Symmetry transform in Faddeev-Jackiw quantization of dual models
C. P. NatividadeInstituto de Fı´sica, Universidade Federal Fluminense, Avenida Litoraˆnea s/n, Boa Viagem, Nitero´i, 24210-340 Rio de Janeiro, Brazil and Departamento de Fı´sica e Quı´mica, Universidade Estadual Paulista, Avenida Ariberto Pereira da Cunha 333,
Guaratingueta´, 12500-000 Sa˜o Paulo, Brazil
A. de Souza Dutra
Departamento de Fı´sica e Quı´mica, Universidade Estadual Paulista, Avenida Ariberto Pereira da Cunha 333, Guaratingueta´, 12500-000 Sa˜o Paulo, Brazil
H. Boschi-Filho
Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, 21945-970 Rio de Janeiro, Brazil
~Received 20 October 1998; published 23 February 1999!
We study the presence of symmetry transformations in the Faddeev-Jackiw approach for constrained sys-tems. Our analysis is based in the case of a particle submitted to a particular potential which depends on an arbitrary function. The method is implemented in a natural way and symmetry generators are identified. These symmetries permit us to obtain the absent elements of the sympletic matrix which complement the set of Dirac brackets of such a theory. The study developed here is applied in two different dual models. First, we discuss the case of a two-dimensional oscillator interacting with an electromagnetic potential described by a Chern-Simons term and second the Schwarz-Sen gauge theory, in order to obtain the complete set of non-null Dirac brackets and the correspondent Maxwell electromagnetic theory limit.@S0556-2821~99!04804-3#
PACS number~s!: 11.10.Kk, 11.15.Tk
I. INTRODUCTION
Dual symmetries play a fundamental role in classical elec-tromagnetic theory as realized since the completion of its equations by Maxwell in the last century. In quantum theory, however, these symmetries were not fully appreciated until the works of Montonen and Olive @1# and more recently Seiberg and Witten @2#in 311 dimensions and the study of Chern-Simons ~CS! theories @3# in 211 dimensions. Since these theories have gauge symmetries, they are naturally constrained.
The study of constrained systems consists of a very inter-esting subject which has been intensively explored by using different techniques @4#, alternatively to the pioneering pro-cedure of Dirac @5#. In that original work, the constraints were classified into two categories which have different physical meanings: first-class constraints related to gauge symmetries and second-class ones which represent a reduc-tion of the degrees of freedom. In addireduc-tion to its applicabil-ity, the Dirac method presents some difficulties when one studies systems presenting only second-class constraints and there one verifies the presence of symmetries despite the gauge fixation. This is what happens with 2D induced grav-ity where the SL(2,R) symmetry was not detected by con-vential methods but by analyzing the anomaly equation by Polyakov@6,7#. Later on, Barcelos-Neto@8#using the Dirac and sympletic methods reobtained this result and also found a Virasoro hidden symmetry in the Polyakov 2D gravity.
From the canonical point of view, the study of symmetries can be approached with the Faddeev-Jackiw ~FJ!sympletic procedure@9#. In this approach, the phase space is reduced in such a way that the Lagrangian depends on the first-order velocities. The advantage of this linearization is that the
momentum of the CS field. This planar system is also inter-esting to use to explore the role of the canonical quantization of a particle under the influence of a gauge field. Conse-quently, it can be interpreted like a laboratory to the dimen-sional reduction approach in other more complicated models @16#. In Sec. IV, we explore the ideas introduced in Sec. II to quantize, from the canonical point of view, the Schwarz-Sen model @17#. The study of symmetry dualities reveals a con-flict between electric-magnetic duality symmetries and Lor-entz invariance at the quantum level in the Maxwell theory @18,19#. In a very interesting way, Schwarz and Sen pro-posed a four-dimensional action by using two gauge poten-tials, such that the duality symmetry is established in a local way. As a consequence, the equivalence between this alter-native theory and Maxwell’s one is demonstrated. Here, we use the features discussed in Sec. II to obtain this equiva-lence. For our convenience, we choose the Coulomb gauge in the treatment of both gauge theories discussed in Secs. III and IV, so that a parallel of the sympletic structure between that two different dual models can be easily traced. Conclu-sions and final comments are presented in Sec. V.
II. SYMMETRY TRANSFORM IN THE FADDEEV-JACKIW APPROACH
In order to show how the symmetry transformations are related to the zero modes of the sympletic matrix in the FJ approach we have made use of a simple case, where a par-ticle has been submitted to a potential which depends on a constrained function. For a review of FJ method and appli-cations we refer to Refs.@9–13#.
Let us start by considering the following Lagrangian:
L~0!5piq˙i1V~q,p,V !, ~2.1!
where the potential is defined as
V~q,p,V !5lV~q,p!2W~q,p!, ~2.2!
such that V(q,p) represent the constraints, l a Lagrange multiplier, andW(q,p) the resultant potential. Following the steps of the sympletic method we must build a sympletic matrix which contains the Dirac brackets. Hence, we begin defining the matrix elements@9–13#
rˆ[~ri j!5
]aj
]ji2
]ai
]jj
, ~2.3!
ji[(qi,pi) being the generalized coordinates andaithe
co-efficients of the velocities in the first-order LagrangianL(0). Therefore we have, by inspectingL(0), thataq
i
[piand then
rq p i j
52 ]aq
i
]pj1
]ap j
]qi52
di j, ~2.4!
since ap j
vanishes. Now, defining the vector ji5(qi,pi,l)
and calculating the respective coefficients, we obtain the ma-trix
rˆ~0!5
S
0 2di j 0
di j 0 0
0 0 0
D
, ~2.5!
which is obviously singular, since detrˆ(0)50. Then, in this case we cannot identify rˆ(0) as the sympletic matrix. This feature reveals that the system under consideration is con-strained@10–12#. A manner to circumvent this problem is to use the constraints conveniently to change the coefficients
ai(j) in the first-order Lagrangian ~2.1! and consequently
obtain a rank-2 tensor which could be identified with the sympletic matrix.
In the present case, we can build up an eigenvalue equa-tion with the matrix rˆ(0) and eigenvectorsvi(0) such that
vi~0!r~0!i j50. ~2.6!
From the variational principle applied to Lagrangian~2.1!we find the condition over the zero modes,
vi~0!]iV~0![x~0!, ~2.7!
which generates the constraint x(0). If we impose thatx(0) does not evolve in time, we arrive at
x˙~0!5~]ix~0!!q˙i, ~2.8!
and since x˙(0) is linear inq˙i, we can incorporate this factor into Lagrangian ~2.1!. This operation means to redefine the coefficientsai
(0)(j) in the form
a ˜i
~0!
~j!→ai~0!~j!1l]ix~0!, ~2.9!
where l is a Lagrange multiplier. Consequently the matrix rˆ(0) becomes
~˜r!i j5
]˜aj
]ji
2]]j˜ai
j
. ~2.10!
After completing this, if det(r˜)ij is still vanishing, we must
repeat the above strategy until we find a nonsingular matrix. As has been pointed out in the Refs. @10–12# for systems which involve gauge fields it may occur that the matrix is singular and the eigenvectors vi(m) do not lead to any new constraints. In this case, in order to obtain an invertible ma-trix, it is necessary to fix some gauge. Such a case will be discussed in the following sections.
Going back to Eq. ~2.5!, we can see that Eq. ~2.8! is satisfied for the eigenvector vi(0)5(0,0,1). On the other hand, from Eq. ~2.7!and the Lagrangian~2.1!, we get
x~0!
5vi~0!]iV~0!5vl~0! ]V~0!
]l 50
[V~p,q!, ~2.11!
L~1!5L~0!uV501h˙V~q,p!
5piq˙i1h˙V~q,p!2W~q,p!. ~2.12!
Hence, the new coefficients which contribute to the matrix are aq
i
5pi andahi5V. Then, the iterated matrixrˆ(1) reads
rˆ~1!5
S
0 2di j
]Vj
]qi
di j 0
]Vj
]pi
2] Vi
]qj 2
]Vi
]pj
0
D
, ~2.13!
which means that detrˆ(1)[$Vi,Vj%
PB. Here, there are two
possibilities. The first is when detrˆ(1)Þ0 and the matrixrˆ(1)
is invertible. The second one occurs when detrˆ(1)50. This case is more interesting, since the eigenvectors
vi~1!5
S
2 ]V ]pi,]V ]qi
,1
D
~2.14!can be identified as the generators of infinitesimal transfor-mations. This feature will be quite explored in our analysis. Going back to the matrix given by Eq. ~2.13!, we notice the absence of the diagonal elements. This is apparently natural since by definition rˆ(m) is a rank-2 tensor, and in general this tensor is antisymmetric. However, there are cases where the system contains duality symmetry, as, for example, in the Chern-Simons theories @3,4#. In order to in-corporate these elements into the iterated matrixrˆ(1), we can suppose that some kind of symmetry transform can be ob-tained from the zero modesvi(m).
Therefore, let us consider the following transformation in the auxiliary coordinate:1
pi→pi1fi⇒dpi5
]fi
]qj
dqj, ~2.15!
where fi5fi(q). Consequently, the modified Lagrangian L
˜(0) is given by
L ˜~0!
5~pi1fi!q˙i1˜V~qi,pi1fi!
5~pi1fi!q˙i1lV˜~qi,pi1fi!2W˜~qi,pi1fi!, ~2.16!
and by implementing the sympletic method here we obtain the matrix
~˜ri j!~1!5
S
fi j di j
]Vj
]qi
2di j 0
]Vj
]pi
2] Vi
]qj 2
]Vi
]pj
0
D
~2.17!
in such a way that, according to Eqs.~2.6!–~2.9!, we arrive at
det~˜ri j!~1!5$V˜i,V˜ j%, ~2.18!
wherefi j[]fj/]qi2]fi/]qj. Now, sincefjis infinitesimal,
we can write
V˜i
~qj,pj1fj!5V˜i~qj,pj!1
S
]V˜i
]pk
D
f50
fk, ~2.19!
which implies that
$V˜i,V˜j%
PB5$Vi,Vj%
F
11S
]V˜i
]pk
D
f50 fkG
[0, ~2.20!
since $Vi,Vj%
50 has been considered here. The above re-sult reveals that the constraint algebra is preserved in front of transformations~2.15!. Consequently, the matrix (r˜(1))i j
re-mains singular and the zero modes in this case become
v~i1!5
S
2 ]V˜i]pj
,]V
˜i
]qj2
]V˜i
]pk
fk,1
D
, ~2.21! which implies thatv~i1!˜ri j~1!5~0,0,$V˜i,V˜j%!, ~2.22!
giving a null vector by virtue of Eq. ~2.20!. On the other hand, the action of the zero modes on the equations of mo-tion yields
vi~1!
F
]L~1!]ji
2 d
dt
S
]L~1!
]j˙i
DG
505h˙$V ˜i,V˜ j%2$V˜i, W˜j%,
~2.23!
as a consequence of Eqs.~2.12!and~2.21!. This means that no new constraints can arise from the equations of motion. From the above equation we can get
$V˜i,W˜ j%
5vi~1!]iW˜, ~2.24!
and by virtue of Eq.~2.20!we conclude that the zero modes are orthogonal to the gradient of the potentialW˜, indicating that they are generators of local displacements on the isopo-tential surface. Consequently, they generate the infinitesimal transformations; i.e., for some quantity A(j) we must have
1The transformation given by this equation has been suggested in
dAa5
S
]A a]ji
vi
D
«, ~2.25!« being an infinitesimal parameter. From Eqs. ~2.21! and ~2.25!we have
dqi52
]V˜i
]pl«l, ~2.26!
dpi5
S
]V˜i
]ql
2]V
˜i
]pl f il
D
«l, ~2.27!
dh5«, ~2.28!
which permit us to show that the LagrangianL˜(1) becomes
L ˜~1!
5L~1!1dL~1!
5L~1!1 d
dt~pidqi1dhV˜ i
!«, ~2.29!
which does not change the original equation of motion. Therefore, the introduction of symmetry transforms into the original LagrangianL(0)leads to some elements of the sym-pletic matrix, which are the Dirac brackets. Notice that this result has been obtained without loss of the formal structure of the constraint algebra and the equations of motion. In the following sections we present explicitly examples in field theory where these ideas will be explored in some depth.
III. OSCILLATOR INTERACTING WITH A CHERN-SIMONS TERM
Let us now consider the problem of charged particles sub-jected to a harmonic oscillator potential moving in two di-mensions and interacting with an electromagnetic field de-scribed by a Chern-Simons term. This problem was inspired by the Dunne-Jackiw-Trugenberger model@15#and has been considered before @14,13# in different situations. Here we want to apply the canonical form of the sympletic method which will lead us to the Dirac brackets but some of them will be missing as we discussed in the previous section. Then we use a convenient transformation to get the complete brackets set. So we start with the Lagrangian2
L5m 2@q˙i~t!q˙
i
~t!2v2qi~t!qi
~t!#
2e
E
d2xA0~t,xW!d~xW2qW!1eE
d2xAi~t,xW!3d~xW2qW!q˙i~t!1u
E
d2xemnrAm~t,xW!]nAr~t,xW!, ~3.1!whereqi(t) is the particle coordinate with charge2eon the
plane (i51,2),Am(t,xW) is the electromagnetic potential (m 50,1,2), and u is the Chern-Simons parameter. In order to implement the sympletic method here we introduce the aux-iliary coordinate pi(t) through the transformation q˙2
→2p•q2p2 @10#, and define an auxiliary field Pi(t,xW)
5ei jAj(t,xW), so that we can write the above Lagrangian as
L~0!5@m pi~t!2eAi~t,qW!#q˙i~t!
2u
E
d2xPi~t,xW!A˙i~t,xW!2V~0!, ~3.2!where Ai(t,qW)5*d2xAi(t,xW)d(xW2qW), and the potential is given by
V~0!5m 2 @pi~t!p
i
~t!1v2qi~t!qi
~t!#
1eA0~t,qW!12u
E
d2x]iPi~t,xW!A0~t,xW!. ~3.3!Since this Lagrangian is linear on the velocities, we can iden-tify the sympletic coefficients
aqi~t! ~0!
5m pi~t!2eAi~t,qW!, ~3.4!
aA
i~t,xW!
~0!
52uPi~t,xW!, ~3.5!
while the others are vanishing, which lead us to the matrix elements
rqipj
~0!
52mdi j, ~3.6!
rq~0iA!j5edi jd~yW2qW!, ~3.7!
rAiAj
~0!
50, ~3.8!
rA~0iP! j5udi jd~xW2yW!. ~3.9!
Defining the sympletic vector to be given by ya
5(qW,pW,AW,PW,A0) we have the matrix
2Our conventions here aree
rab~0! 5
S
0 2mdi j edi jd~yW2qW! 0 0
mdi j 0 0 0 0
2edi jd~yW2qW! 0 0 udi jd~xW2yW! 0
0 0 2udi jd~xW2yW! 0 0
0 0 0 0 0
D
, ~3.10!
which is obviously singular. The zero modes come from the equation
]V~0!
]A0~t,xW!5
ed~xW2qW!12u]iP
i, ~3.11!
which implies the primary constraint x(0)
5ed(xW2qW)12u]iP
i ~Gauss law!. Using this constraint we can build up the
Lagrangian
L~1!5L~0!1l˙x~0!, ~3.12!
wherelis a Lagrange multiplier, and now the potential reads
V~1!5V~0!ux~0!505 m
2 @pi~t!p
i
~t!1v2q
i~t!qi~t!#. ~3.13!
The new non-null velocity coefficient is given by al(1)5x(0)5ed(xW2qW)12u]iPi, so that we have new matrix elements
r(1)Ail50 andr(1)Pjl52u]jd(xW2yW), which lead us to the matrix@ya5(qW,pW,AW,PW,l)#
rab~1!5
S
0 2mdi j edi jd~yW2qW! 0 0
mdi j 0 0 0 0
2edi jd~yW2qW! 0 0 udi jd~xW2yW! 0
0 0 2udi jd~xW2yW! 0 2u]id~xW2yW!
0 0 0 22u]jd~xW2yW! 0
D
, ~3.14!
which is still singular. The zero modes will not lead to any new constraints, and so we have to choose a gauge, which will be the Coulomb one (¹W•AW50), and we include it in the Lagrangian via another Lagrange multiplierh:
L~2!5L~1!1h˙¹W•AW
5L~0!1l˙x~0!
1h˙¹W•AW1V~1!
5@m pi~t!2eAi~t,qW!#q˙i~t!2u
E
d2xPi~t,xW!A˙i~t,xW!1l˙@ed~xW2qW!12u]iP
i#1h˙¹W•AW1 m
2 @pi~t!p
i~
t!1v2qi~t!qi~
t!#, ~3.15!
which implies the additional coefficientah(2)5¹W•AW, and the new elementr(2)Aih5]id(xW2yW). The sympletic tensor can then be identified with the matrix, ya5(qW,Wp,AW,PW,l,h),
rab~2! 5
S
0 2mdi j edi jd~yW2qW! 0 0 0
mdi j 0 0 0 0 0
2edi jd~yW2qW! 0 0 udi jd~xW2yW! 0 ]id~xW2yW!
0 0 2udi jd~xW2yW! 0 2u]id~xW2yW! 0
0 0 0 22u]jd~xW2yW! 0 0
0 0 2]jd~xW2yW! 0 0 0
D
, ~3.16!
~r~2!!ab 5
¨
0 1
mdi j 0 0 0 0
2 1
mdi j 0 0 2
e
muDi jd~xW2qW! 0 2
e m
]i
¹2d~xW2qW!
0 0 0 21
uDi jd~xW2yW! 0 2
]i
¹2d~xW2yW!
0 e
muDi jd~xW2qW! 1
uDi jd~xW2yW! 0 2
1 2u
]i
¹2d~xW2yW! 0
0 0 0 1
2u ]i
¹2d~xW2yW! 0 2 1 2
1
¹2d~xW2yW!
0 e
m
]j
¹2d~xW2qW! ]j
¹2d~xW2yW! 0
1 2
1
¹2d~xW2yW! 0
©
,~3.17!
where Di j5di j2]i]j/¹2. From this result we can write
down the following Dirac brackets of the theory:
$qi,pj%5
1
mdi j, ~3.18!
$pi,Pj%52 e
muDi jd~xW2qW!, ~3.19!
$pi,h%52e
m
]i
¹2d~xW2qW!, ~3.20!
$Ai,Pj%52
1
uDi jd~xW2yW!, ~3.21!
$Ai,h%52
]i
¹2d~xW2yW!, ~3.22!
$Pi,l%52
1 2u
]i
¹2d~xW2yW!, ~3.23!
$l,h%521 2
1
¹2d~xW2yW!. ~3.24!
However, as was anticipated in Sec. II, some non-null brack-ets are missing. To overcome this situation let us consider the transformation
Pi~t,xW!→Pi
8
~t,xW!2ei jAj~t,xW! ~3.25!on the Lagrangian of the system. So following the same steps as shown above we find a LagrangianL(2)8which is identical toL(2) except for the substitutions
2u
E
d2xPi~t,xW!A˙i~t,xW!→2uE
d2xPi8
~t,xW!A˙ i~t,xW!
1u
E
d2xei jAj~t,xW!A˙i~t,xW!and
2ul˙]iP
i~t,xW!→2ul˙]iPi
8
~t,xW!22ul˙ei j]iAj~t,xW!,so that we find the coefficients
a~2!8Ai~t,xW!52uPi
8
~t,xW!1uei jAj~t,xW!, ~3.26!a~2!8l5ed~xW2qW!12u]iP
8
i~t,xW!22uei j]iAj~t,xW!~3.27!
and the matrix elements
r~2!8
Ai~t,xW!Aj~t,yW!522uei jd~xW2yW!, ~3.28!
r~2!8
Ai~t,xW!l~t,yW!52uei j]jd~xW2yW!. ~3.29!
These new elements imply another sympletic tensor which can also be inverted. From this inverse we can reob-tain the above Dirac brackets and also others which were missing before in our treatment,
$Ai,Aj%52
1
2uei jd~xW2yW!, ~3.30!
$Ai,l%5
1
2uei j]j¹22d~xW2yW!. ~3.31!
At this point we can identify l˙ withA0 so that we have
IV. SCHWARZ-SEN DUAL MODEL
Let us begin by introducing the basic idea of the Schwarz-Sen dual model@17#. The proposal is to treat the problem of the conflict between electric-magnetic duality and manifest Lorentz invariance of the Maxwell theory. We mention, for instance, that by using the Hamiltonian formalism it can be shown that a nonlocal action emerges when one imposes the manifest Lorentz invariance and tries to implement the dual-ity symmetry @18#. In order to circumvent this difficulty Schwarz and Sen proposed the introduction of one more gauge potential into the theory.
In this sense, the model is described by an action that contains two gauge potentials Ama (1<a<2 and 0<m<3) and is given by3
S521 2
E
d4x~Ba,ieab
Eb,i1Ba,iBa,i!, ~4.1!
where Ea,i
52Fa,0i, while Ba,i
52(1/2)ei jkF jk a
, and Fmna
5]mAna2]nAma. This action is separately invariant under lo-cal gauge transformations
dAa,05ca; eAa,i52]iLa ~4.2!
and duality transformations
Aa,m→eabAb,m. ~4.3!
In terms of the gauge potentials, the corresponding Lagrang-ian density is given by
L51
2e
i jk~] jAk
a
!eab~A˙i b
!
21 2e
i jk
~]jAk a
!eab~]iA0
b
!21 4F
a,jk Fjk
a
. ~4.4!
Now, the above Lagrangian density is of first order in time derivative. In order to implement the sympletic method we can define an auxiliary field to make more simple the subse-quent calculations. Hence, let us consider the field
Pa,i
5eabei jk~]jAk b
!
[PWa
5eab¹3AWb, ~4.5!
and the Lagrangian density ~4.4!becomes
L~0!51
2PW
a• AW˙a21
2PW
a•¹ A0a21
2PW
a•PWa
[1 2PW
a•
AW˙a2V~0!, ~4.6!
where V(0)512PW a•¹
A0a1(12)PW a•PWa
. Therefore, the sym-pletic vector will be given as jW(0)
5(AWa,PWa,
A0a). From the
developments of Sec. II, it is easy to verify that the sympletic matrix correspondent toL(0)is singular. The zero-mode
vec-tor in this case will be v(0)5(0,0,vA
0
(0)) and the use of Eqs.
~2.6!–~2.8!will give origin to the constraint
x5¹•PWa50, ~4.7!
which can be incorporated into the new Lagrangian density via a Lagrange multiplier. Consequently, we have
L~1!5L~0!ux
501l˙¹•PWa 51
2PW
a•
AW˙a1l˙¹•PWa
21 2PW
a•PWa, ~4.8!
which leads to another singular matrix
~rab
~1!!
i j5
S
0 2di j 0
di j 0 2]i
x
0 ]j x
0
D
dabd~xW2yW!, ~4.9!
where the sympletic vector has components j(1) 5(AWa,PWa
,l). On the other hand, the use of Eq.~2.7!implies that
vWAWa2¹vl50, ~4.10! and no new constraints are generated. Here, we remark that the above relation can be used to derive the well-known gauge symmetry
dAWa5¹l, dPWa
50, dA0a5l˙. ~4.11!
Hence, in this step it is necessary to impose a gauge fixing. If we adopt the Coulomb gauge, as was done in the previous section, the new Lagrangian density becomes
L~2!5L~1!u¹•AWa501h˙~ ¹•AWa!
51 2PW
a•
AW˙a1l˙~ ¹•PWa!
1h˙~ ¹•AWa!21 2PW
a•PWa
, ~4.12!
and the sympletic matrix, given by
~rab
~2!!
i j5
S
0 2di j 0 2]i x
di j 0 2]i
x
0
0 ]j x
0 0
]j x
0 0 0
D
dabd~xW2yW!, ~4.13!
can be inverted to give
~rab
~2! !i j2
1 5
S
0 2dabDi j 0 ]i x¹22
dabDi j 0 ]i
x
¹22 0
0 2]j
x¹22 0 ¹22
2]j x
¹22 0
2¹22 0
D
3d~xW2yW!, ~4.14!3Our conventions aree12
where Di j5di j1]i x]
j
x¹22. The sympletic vector now is
j(2)
5(AWa,PWa,l,h); therefore, from the above matrix we get
$AWa~xW!,PWb~
yW!%D52dab
S
di j1]ix] j x
¹2
D
d~xW2yW!, ~4.15!which agrees with the result in Ref.@19# obtained from the Dirac procedure. It is important to notice that the matrix ~4.14!presents only one bracket, since by virtue of the dual symmetry it must contain diagonal elements like $AWa(xW),AWb(yW)%D. This feature can be interpreted by
consid-ering the bracket~4.15!as a dynamical one. The part of the sympletic matrix ~4.14! related to symmetries cannot be identified directly. However, this term can be generated by means of a convenient symmetry transformation.
In order to implement this, let us consider the following transformation into the Lagrangian density~4.6!:
PWa→PWa
8
2eab¹3AWb, ~4.16!so that we rewrite it as
L~0!8511
2~ PW
a
8
2eab¹3AWb!•~AW ˙a2¹A0
a
!
21 2~ PW
a
8
2eab¹3AWb!2. ~4.17!Now, by using Eqs. ~2.6! and~2.7!, it is easy to verify the presence of the constraint
x
8
5¹•PWa8
~4.18! and, consequently,L~1!85L~0!8ux
8501a˙~ ¹•PWa
8
!. ~4.19! The singular matrix corresponding to the above Lagrangian density is given by~Gab~1!!i j5
S
eabei jk]k 2dabdi j 0
dabdi j 0 2dab]i
x
0 dab]j x
0
D
d~xW2yW!.
~4.20!
Then, from Eq.~2.7!we obtain the zero modes
v
WAWa5¹vaa, vWPWa5¹3~eabvWAWb!, ~4.21!
which confirm the two expected symmetries
dAWa5¹aa ~gauge!, ~4.22!
dPWa
8
5¹3~eabAb! ~dual!, ~4.23! and no new constraints are generated. Therefore, we can adopt a gauge fixing. Choosing again the Coulomb gauge we arrive atLGF51
2@~AW
˙a
2¹a˙a2PWa
8
1eab¹3AWb!•PWa8
2~ ¹3AWa!22~A˙a2¹a˙a!eab¹3AW b
#¹•AWa50. ~4.24! Identifying a˙a[A
0
a
, we get EWa
52A˙a
1¹A0a, and conse-quently the gauge-fixed Lagrangian density becomes
LGF
8
512~2EW
a
2PWa
8
1eab¹3AWb!•PWa8
21 2~ ¹3AW
a!2
2dLGF ~4.25!
and the gauge-fixing term can be written as
dLGF51
2~EW
a
1PWa
8
!dPWa8
51 2~EW
a
1PWa
8
!d~eab¹3AWb!5212¹3~EWa
1PWa
8
!•d~eabAWb!1surface terms,~4.26!
where we used Eq.~4.23!.
Before going on, it is important to make some remarks. First of all, we mention that from Eq.~4.25!it is easy to infer that the Dirac bracket between the gauge fields in this case is given by
$AWa~xW!,AWb~yW!%D5eab¹22¹3d~Wx2yW!, ~4.27!
which gives rise to a nonlocal commutation relation for the
AWa field. This relation was obtained here within the context of the sympletic methodology starting from the use of the symmetry transform given by Eq.~4.16!.
Another interesting feature is that from the use of Eq. ~4.16!and of the gauge-fixed Lagrangian density ~4.25!we can show the equivalence between the Schwarz-Sen model and Maxwell theory. From Eq. ~4.21!, we notice that the variations overeabAW
b
lead to
¹3~ PWa
8
1EWa!50, ~4.28!
and since at this stage the Gauss law can be used, we con-clude that
dPWa
8
52dEWa5¹3~eabAWb!. ~4.29! Going back to Eq.~4.30!, takinga51, b52, we haveLGF
8
→ LM512~EW 1•EW1
2BW1•BW1!, ~4.30! which is the Maxwell Lagrangian density. From the Gauss law ¹•PWa
8
50 and Eq. ~4.28! we find that the vector uWa5PWa
that thePWa
8
(xW) field is not the canonical momentum of the electromagnetic theory but it represents here an auxiliary field in order to implement the Faddeev-Jackiw method.V. COMMENTS AND CONCLUSIONS
In this work, we study the role of the symmetry transfor-mations in the Faddeev-Jackiw approach. We verify that the generators of such a transformation can be represented in terms of the zero-mode vectors of the singular presympletic matrix. Since the inverse of the sympletic matrix contains elements which define the Dirac brackets of the constrained system, it is natural to ask what happens when some brackets do not appear in this inverse matrix. In our interpretation, these elements are associated with some kind of symmetry transform which, on the other hand, are generated by zero-mode vectors. Hence, after a convenient symmetry transfor-mation we can complete the set of the fundamental brackets of the models in question.
Here, we explore this strategy in three different situa-tions: first, for the case where a particle is submitted to a ‘‘constrained potential.’’ After we discuss the case of an os-cillator in two space dimensions coupled to a Chern-Simons gauge field and finally the Schwarz-Sen dual model for which our main goal was to obtain the corresponding Dirac brackets and how to describe its equivalence with the Max-well theory. In this point the zero modes played a very im-portant role.
ACKNOWLEDGMENTS
The authors acknowledge partial financial support from Conselho Nacional de Desenvolvimento Cientı´fico e Tecno-lo´gico, CNPq, Brazil. A.S.D also acknowledges partial fi-nancial support from Fundac¸a˜o de Aruparo a` Pesquisa do Estado de Sa˜o Paulo, FAPESP.
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