1.
PHYSICAL REVIEW D 76, 025025 (2007)
Gauge invariance of the action principle for gauge systems with noncanonical
symplectic structures
Vladimir Cuesta* and Merced Montesinos†
´ ´
Departamento de Fısica, Cinvestav, Avenida Instituto Politecnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero,
´ ´
Ciudad de Mexico, Mexico
Jose David Vergara‡
´
´ ´ ´ ´
Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, 70-543, Ciudad de Mexico, Mexico
(Received 9 April 2007; published 24 July 2007)
The effect of the gauge transformation in the action principle for Hamiltonian gauge systems
formulated in terms of noncanonical symplectic structures is studied and, particularly, the compatibility
between gauge conditions and boundary conditions is analyzed. It is shown that the complete set of
commuting observables at the time boundary is now ﬁxed by the boundary term and the symplectic
structure. The theory is applied to two nontrivial models having SL…2; R† and SU…2† gauge symmetries,
respectively, whose extended phase spaces are endowed with new interactions produced by noncanonical
symplectic structures.
DOI: 10.1103/PhysRevD.76.025025 PACS numbers: 11.10.Ef, 03.65.Ca
I. INTRODUCTION action through the systematic implementation of Dirac’s
method, i.e., the starting point is a Lagrangian action from
Hamiltonian systems with constraints, in the sense in-
which the momenta pi canonically conjugate to the con-
troduced by Dirac, are described by an action principle of
ﬁguration variables qi are deﬁned. From the deﬁnition of
the form [1,2] (summation convention over repeated in-
the momenta, primary constraints usually arise which are
dices is used throughout)
evolved until all the constraints are obtained, which are
Z 2 then classiﬁed into ﬁrst class and second class. By con-
S‰qi ; pi ; a ; Š ˆ …pi qi ÿ H ÿ a
a ÿ †d;
_
1 struction, the canonical symplectic structure (5) plays a
(1) central role in Dirac’s method.
Nevertheless, the action (1) is not the most general
i ˆ 1; . . . ; N, where H is taken to be a ﬁrst-class action allowed to describe a constrained Hamiltonian sys-
Hamiltonian and the
’s are ﬁrst-class constraints, while tem. In fact, constrained dynamical systems with a ﬁnite
the ’s are second class, i.e., [3] number of degrees of freedom can be written in a
Hamiltonian form by means of the dynamical equations
f
a ;
b g ˆ Cab c
c ‡ Tab
10.
CUESTA, MONTESINOS, AND VERGARA PHYSICAL REVIEW D 76, 025025 (2007)
f
a ;
b g! ˆ Cab c
c ‡ Tab
11.
; (9a) preted: as a Hamiltonian action or as a Lagrangian action.
These two possible interpretations of the action lead to two
f
a ; g! ˆ Ca b
b ‡ Ca
12.
; (9b)
alternative approaches. In this paper the action will be
fH;
a g! ˆ Va b
b ‡ Va
13.
; (9c) interpreted as a Hamiltonian one, and in the Concluding
fH; g! ˆ V b
b ‡ V
14.
: (9d) Remarks section some comments regarding the
Lagrangian viewpoint will be made.
Also, Therefore, it is assumed that the action (12) has the
f ;
17.
† Þ 0; (10) Hamiltonian form from the very beginning. For the sake
of completeness, a function B…x; † will be added at the
where the Poisson brackets involved in Eqs. (9) and (10) time boundary
are computed using the symplectic structure ! on ÿ:
SB :ˆ S‰x ; a Š ÿ B…x; †j2 ;
1 (13)
@f @g
ff; gg! ˆ ! …x† : (11)
@x @x just to choose the variables that are going to be ﬁxed at 1
The dynamical equations of motion (6) and the constraints and 2 . The change of the action (13) under the inﬁnitesi-
(7) can be obtained from the action principle mal gauge transformation of the x’s,
Z 2 @G
S‰x ; a ; Š ˆ … …x†x ÿ HE †d;
_ (12) x ˆ fx ; a
a g! ˆ ! …x† ; (14)
1 @x
with ! ˆ d where ˆ …x†dx is the symplectic po- where G :ˆ a
a , and a are the inﬁnitesimal gauge pa-
tential 1-form. rameters, and the inﬁnitesimal gauge transformations of
Once the differences between these two approaches to the Lagrange multipliers a and is
Hamiltonian systems have been recalled, the problem Z 2
studied in this paper is set down. In Refs. [4 –6] the issue S B ˆ ‰… †x ‡ x ÿ H ÿ
a a
_ _
1
of the change of the action of Eq. (1) due to the change of
the canonical variables induced by the gauge transforma- ÿ a
a ÿ ÿ Šd ÿ Bj2 :
1
tion generated by the ﬁrst-class constraints was analyzed. (15)
There, the analysis was restricted to inﬁnitesimal gauge
transformations. Later on, the analysis was extended to Integrating by parts the term x , by plugging (6) and
_
include the full change of the action induced by ﬁnite
gauge transformations [7], which are relevant both classi- @
18.
@G
ˆ f ; Gg! ˆ ! ;
cally and quantum mechanically because the latter include @x @x
19.
(16)
also the ‘‘large’’ gauge transformations that are not in- _ @ @HE
cluded in the inﬁnitesimal procedure developed in ˆ f ; HE g! ˆ !
20.
;
@x @x
Refs. [4 –6]. Moreover, the inﬁnitesimal case was also
developed in Ref. [7], where some new aspects of this into the right-hand side of SB and using ! ˆ @ ÿ
case were reported, among others the differential equation @ , leads to
that the boundary term must satisfy in order to have gauge- Z 2
invariant actions. In all these papers, the analysis was S B ˆ ‰fHE ; Gg! ÿ H ÿ
a a ÿ a
a
carried out by using the action (1), i.e., the extended phase 1
space ÿ is endowed with canonical symplectic structures ÿ ÿ Šd ‡ … x ÿ B†j2 :
1
from the very beginning. Moreover, the analysis has been (17)
applied to general relativity formulated in terms of
Ashtekar variables and Polyakov’s action deﬁned on mani- Using (9) to write explicitly the right-hand side of H ˆ
folds M having the topology M ˆ R with com- fH; Gg! ˆ a fH;
a g! ‡
a fH; a g! and inserting the re-
pact and without a space boundary [8,9]. sult (and doing the same for
a and ), leads to
In this paper, the issue of the gauge invariance of the Z 2
action principle is analyzed for an action of the form (12). S B ˆ ‰a fHE ;
a g! ÿ a …Va b
b ‡ Va
21.
†
This topic is relevant for the path integral quantization of 1
gauge systems [3–6,10], when new interactions are intro- ÿ a b …Cab c
c ‡ Tab
22.
† ‡ a …Ca b
b
duced through the noncanonical symplectic structure.
‡ Ca
23.
† ÿ
a a ÿ Šd
II. THEORETICAL FRAMEWORK ‡ … x ÿ B†j2 :
1 (18)
The gauge invariance of the action can be analyzed from On the other hand, if the Lagrange multipliers a are
two viewpoints depending on how the action (12) is inter- simultaneously transformed as [3]
025025-2
24.
GAUGE INVARIANCE OF THE ACTION PRINCIPLE FOR . . . PHYSICAL REVIEW D 76, 025025 (2007)
a beginning. Thus the roles of the symplectic potential
d
a ˆ ‡ c b Cbc a ‡ b Cb a ÿ b Vb a ; and the symplectic structure were not fully
d (19)
c b
29.
;
(3) Equation (22) was obtained allowing the possibility
then that the gauge parameters could depend on the
phase space variables x: F…x† ˆ fF; Gg! ˆ
Z 2 da
S B ˆ a fHE ;
a g! ÿ
a d a fF;
a g! ‡
a fF; a g! . Usually, however, the
1 d gauge parameters are allowed to depend on only
‡ … x ÿ B†j2 : (20) and so F ˆ fF; Gg! ˆ a fF;
a g. In this last
1
case, Eq. (22) is unaltered and reduces to
Integrating by parts the second integrand leads to
S B ˆ a …† …x† ÿ @B
Z 2
S B ˆ …a fHE ;
a g! ‡ a
a †d
_ @x
34.
1
Z 2 @
a
ˆ a d ‡ … x ÿ G ÿ fB; Gg! †j2 :
1 Under this assumption, the boundary term vanishes
1 @ because either (a) the gauge parameters vanish at the
(21) time boundaries, a …1 † ˆ 0 ˆ a …2 †, (b) the terms
inside the curly brackets vanish at the time bounda-
Assuming that the
’s do not depend explicitly on , ries, or (c) a combination of both (a) and (b).
@
a
@ˆ 0.1 So, by using (14), ﬁnally (4) regarding the particular case given in Eq. (23) where
43.
the variables x provided that an appropriate choice
@x @x
44.
1
for the geometrical objects ( , B, ! , and
a ) is
is the change of the action (13) at ﬁrst order in the gauge made, i.e., gauge-invariant actions Sinv can be con-
parameters a induced by the gauge transformation of the structed by properly handling these geometrical ob-
dynamical variables generated by the ﬁrst-class constraints jects. (See the examples in the next section.)
a . Equation (22) clearly expresses the fact that there are (5) if the original choice of the gauge potential and
ﬁve objects which contribute to the boundary term (22): the the function B is such that the action is not gauge
symplectic potential ˆ …x†dx , the inverse of the invariant, it is still possible to add another function
symplectic structure ! , the gauge parameters a , the at the time boundary in such a way that the new
ﬁrst-class constraints
a , and the boundary term ÿBj2 .1 action is invariant (assuming that ! and
a have
Some remarks follow: been ﬁxed).
(1) if B in (13) does not depend explicitly on , B…x†, (6) note that the complete set of commuting observables
then its contribution to the action (13) and therefore ﬁxed at the time boundaries 1 and 2 depend not
to the boundary term (22) can be absorbed by choos- just on the boundary term B but also on the sym-
ing the new symplectic potential # …x† ˆ plectic structure, a property not fully appreciated
ÿ @B=@x [the symplectic potential, by hy- when a canonical symplectic structure is used.
pothesis, does not depend explicitly on ; see Coming back to the general discussion, once the gauge
Eq. (12)]. invariance of the action has been analyzed, it just remains
(2) in the previous approaches to the subject [5–7], the to say some words about the compatibility between the
discussion about the objects that contribute to the gauge conditions and the boundary conditions. Note that
boundary term was focused on the dependency of good gauge conditions must take into account the sym-
the ﬁrst-class constraints
a , the gauge parameters, plectic structure in the sense that the matrix of their
and the boundary term ÿBj2 , simply because the
1 Poisson brackets with the ﬁrst-class constraints must
symplectic structure and the potential were ﬁxed have a nonvanishing determinant. Like in the case when
and tied to canonical coordinates from the very symplectic structures and symplectic potentials having the
canonical form are employed [4 –7], the boundary condi-
1 tions of the action (13) might not be compatible with the
Alternatively, the -dependence of the
’s can be handled by
parametrizing the theory and considering …; p † as new varia- choice of a particular gauge condition without it mattering
bles thus enlarging the phase space as it was done in the if the action is invariant or if it is not. If this were the case,
canonical case [11]. the procedure for how to achieve such compatibility is
025025-3
45.
CUESTA, MONTESINOS, AND VERGARA PHYSICAL REVIEW D 76, 025025 (2007)
essentially the same as that discussed in Refs. [4 –7]. For …! † ˆ fx ; x g
the beneﬁt of the readers, such a procedure is applied to the 0 1
examples contained in the next section. 0 0 0 1 0 0 0
B ÿ 0
B 0 0 0 1 0 0CC
Finally, for the sake of completeness, Noether’s theorem B C
is discussed. If the action (13) transforms (without using B 0
B 0 0 0 0 1 0CC
B 0 0 ÿ 1C
ˆB C; (28)
the equations of motion) at ﬁrst order as 0 0 0 0
B
B ÿ1 0 C
B 0 0 0 0 0 0CC
B 0 ÿ1
B 0 0 0 0 0 0CC
Z 2 dF B C
SB ˆ d (24) @ 0 0 ÿ1 0 0 0 0 0A
1 d 0 0 0 ÿ1 0 0 0 0
under the inﬁnitesimal transformations
where and are ﬁxed constant real parameters. The
difference between the two models comes from the con-
x0 …0 † ˆ x …† ‡ x ; 0a …0 † ˆ a …† ‡ a ; crete form for the constraints that the phase space variables
0 …0 † ˆ …† ‡ ; 0 ˆ ‡ ; (25) must satisfy in each case.
One of the lessons learned from these models is that
gauge-invariant actions can be built in spite of the fact that
L ~ 0
then the corresponding Noether’s condition x x ‡ the constraints are quadratic in the phase space variables
0 0 0
L ~ a L ~ d @L ~ 0 provided that an appropriate choice for the symplectic
a ‡ ‡ d …@x x ‡ L ÿ F † ˆ 0 with
_
~ dx
x ˆ x ‡ d (and so on for the other variables) potential is made.
0
and L ˆ … ÿ @x †x ÿ @B ÿ H ÿ a
a ÿ be-
@B
_ @
comes A. SL…2; R† model
This model was introduced in Ref. [12] in the context of
noncommutative quantum theory. Here, however, the
@HE ~ ~ ~
! x ÿ x ÿ
a a ÿ
_
model is interpreted as a usual gauge system of the type
@x
mentioned in Secs. I and II.
d @B ~ 0 ÿ F ˆ 0; (26) The phase space variables must satisfy the constraints
‡ ÿ x ‡ L
d @x
1
and so, if the equations of motion (6) and the ﬁrst- and C1 :ˆ ‰…p1 †2 ‡ …p2 †2 ÿ …v1 †2 ÿ …v2 †2 Š ÿ v1 2
2
second-class constraints are satisﬁed,
1
ÿ 2 …2 †2 0;
2
@B ~ 1
O ˆ ÿ x ÿ F C2 :ˆ ‰…1 †2 ‡ …2 †2 ÿ …u1 †2 ÿ …u2 †2 Š ÿ u1 p2
@x 2
@B @HE @B 1
‡ ÿ ! ÿ H ÿ (27) ÿ 2 …p2 †2 0;
@x @x @ 2
V :ˆ ui pi ÿ vi i ‡ p1 p2 ÿ 1 2 0; (29)
is constant along evolution in . It is understood that all
terms in the right-hand side of the last equation are eval-
uated on the surface of ﬁrst- and second-class constraints. with i ˆ 1, 2. It turns out that the constraints of Eq. (29) are
ﬁrst class with respect to the Poisson brackets computed
with the symplectic structure of Eq. (28). The resulting
III. EXAMPLES THAT INVOLVE NONCANONICAL algebra of constraints is
SYMPLECTIC STRUCTURES
In this section, the ideas developed in Sec. II are applied
to two nontrivial models. In both cases, the extended phase fC1 ; C2 g! ˆ V ; fC1 ; V g! ˆ ÿ2C1 ;
(30)
space is ÿ ˆ R8 , and its points are labeled by …x † ˆ fC2 ; V g! ˆ 2C2 ;
…x1 ; x2 ; . . . ; x8 † ˆ …u1 ; u2 ; v1 ; v2 ; p1 ; p2 ; 1 ; 2 †. The sym-
plectic geometry on ÿ ˆ R8 is given by the nondegenerate
closed 2-form ! ˆ 1 ! dx ^ dx ˆ dp1 ^ du1 ‡
2 which is isomorphic to the sl…2; r† Lie algebra.
dp2 ^ du2 ‡ d1 ^ dv1 ‡ d2 ^ dv2 ‡ dp1 ^ dp2 ‡ By plugging (28) and (29), …1 ; 2 ; 3 † ˆ …N; M; †, and
d1 ^ d2 . Equivalently, the inverse of the noncanoni- …
1 ;
2 ;
3 † ˆ …C1 ; C2 ; V † into Eqs. (6), the dynamical
cal symplectic structure is Eqs. (6) acquire the form
025025-4
46.
GAUGE INVARIANCE OF THE ACTION PRINCIPLE FOR . . . PHYSICAL REVIEW D 76, 025025 (2007)
u1 ˆ Np1 ÿ Mu2 ‡ …u1 ‡ 2p2 †;
_ under the boundary conditions
u2 ˆ Np2 ‡ u2 ;
_ …u1 ‡ p2 †… † ˆ U ;
1
u2 … † ˆ U ;
2
v1 ˆ M1 ÿ …v1 ‡ 22 † ÿ Nv2 ;
_ …v1 ‡ 2 †… † ˆ V ;
1
v2 … † ˆ V ;
2
ˆ 1; 2;
v2
_ ˆ M2 ÿ v2 ; (37)
1 2 1 2
p1 ˆ
_ M…u1 ‡ p2 † ÿ p1 ; (31) with U , U ,andV , speciﬁed real numbers.
V
The change of the action (36) under the ﬁnite gauge
p2 ˆ Mu2 ÿ p2 ;
_
transformation (33)–(35) is
1 ˆ N…v1 ‡ 2 † ‡ 1 ;
_
S‰x0 ; N 0 ; M0 ; 0 Š ˆ S‰x ; N; M; Š ‡
47.
…u p ‡ v
~ ~ ~ ~
2
2 ˆ Nv ‡ 2 :
_
‡ p1 p2 ‡ 1 2 †
Using (31), the evolution of the constraints (29) yields 1
_ _ ‡ …
†‰u u ‡ ‡ 2u1 p2
~ ~ ~ ~
C 1 ˆ MV ÿ 2C1 ; C2 ˆ ÿNV ‡ 2C2 ; 2
(32) 1
_
V ˆ ÿ2MC2 ‡ 2NC1 ; ‡ 2 …p2 †2 Š ‡ …
48.
†‰v v ‡ p p
~ ~ ~ ~
2
in agreement with the sl…2; r† Lie algebra of Eq. (30). ‡ 2v1 2 ‡ 2 …2 †2 Š; (38)
Gauge transformation. The ﬁnite gauge transformation
of the dynamical variables is and so the action is not invariant. Independently of this
0 01 1 0 10 1 1 property of the action (36), the choice of speciﬁc gauge
u ÿ
49.
… ÿ † u conditions could imply a gauge orbit whose end points at
B u02 C B 0
B CˆB 0
50.
CB u2 C
CB C; (33)
B 0 C B CB C 1 and 2 might additionally not satisfy the boundary
@ p1 A @
0
A@ p1 A conditions (37) already speciﬁed [4 –7]. For instance, the
p0
2 0
0 p2 gauge conditions
and C1 :ˆ u1 ‡ p2 ˆ 0; C2 :ˆ v1 ‡ 2 ˆ 0;
0 1 0 10 1 (39)
01
52.
CB 2 C
CB C; (34)
B 01 C B
@v A @
CB C
… ÿ † ÿ
53.
A@ v1 A with c3 a constant, are in general incompatible with the
v02 0
0 v2 boundary conditions (37). In fact, Eqs. (39) are good gauge
conditions in the sense that they ﬁx the Lagrange multi-
where ,
55.
ˆ 1, while the pliers to be N ˆ M ˆ ˆ 0 (and thus the dynamics is
Lagrange multipliers transform as ‘‘frozen’’). Moreover, the Poisson brackets between the
_
N 0 ˆ 2 N ÿ
62.
:
_ p2 0 u2
Notice that when the parameters and are turned off, 0 1
p1 0 0
ˆ 0 ˆ , the current model reduces to the SL…2; R† ˆ @ 0 1 0 A; (40)
model introduced in Ref. [13]. p2 0 c3
It is time to implement on this model the theory devel-
oped in Sec. II. This will be done in the following two where the gauge conditions (39) were used to get the
subsections by choosing different symplectic potentials second equality. The determinant of this matrix is
which amounts to choosing different boundary conditions. c3 p1 1 , and does not vanish for generic values of the
variables.
1. A noninvariant action The incompatibility between the gauge conditions (39)
and the boundary conditions (37) can be removed by
The equations of motion of the model (29) and (31) can, modifying both the action as well as the boundary con-
for instance, be obtained from the action principle ditions. The idea is to impose the gauge conditions in the
Z 2 gauge-transformed variables
S‰x ; N; M; Š ˆ d‰p1 …u1 ‡ p2 † ‡ p2 u2
_ _ _
1 u01 ‡ p0 2 ˆ 0; v01 ‡ 0 2 ˆ 0; u02 ˆ c3 Þ 0;
_1 _2
‡ 1 …v ‡ 2 † ‡ 2 v ÿ NC1
_ (41)
ÿ MC2 ÿ V Š (36) i.e., the gauge conditions retain their functional form in the
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63.
CUESTA, MONTESINOS, AND VERGARA PHYSICAL REVIEW D 76, 025025 (2007)
gauge-transformed variables. The substitution of (33) and In fact, a straightforward computation shows that
(34) into the left-hand side of (41) implies precise forms
Sinv ‰x0 ; N 0 ; M0 ; 0 Š ˆ Sinv ‰x ; N; M; Š (46)
for the gauge parameters:
c3 p1 under the gauge transformation (33)–(35). Therefore,
ˆÿ 1 ; Sinv ‰x ; N; M; Š is fully invariant because the noninvariant
…u ‡ p2 †p2 ÿ p1 u2
action (36) and the added noninvariant boundary term
c3 …u1 ‡ p2 † combine exactly to make Sinv ‰x ; N; M; Š strictly
64.
ˆ ;
…u1 ‡ p2 †p2 ÿ p1 u2 invariant.
(42) Equivalently, due to the fact the boundary term in (45) is
…v1 ‡ 2 †‰…u1 ‡ p2 †p2 ÿ p1 u2 Š
ˆÿ ; -independent, its contribution can be understood as a
c3 ‰…u1 ‡ p2 †…v1 ‡ 2 † ÿ p1 1 Š
different choice for the symplectic potential. By introduc-
1 ‰…u1 ‡ p2 †p2 ÿ p1 u2 Š ing this boundary term into the integrand,
ˆ :
c3 ‰…u1 ‡ p2 †…v1 ‡ 2 † ÿ p1 1 Š Z 2
Sinv ‰x ; N; M; Š ˆ d‰ x ÿ NC1 ÿ MC2 ÿ V Š;
_
On the other hand, by using (33) and (34), the unprimed 1
variables in the left-hand side of the boundary conditions (47)
(37) are replaced in terms of the primed variables and the
gauge parameters. In the expressions thus obtained, with
1
…u01 ‡ p0 2 ÿ
66.
p0 2 †… † ˆ U ;
2
ÿ …v1 ‡ 2 †d1 ‡ …1 ÿ v2 †d2 Š: (48)
0 02 2
…ÿ
2 ‡ v †… † ˆ V ;
That is to say, if the dynamical system were deﬁned by the
the parameters given in (42) must be inserted to obtain the action (47) from the very beginning, there would be no
right boundary conditions compatible with the gauge con- need to add a boundary term because such an action is
ditions and with the corresponding action already invariant under the gauge transformation, in com-
plete agreement with Eq. (23):
S‰x0 ; N 0 ; M0 ; 0 Š ÿ ÿ…
67.
†…u0 p0 ‡ v0 0 ‡ p0 1 p0 2
~ ~ ~ ~ @
a
! ÿ
a ˆ 0: (49)
1 @x
‡ 1 2 † ‡ …
†…u0 u0 ‡ 0 0 ‡ 2u01 p0 2
0 0 ~ ~ ~ ~
2 The action (45) or (47) yields the equations of motion
1 provided that the boundary conditions
‡ 2 …p0 2 †2 † ‡ …
68.
†…v0 v0 ‡ p0 p0
~ ~ ~ ~
2 1 2
1 u ‡ p2 1 u
ln … † ˆ Q1 ;
ln … † ˆ Q2 ;
‡ 2v01 0 2 ‡ 2 …0 2 †2 † ; (44) 2 p1 2 p2
1 2
1 v ‡ 2 1 v
which is (the original action S‰x ; N; M; Š but expressed ln … † ˆ Q3 ;
ln … † ˆ Q4 ;
2 1 2 2
in terms of the primed variables and it is) obtained from
(38) [7]. ˆ 1; 2 (50)
are satisﬁed.
2. An invariant action Like in the the case of the noninvariant action, the choice
As mentioned in Sec. II, there exists the possibility of of a particular gauge condition might lead to a solution of
adding a boundary term to the action (36) in such a way the equations of motion (a gauge orbit) whose values at the
that the resulting action is gauge invariant [7]. The simplest end points 1 and 2 might not match the speciﬁed values
action is in the right-hand side of the boundary conditions (50) (see
Z 2 Fig. 1). For instance, the gauge conditions
Sinv ‰x ; N; M; Š ˆ d‰p1 …u1 ‡ p2 † ‡ p2 u2
_ _ _
1 C1 :ˆ u1 ‡ p2 ˆ c1 Þ 0;
‡ 1 …v1 ‡ 2 † ‡ 2 v2 ÿ NC1
_ _ _ C2 :ˆ v1 ‡ 2 ˆ c2 Þ 0; (51)
1 C3 :ˆ u2 ÿ p2 ˆ 0;
ÿ MC2 ÿ V Š ÿ ‰…u1 ‡ p2 †p1
2
with c1 and c2 constants, are not compatible with the
‡ u p2 ‡ …v ‡ 2 †1 ‡ v2 2 Šj2 :
2 1
1
boundary conditions (50). These conditions are good gauge
(45) conditions in the sense that they ﬁx the Lagrange multi-
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GAUGE INVARIANCE OF THE ACTION PRINCIPLE FOR . . . PHYSICAL REVIEW D 76, 025025 (2007)
and (34), the original boundary conditions (50) are rewrit-
ten in terms of the primed variables and the gauge parame-
ters:
01 1
1 u ‡ p0 2 1
71.
… † ‡ … †e
FIG. 1. The path in the right-hand side (RHS) matches the ln … † ˆ ln ;
2 0 1 2 3
… † ‡
… †e2Q
gauge conditions, but its end points are incompatible with the
02 4
original boundary conditions for the action principle. Keeping 1 v 1
72.
… † ‡ … †e2Q
the gauge conditions (and so the path in the RHS) forces us to ln 0 … † ˆ ln 4 ;
modify the boundary conditions. The right modiﬁcation can be
2 2 2 … † ‡
… †e2Q
achieved by using the path in the left-hand side (LHS) which is
where ˆ 1, 2, which are by construction compatible with
obtained from the path in the RHS by a gauge transformation
that does not vanish at the end points. The path in the LHS is not
the gauge conditions. The corresponding action is the same
compatible with the gauge conditions, but its end points are one given in (45) or (47) but expressed in terms of the
compatible with the original boundary conditions. primed variables: Sinv ‰x0 ; M0 ; M0 ; 0 Š. Once the goal has
been achieved, one can simply drop the apostrophe ‘‘0 ’’
both in the action as well as in the boundary conditions to
pliers to be ˆ N ˆ M ˆ 0. Additionally, have an action principle written in the usual form.
0 1 The difference between the case of the noninvariant
p1 0 u1 ‡ p2
…fCa ;
b g! † ˆ B 0 1 ÿ…v1 ‡ 2 † C
action and the case of the fully gauge-invariant action
@ A
lies in the fact that in the latter there is no need to modify
p2 0 u2
the action, but just to handle the boundary conditions.
0 1
p1 0 c1
ˆ@ 0 1 ÿc2 A; (52) B. SU…2† model
p2 ÿp2 2p2
Now, a second model having an SU…2† gauge symmetry
where the gauge conditions were used to get the second is given. The dynamical variables must satisfy the con-
equality. The determinant of the matrix is p2 ‰p1 …21 ÿ straints
c2 † ÿ c1 1 Š.
Such a compatibility can be achieved by imposing the H1 :ˆ u1 u2 ‡ v1 v2 ‡ p1 p2 ‡ 1 2 ‡ u2 p2 ‡ v2 2 0;
gauge conditions in the gauge-related variables H2 :ˆ …u1 †2 ‡ …v1 †2 ‡ …p1 †2 ‡ …1 †2 ÿ …u2 †2 ÿ …v2 †2
u01 ‡ p0 2 ˆ c1 Þ 0; v01 ‡ 0 2 ˆ c2 Þ 0; ÿ …p2 †2 ÿ …2 †2 ‡ 2u1 p2 ‡ 2v1 2 ‡ 2 …p2 †2
(53)
u02 ÿ p0 2 ˆ 0; ‡ 2 …2 †2 0;
from which, using (33) and (34), the gauge parameters are D :ˆ u1 p2 ‡ v1 2 ÿ u2 p1 ÿ v2 1 ‡ …p2 †2 ‡ …2 †2 0:
ﬁxed: (57)
p2 ‰c1 …v1 ‡ 2 † ÿ c2 p1 Š The Poisson brackets computed with respect to the sym-
ˆ
‰…u1 ‡ p2 †p2 ÿ p1 u2 Š…v1 ‡ 2 † plectic structure of Eq. (28) among these constraints yield
p1 ‰u2 …v1 ‡ 2 † ÿ p2 1 Š
ÿ ; (54) fH1 ; H2 g! ˆ ÿ4D; fH1 ; Dg! ˆ H2 ;
‰…u1 ‡ p2 †p2 ÿ p1 u2 Š…v1 ‡ 2 † (58)
fH2 ; Dg! ˆ ÿ4H1 ;
c2 …u1 ‡ p2 †p2 ÿ c1 u2 …v1 ‡ 2 † and so they are ﬁrst class. The algebra turns out to be
73.
ˆ
‰…u1 ‡ p2 †p2 ÿ p1 u2 Š…v1 ‡ 2 † isomorphic to the su…2† Lie algebra. This can be easily
…u1 ‡ 2 †‰u2 …v1 ‡ 2 † ÿ p2 1 Š accomplished by rescaling the constraints J1 :ˆ H1 =2,
‡ ; (55) J2 :ˆ ÿH2 =4, and J3 :ˆ D=2, which satisfy fJi ; Jj g ˆ
‰…u1 ‡ p2 †p2 ÿ p1 u2 Š…v1 ‡ 2 †
c2 ÿ
1 ij k Jk with ijk the three-dimensional Levi-Civita symbol
ˆ 1 123 ˆ ‡1.
…v ‡ 2 †
Like in the noncommutative SL…2; R† model, when the
(recall that ÿ
74.
ˆ 1). On the other hand, using (33) noncommutative parameters are turned off, ˆ 0 ˆ ,
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75.
CUESTA, MONTESINOS, AND VERGARA PHYSICAL REVIEW D 76, 025025 (2007)
the resulting model still involves an SU…2† gauge symmetry.
By plugging (28) and (57), …1 ; 2 ; 3 † ˆ …N; M; †, and …
1 ;
2 ;
3 † ˆ …H1 ; H2 ; D† into Eqs. (6), the dynamical
equations become
u 1 ˆ N…p2 ‡ u1 ‡ 2 p2 † ‡ 2M…p1 ÿ u2 † ÿ …u2 ‡ p1 †;
_ u2 ˆ Np1 ÿ 2Mp2 ‡ …u1 ‡ p2 †;
_
v1 ˆ N…2 ‡ v1 ‡ 2 2 † ‡ 2M…1 ÿ v2 † ÿ …v2 ‡ 1 †;
_ v2 ˆ N1 ÿ 2M2 ‡ …v1 ‡ 2 †;
_
(59)
p1 ˆ ÿNu2 ÿ 2M…u1 ‡ p2 † ÿ p2 ;
_ p2 ˆ ÿN…u1 ‡ p2 † ‡ 2Mu2 ‡ p1 ;
_
1 ˆ ÿNv2 ÿ 2M…v1 ‡ 2 † ÿ 2 ;
_ 2 ˆ ÿN…v1 ‡ 2 † ‡ 2Mv2 ‡ 1 :
_
Gauge transformation. The ﬁnite gauge transformation where a, b, c, d 2 R with a2 ‡ b2 ‡ c2 ‡ d2 ˆ 1, while
of the phase space variables is the Lagrange multipliers transform as
X0 ˆ AX; Y 0 ˆ BY;
0 11 0 11
u v
B u2 C
B C; B v2 C
B C; (60)
XˆB C
@ p1 A YˆB C @ 1 A N 0 ˆ ‰1 ÿ 2…b2 ‡ c2 †ŠN ‡ 4…ab ‡ cd†M ‡ 2…ac ÿ bd†
p2 2 _ _ _
‡ ab ‡ da ÿ ad ÿ bc; _
where the matrix A and B are given by M0 ˆ ‰1 ÿ 2…b2 ‡ d2 †ŠM ÿ …bc ‡ ad† ‡ …cd ÿ ab†N
0 1
a ÿ d ÿb ‡ c ÿc ÿ b ÿd…1 ‡ 2 † 1 _ _
B b c ‡ b C ‡ …ca ‡ bd ÿ db ÿ ac†;
_ _
AˆB C;
a ÿd 2
B
@ c C
d a ÿb ‡ c A 0 ˆ ‰1 ÿ 2…c2 ‡ d2 †Š ‡ 4…ad ÿ bc†M ÿ 2…bd ‡ ac†N
d ÿc b a ‡ d
0 1 _ _
‡ ab ‡ cd ÿ ba ÿ dc:
_ _ (62)
a ÿ d ÿb ‡ c ÿc ÿ b ÿd…1 ‡ 2 †
B b c ‡ b C
BˆB C;
a ÿd
B
@ c C
d a ÿb ‡ c A
d ÿc b a ‡ d Dirac Observables. The following functions are invari-
(61) ant under the gauge transformation (60)–(62):
1 1
O 1 ˆ ‰…u1 †2 ‡ …u2 †2 ‡ …p1 †2 ‡ …p2 †2 Š ‡ u1 p2 ‡ 2 …p2 †2 ;
2 2
O2 ˆ u v ‡ u v ‡ p1 1 ‡ p2 2 ‡ u 2 ‡ v p2 ‡ …p2 †2 ;
1 1 2 2 1 1
O3 ˆ u1 v2 ÿ u2 v1 ‡ p2 1 ÿ p1 2 ‡ v2 p2 ÿ u2 2 ;
(63)
O4 ˆ u1 1 ‡ u2 2 ÿ v1 p1 ÿ v2 p2 ‡ p2 1 ÿ v1 2 ;
O5 ˆ u1 2 ÿ u2 1 ÿ v1 p2 ‡ v2 p1 ‡ p2 2 ÿ p2 2 ;
1 1
O6 ˆ ‰…v1 †2 ‡ …v2 †2 ‡ …1 †2 ‡ …2 †2 Š ‡ v1 2 ‡ 2 …2 †2 :
2 2
The Poisson brackets among them are 1. A noninvariant action
fO1 ; O2 g! ˆ O4 ; fO1 ; O3 g! ˆ ÿO5 ; Now, the effect of the gauge transformation in the action
principle will be analyzed. The equations of motion of the
fO1 ; O4 g! ˆ ÿO2 ; fO1 ; O5 g! ˆ O3 ; model (57) and (59) can, for instance, be obtained from the
fO2 ; O4 g! ˆ 2O1 ÿ 2O6 ; fO2 ; O6 g! ˆ O4 ; (64) action principle
Z 2
fO3 ; O5 g! ˆ 2O1 ‡ 2O6 ; fO3 ; O6 g! ˆ O5 ; S‰x ; N; M; Š ˆ d‰p1 …u1 ‡ p2 † ‡ p2 u2
_ _ _
1
fO4 ; O6 g! ˆ ÿO2 ; fO5 ; O6 g! ˆ ÿO3 : (65)
‡ 1 …v ‡ 2 † ‡ 2 v2 ÿ NH1
_1 _ _
A straightforward computation shows that this algebra of ÿ MH2 ÿ DŠ;
observables is isomorphic to the su…2† so…2; 1† Lie al-
gebra [14]. under the boundary conditions
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76.
GAUGE INVARIANCE OF THE ACTION PRINCIPLE FOR . . . PHYSICAL REVIEW D 76, 025025 (2007)
…u1 ‡ p2 †… † ˆ U ;
1
u2 … † ˆ U ;
2 @
a
! ÿ
a ˆ 0; (72)
@x
…v1 ‡ 2 †… † ˆ V ;
1
v2 … † ˆ V ;
2
ˆ 1; 2; (66) as expected.
The analysis of a possible incompatibility between the
1 2 1 2
with U , U , V , and V speciﬁed real numbers. boundary conditions
The change of the action (65) under the ﬁnite gauge
transformation (60)–(62) yields 1 2
1 u ‡ p2 1 u
ln … † ˆ Q1 ;
ln … † ˆ Q2 ;
…ac ‡ bd† 2 p1 2 p2
S‰x0 ; N 0 ; M0 ; 0 Š ˆ S‰x ; N; M; Š ‡ 1 2
2 1 v ‡ 2 1 v
ln … † ˆ Q3 ;
ln … † ˆ Q4 ;
‰H2 ‡ 2…p1 †2 ÿ 2…p †2 ‡ 2… †2
2 1 2 1 2 2
ÿ 2…2 †2 Š ‡ …ad ÿ bc†‰H1 ÿ 2p1 p2 ˆ 1; 2; (73)
ÿ 21 2 Š ÿ …c2 ‡ d2 †‰u
~ p‡v
~ ~ ~
‡ p1 p2 ‡ 1 2 Š; (67) of the action (68) or (70) with chosen gauge conditions can
be carried out along the same steps made for the case of the
SL…2; R† model.
and so the action (65) is not gauge invariant.
2. An invariant action IV. CONCLUDING REMARKS
Once again, it is possible to build gauge-invariant ac- The issue of the gauge invariance of the action principle
tions, the simplest of which is for Hamiltonian gauge systems whose extended phase
space is described in terms of arbitrary symplectic struc-
Z 2 tures has been studied assuming that the action is already in
Sinv ‰x ; N; M; Š ˆ d‰p1 …u1 ‡ p2 † ‡ p2 u2
_ _ _
1 a Hamiltonian form. One of the main results reported in
this paper is the fact that an action featuring ﬁrst-class
‡ 1 …v1 ‡ 2 † ‡ 2 v2 ÿ NH1
_ _ _
constraints quadratic in the phase space variables can be
1 strictly gauge invariant. The gauge invariance of the action
ÿ MH2 ÿ DŠ ÿ ‰…u1 ‡ p2 †p1
2 (13) can also be analyzed from a Lagrangian viewpoint. In
‡ u2 p2 ‡ …v1 ‡ 2 †1 ‡ v2 2 Šj2 : this last approach, the action (13) is assumed to have a
1
Lagrangian form and Dirac’s method is applied systemati-
(68) cally, i.e, one ﬁrst deﬁnes momenta canonically conjugate
to the variables x , a , and , increasing the number of
In fact, a straightforward computation using (60)–(62) variables that label the points of the extended phase space
shows that which is also equipped with a symplectic structure having,
by construction, the usual canonical form. If this approach
Sinv ‰x0 ; N 0 ; M0 ; 0 Š ˆ Sinv ‰x ; N; M; Š: (69) were followed, the gauge invariance of the resulting action
could be handled with the tools developed in Refs. [4 –7].
Therefore, Sinv ‰x ; N; M; Š is strictly invariant. Like in the
Sometimes, however, it is not convenient to enlarge the
case of the SL…2; R† model, introducing the boundary term
phase space, but to work the theory assuming that the
into the integrand
action already has a Hamiltonian form, and then the analy-
Z 2 sis of the gauge invariance of the action must be carried out
Sinv ‰x ; N; M; Š ˆ d‰ x ÿ NH1 ÿ MH2 ÿ DŠ;
_ along the ideas studied in this paper. (An example of the
1
Hamiltonian viewpoint can be found in the three-
(70) dimensional Chern-Simons theory.)
Finally, the results of this paper can also be used to
with extend the analyses developed in Refs. [15–17] by intro-
ducing new interactions through the symplectic structure.
1
ˆ ‰p1 du1 ‡ p2 du2 ‡ 1 dv1 ‡ 2 dv2
2 ACKNOWLEDGMENTS
ÿ …u1 ‡ p2 †dp1 ‡ …p1 ÿ u2 †dp2
J. D. V. acknowledges partial support from SEP-
ÿ …v1 ‡ 2 †d1 ‡ …1 ÿ v2 †d2 Š; (71) CONACYT Project No. 47211-F and DGAPA-UNAM
Grant No. IN109107. M. M. acknowledges ﬁnancial sup-
leads to the introduction of a new symplectic potential and ´
port from CONACyT, Mexico, Grant No. 56159-F.
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77.
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