DB's become (anti) commutators acting on vec- tors of an extended space where the physical states are annihilated by the first class constraints (written as oper- ators). Second class constraints vanish as operatorial identities. For systems which are not simple, however, it can be difficult to find representations of the DB's at quantum level [2]. The same kind of difficulties are of course present in the BRST operatorial quantization for systems with constraints of both first and second classes.

Regarding the functional integration, the difficulties associated in representing DB's as (anti) commutators manifest themselves, for instance, when one tries to con- struct the kernel of the evolution operator in coordinate representation [2]. Also, due to the form of the measure, the functional integral (8) is not in a proper form for starting a diagrammatic calculation. At last, it may be convenient to unify the method of quantization for both sets of constraints.

A way of circumventing the difficulties cited above is to apply the ideas of BFT [6], which essentially consist in enlarging the phase space by introducing additional vari- ables (one for each second class constraint) in such a way that the second class constraints become first class (actual- ly Abelian) and constant when the evolution is generated

by a properly extended Hamiltonian. The constraints which were initially first class have also to be modified in such a way that the algebra associated to them becomes essentially invariant by the proposed phase space exten- sion. The way of implementing this process is by assuming that all the transformed quantities can be written as Taylor series in the new variables (BF variables 0~). It is them imposed that the ex-second class constraints become Abelian and involutive with respect to the extended Hamiltonian. These conditions are grouped order by or- der in 0 ~, giving an infinite set of coupled equations which can be formally solved [6].

In [7] it has been proved that when the second class constraints are linear in the canonical variables, the infi- nite set of coupled equations generated by the BFT pro- cedure can be integrated in a generically way, giving as output a kind of shift in the canonical variables. The functions of the shifted variables have null PB's, in the BFT extended phase space, with all Abelian converted constraints. The process of conversion of the constraints is itself done by this kind of shift.

Although it seems to be very restrictive considering only linear second class constraints, most of the works appearing in literature are devoted just to this kind of situation [8]. This justify to look for a general solution of the BFT method for systems with linear second class constraints. In this section we are going to present a brief review of the results given in ref. [7] and after that show how the algebra (7) is generically transformed. With these results it will be possible to discuss the process of quantiz- ation for such systems in the next section.

In order to implement the BFT procedure, we assume that the BFT variables 0~,(c~ = 1, 2, ... ,2n) have the Poisson bracket structure [6]

{0 ~, 0 ~} = co =~, (16)

where co ~ is a constant, antissymmetric and invertible matrix. From (16) we see that in the BFT extended phase space, the PB between two quantities A(y, 0) and B(y, 0) is given by

~?A u~ ~?B OA OB = _ _ ~a (17) {A, B} ~7;y~f ~y~ + ~0co ~?0a ,

since the two sectors of the extended phase space are by construction independent.

Now it is possible to define new constraints

2~ = z~ + x ~ 0 p. (18)

By requiring that

{)~, 2a} = 0, (19)

we see from (16-19) that

X~co'>eXa~ = - A~a, (20)

if X~ can be chosen as independent of the variables y. In fact, from the first of equations (3) and by the fact that the constraints X~ are linear, X=e can always be taken as a constant, invertible matrix and the original constraints can be converted to Abelian ones. This is possible because for linear second class constraints the constraint matrix A =a is independent of the canonical variables.

698

Let us now consider the quantities

B~ = ~ x ~ { z . . y q

y~ = ~,~B~

z u = y~ - y~. (21)

where X ~ and c%~ are respectively the inverses of X ~ and o~ ~ introduced in (16) and (18). Corresponding to any quantity A = A(y") , it can then be defined a new quantity

= A(zU). (22)

It is easy to see that A is in involution with )~. Actually,

{~o, 4} = {z~, a(z) } + { X ~ , A ( ~ ) }

~ A ( z ) . ~ u _ ~A(z) {z~, y"} + x ~ - { O ~, Y }

OA(z) - ~ ({z~, y } - x ~ J ~ B , ' )

= 0 (23 )

The results quoted above are just those which would be obtained through the direct application of the method of BFT. We note that the conversions (18) and (22) are done by the action of the operator

G = e x p ( y ~ y , ) (24)

on the old corresponding quantities. It is straight forward to verify that

)~ = z~(z ) = ~ z~(y)

= A ( z ) = G A(y). (25)

The Hamiltonian H(y) is transformed in the same way, then conserving automatically the Abelian converted con- straints )~.

Let us consider now the fundamental PB's among the shifted variables z u previously defined in (21). We see that

{~', ~'} = { / , / } + {Y, 7 }

= { / , / } - { / , x ~ } ~ % z ~ , / }

= { / , / } ,

= f."~, (26)

where we have used expressions (4), (20) and (21). We note that the deformed symplectic structure (5) is

reproduced here in terms of the PB's of the BFT extended phase space. This fact guarantees that the algebra (7) keeps the same form when we replace the old quantities A = A(y) by the extended functions J~ = A(z) and at the same time use the PB's defined in (17) in place of the DB's that appear in (7). To see how this important result comes, it is enough to observe that, from (26), the PB between two any quantities J~ and/~ can be written as

{ft, B} - c?A(z) t 'u~B(z) (27)

By comparing (27) and (6), and noting that for linear constraints the deformed symplectic matrix f . ~ is actually

independent of the phase space variables, we see that if {A(y), B(y)}. = C(y), whatever C(y)is, then {4, /~} = C(z). It is implicit that in the last identity we are using the PB defined in the BFT extended space. As this correspond- ence will be of great importance in what follows, let us state it as a theorem:

{A(y), B(y)}, = C(y) ~ {ft, B} = C(z) = C (28)

for linear second class constraints. As a consequence of (28), we can rewrite the algebra (7)

a s

{~, 2~} = 0,

G , ~o} : 0,

G , FI} -~ = VoTb + YYYde,

{2~,/q} = 0, (29)

in terms of PB's (17). In the above expressions,

~ = V~(z)

~ J = Vy(z), (30)

have the same functional form as the structure functions appearing in (7).

Now it is easy to see that the first order action

dt[�89 yUfu~v + �89 ~ % ) ~ _ / ~ + )f]a + )72~] (31)

is invariant under the transformations generated by the by now first class set of constraints ~7~, 2~, once the Lagrange multipliers 2 transform in the proper way [2].

4 Quantization

Once the BFT procedure has been implemented, as de- scribed in last section, we get a dynamical system with only first class constraints and Hamiltonian, satisfying a closed algebra given by (29). The BFV treatment of such system will be achieved in a phase space spanned by the old canonical coordinates y", the BF variables ~ , the old ghost sector associated with the original first class con- straints and described by the pairs Pa, rfl and by a new ghost sector associated to the new Abelian converted constraints 2~ which will be here designed by P~, rf.

Due to (28-29), the new BRST generator for the BFT converted system will be given by

= GO. (32)

In (32) ~ has the same functional form as the old BRST generator defined in (14), but with yU replaced by z u wher- ever it occurs. In the definition of 0, the second term in the right has this trivial form since it is associated to an Abelian constraint sector. Once the nilpotency of ~2 (see (12)) occurs in terms of DB's, the nilpotency of ~ also occurs, but realized in terms of PB~ This can be seen by

699

the same kind of results given by (28). tn more details, we note that the nilpotency of ~2 in expression (12) is formally achieved as a sum of two terms, one due to deformed symplectic structure of the original phase space and the other one due the symplectic structure of the ghost sector (gs) spanned by tfl, P~:

8Q ~ 8Q {ga, a } , : Tf . ,* + {a,

=- T t (y , tl, P) + T2(y, t/, P). (33)

For rank zero theories, both terms are identically null. For higher ranks, the architecture of (2 is such that T~ is equal minus T2. Now, when we shift the y variables to the z ones, due to (28), we have that

{o, o} aa( ) f,. aa(z) = a z . a z +

= T, (z , r I, P) + T2(z , rl, P). (34)

where the dependence of f2 in the ghosts is implicit. Ex- pression (34) means that if {f2, ~2}. = 0, then {0, 0} = 0, since T~ and T2 keep the same functional form. Once f2 is seem to be nilpotent, the nilpotency of 0 follows trivially, due to the Abelian character of )~, and the fact that both ghost sectors are disjoint.

By now it becomes clear how the process of conversion is applied to the remaining quantities appearing in the BFV description of the vacuum integral functional. The gauge fixed BRST-invariant Hamiltonian (see expression (11)) is now defined as

/Tr,p =/71 + {~, 0}, (35)

where

ffI1 = G HI(y , tl a, Pa)

= HI(Z , rl a, P~)

and ~ is a new gauge fixing fermion which takes in account the added symmetries associated to the gener- ators )~. Observe that /ta is independent of the ghost sector associated to )~, since it is, by construction, involu- tive with respect to them.

The vacuum functional (see (8)) is rewritten as 1

a,p = ~ [dy u] Ide t f l -~[dP , ] [dr/~] [dP=] 1

x [dr/~] [d~ ~ ] Idet col -~

exp {i f dt [�89 yUF,~ 1)~ + �89

+ P=O ~ + Pffl'- - /~ ,p] }, (37)

where again we have introduced the determinants of f"~ and co~e (see (1) and (17)) in order to keep the invari- ance of the measure in the extended phase space. Actually these determinants go to 1 when the usual splitting in coordinates and momenta are done.

As (37) is now written as an usual BFV first class functional integral, invariance of it by redefinitions of ~ is achieved in the usual way [2] [4]. By the same reason, general Ward identities can also be derived along the usual lines.

Of course the results of Fradkin and Fradkina [5] can be reobtained in the unitary gauge, which implements the vanishing of the BFT variables ~ . To verify this, it is necessary to look a bit more closer to (37). First of all, it is necessary to explicitate from the original canonical sector the Lagrange multipliers 2 ~ and their conjugated momen- ta ~,. As these last quantities are first class constraints, we also double the original ghost sector defined originally by q= and P~. As usual [2], this can be done by the splitting

yU___,(y~', U, 7c~)

_ i p c a)

P~--,(iC~, P~), (38)

where the (P, C) and (C,/5) form two sets of canonical conjugated ghost variables. The BRST generator (32), because of the splitting (38), is written as

(2 = 0 + 2~C ~ - lurch. (39)

The gauge fixing fermion can be defined as

= ~ + iC~G ~ + P~2 , (40)

where G ~ are gauge fixing conditions. It is easy to see that

{(2, = {0,

+ iC=Cp {2~, GB} -- 2=U -- ~ G ~ + iUP~. (41)

Regarding expressions (35), (37) and (41), integrations over 2 and 7c introduce the delta functionals o f ~ and G ~ in the measure. Integration in P , /5 gives a trivial unity factor. Integration in C, C gives in the measure the deter- minant of {2~, Ga} �9 Of course this only means that we can represent the Faddeev integral [13] without using ghosts.

At this stage, the unitary gauge is implemented by letting ~ vanish. This is done by identifying G ~ with ~ . As a consequence (see (16), (18))

det {~, G v} = Xco, (42)

where X = det X=~ and co = det co ~. Now, by using (20), it is trivial to verify that actually

1

I Xcol = I Aco I (43)

In the above expression, A = detA~, A~, given by (18). The co factor, by (42), (43), is canceled from the measure in (37) and the integration in ~b is done trivially. As identity (28) implies in

{0, }lo=o = (44)

/ t e becomes identical to H,v in the unitary gauge and we finally recover the original path integral of Fradkin and Fradkina given by expression (8).

5 Conclusions

In this work we have considered the quantization of gauge invariant systems with second class constraints. Initially, we have reviewed some of the results due to Fradkin and Fradkina, which describes a functional formalism based on Dirac brackets to strongly eliminate the second class

700

constraints. Fol lowing that we have established general results which come from the Batalin, Fradkin and Tyut in formalism when the second class constraints are con- sidered to be linear in the phase space variables. Specifi- cally, we have shown the correspondence between the usual constraint algebra written in terms of Dirac brackets and its extended version, by using BFT variables and Poisson brackets in the extended phase space. In the last section we have considered some quan tum aspects of the formalism, establishing a generic form for the vacuum functional when the linear second class constraints are Abelian converted by the method of BFT. The corres- pondence between this formalism and the Fradkin and Fradkina one has been done in details.

Finally, we observe that the BFT -BFV formalism can be a good algori thm for the calculation of Wess-Zumino Lagrangians [8]. These terms can be generated by break- ing the "Hamil tonian covariance" we are working with, and then integrating over all the momenta . Al though it is in some sense arbitrary, it is always possible to keep the configurat ion space coordinates which would be used to define a Lagrangian formalism. The Wess-Zumino terms then come as the difference between the effective Lagran- gians and the original ones. At least in some cases, we have shown that Wess-Zumino terms can be generated also by a shift mechanism constructed at Lagrangian level [14]. Generic results concerning this and related points are presently under considerat ion and will be published else- where [-15].

Acknowledgment. This work is supported in part by Conselho Nacional de Desenvolvimento CientlfiCO e Tecnol6gico - CNPq (Brazilian Research Agency). I would like to thank J. Barcelos-Neto and N. R. F. Braga for fruitful discussions. I thank also L. E. Souza and R. Thibes for the reading of the manuscript.

References

1. P.A.M. Dirac: Can. J. Math. 2 (1950) 129; Lectures on quantum mechanics (Yeshiva University, New York, 1964); A. Hanson, T. Regge and C. Teitelboim: Constrained Hamiltonian Systems, Accad. Naz. dei Lincei (1976)

2. Henneaux, C. Teitelboim: Quantization of gauge systems (Prin- ceton University Press, Princeton, 1992)

3. C. Becchi, A. Rouet, S. Stora: Ann. Phys. (NY) 98 (1976) 287; I. V. Tyutin, Lebedev preprint FIAN-39/1975, unpublished

4. E.S. Fradkin, G.A. Vilkovisky: Phys. Lett. B55 (1975) 224; I.A. Batalin, G.A. Vilkovisky: Phys. Lett. B69 (1977) 309

5. E.S. Fradkin, T.E. Fradkina, Phys. Lett. B72 (1978) 343 6. I.A. Batalin, E.S. Fradkin: Nucl. Phys. B279 (1987) 514; I. A.

Batalin, L.V. Tyutin: Int. Jour. Mod. Phys. A6 (1991) 3255 and references cited therein

7. R. Amorim, A. Das: A note on Abelian Conversion of Con- straints, pre-print UR-1369, ER 40685-819 (1994) To appear in Mod. Phys. Lett. A

8 T. Fujiwara, Y. Igarashi, J. Kubo: Nucl. Phys. B341 (1990) 695; Y. Kim, S. Kim, W. Kim, Y. Park, K. Kim, Y. Kim: Phys. Rev. D46 91992) 4574; R. Banerjee: Phys. Rev. D48 (1993) R54467; R. Banerjee, H.J. Rothe, K D. Rothe: Phys. Rev. 49D (1994) 5438; R. Amorim, J. Barcelos-Neto: Phys. Lett. B333 (1994) 413; R. Amorim and J. Barcelos-Neto: BFT quantization of chiral- boson theories pre-print IF-UFRJ-20/94; R. Amorim, L.E.S. Souza R. Thibes: BFT formalism and first order systems, to appear in Z. Phys. C (1994)

9. L.E. Souza, P. Sacom R. Thibes: work in progress. t0. J. Alfaro, P.H.Damgaard: Nucl. Phys. B404 (1993) 751 11. I.A. Batalin, G.A. Vilkovisky: Phys. Lett. 120B (1981) 27; I.A.

Batalin, G.A. Vilkovisky: Phys. Rev. D28 (1983) 2567 [E: D30 (1984) 508]; for a review, see F. De Jonghe: PhD Thesis, Katholicke Universiteit Leuven (1993). See also M. Henneaux, C. Teitelboim in ref. [2]

12. P. Senjanovick: Ann. Phys. (N.Y.) 100 (1976) 277 13. L.D. Faddeev: Theor. Math. Phys. 1 (1970) 1 14. See Amorim: Souza, Thibes in ref. [8]. 15. R. Amorim: work in progress