Gauge Invariance Of The Action Principle For Gauge Systems With Noncanonical Symplectic Structures

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

2. ; (2a) f a ; g ˆ Ca b b ‡ Ca

3. ; (2b) @H a @ a @ x ˆ ! …x† _ ‡ ‡ fH; a g ˆ Va b b ‡ Va

4. ; (2c) @x @x @x b

5. @HE fH; g ˆ V b ‡ V

6. : (2d) ˆ ! …x† ; ; ˆ 1; 2; . . . ; 2N; (6) @x Also, where HE ˆ H ‡ a a ‡ is the extended f ;

7. g ˆ C

8. ; det…C

9. † Þ 0: (3) Hamiltonian and The Poisson brackets in Eqs. (2) and (3), a …x† 0; …x† 0; (7) @f @g @f @g ff; gg ˆ ÿ ; (4) are the constraints, which define the constraint surface @qi @pi @pi @qi embedded in the extended phase space ÿ. ÿ is a symplectic are computed with respect to the canonical symplectic manifold endowed with the symplectic structure structure 1 ! ˆ ! …x†dx ^ dx ; (8) ˆ dpi ^ dqi : (5) 2 The Hamiltonian action (1) is obtained from a Lagrangian where …x † are the coordinates that locally label the points of ÿ, which is considered as a single entity, i.e., ÿ need not *vcuesta@fis.cinvestav.mx be necessarily interpreted as the cotangent bundle of a † merced@fis.cinvestav.mx configuration space C. Now, instead of Eqs. (2) and (3), ‡ vergara@nucleares.unam.mx the following equations hold: 1550-7998= 2007=76(2)=025025(10) 025025-1 © 2007 The American Physical Society

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 ;

15. g! ˆ C

16. ; det…C

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 !

21. ; @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

22. † 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

23. † ‡ a …Ca b b duced through the noncanonical symplectic structure. ‡ Ca

24. † 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

25. 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

28. b appreciated. ˆ Tbc

29. ÿ Vb

30. ‡ Cb

31. ; (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

33. 2 1 @ a

34. ! …x† ÿ a

37. (23) ‡ … x ÿ G ÿ B†j2 1 @x

38. 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

40. the gauge parameters depend on only, if the gauge …x† @G ÿ G ÿ fB; Gg

42. transformation is allowed at the time boundaries, SB ˆ …x†! !

45. 1 @x a …2 † Þ 0 Þ a …1 †, then the boundary term can

47. vanish even for ﬁrst-class constraints a quadratic in @B…x; † @G

48. ˆ …x† ÿ ! …x† ÿ G

49. (22)

51. the variables x provided that an appropriate choice @x @x

52. 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

53. 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 symplectic 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

54. 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 ‡

55. …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 ‡ …

56. †‰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

57. … ÿ † u conditions could imply a gauge orbit whose end points at B u02 C B 0 B CˆB 0

58. 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

60. 0 1 C3 :ˆ u2 ˆ c3 Þ 0; B 0 C B 0 B 2CˆB 0

61. CB 2 C CB C; (34) B 01 C B @v A @ CB C … ÿ † ÿ

62. 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 ,

63. , , 2 R with ÿ

64. 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

65. 2 M ÿ 2

66. ‡

67. ÿ

68. ; _ gauge conditions and the ﬁrst-class constraints are 0 1 _ M0 ˆ ÿ 2 N ‡ 2 M ‡ 2 ‡ ÿ ; _ (35) p1 0 u1 ‡ p2 …fCa ; b g! † ˆ B 0 1 ÿ…v1 ‡ 2 † C @ A _ 0 ˆ ÿ N ‡

69. M ‡ … ‡

70. † ‡ ÿ

71. : _ 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 developed 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 025025-5

72. 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

73. 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

74. p01 †… † ˆ U ; 1 ˆ ‰p1 du1 ‡ p2 du2 ‡ 1 dv1 ‡ 2 dv2 2 …v01 ‡ 0 2 ÿ 0 1 †… † ˆ V ; 1 ÿ …u1 ‡ p2 †dp1 ‡ …p1 ÿ u2 †dp2 (43) …u0 2 ÿ

75. 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 complete agreement with Eq. (23): S‰x0 ; N 0 ; M0 ; 0 Š ÿ ÿ…

76. †…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 † ‡ …

77. †…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- 025025-6

78. 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 parameters: 01 1 1 u ‡ p0 2 1

79. … † ‡ … †e2Q ln … † ˆ ln ; 2 p0 1 2 1 … † ‡ … †e2Q 02 1 u ln 0 … † ˆ 0; 2 p2 (56) 01 0 2Q3 1 v ‡ 2 1

80. … † ‡ … †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

81. … † ‡ … †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

82. 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

83. ˆ 1). On the other hand, using (33) noncommutative parameters are turned off, ˆ 0 ˆ , 025025-7

84. 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 algebra [14]. under the boundary conditions 025025-8

85. 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 introducing 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. 025025-9

86. CUESTA, MONTESINOS, AND VERGARA PHYSICAL REVIEW D 76, 025025 (2007) [1] P. A. M. Dirac, Can. J. Math. 2, 129 (1950); 3, 1 (1951); [9] M. Montesinos and J. D. Vergara, in Membranes 2000, Proc. R. Soc. A 246, 326 (1958). Fourth Mexican School on Gravitation and Mathematical [2] P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer ´ ´ Physics, edited by N. Breton, J. S. Garcıa, and H. Quevedo Graduate School of Science, New York, 1964). ´ [Rev. Mex. Fıs. 49 S1, 53 (2003)]. [3] M. Henneaux and C. Teitelboim, Quantization of Gauge [10] C. Simeone, Deparametrization and Path Integral Systems (Princeton University Press, Princeton, 1992). Quantization of Cosmological Models (World Scientiﬁc, [4] M. Henneaux and J. D. Vergara, in Sakharov Memorial Singapore, 2001). Lectures in Physics, edited by L. V. Keldysh and V. Ya. ´ [11] J. A. Garcıa, J. D. Vergara, and L. F. Urrutia, Phys. Rev. D Fainberg (Nova Science Publishers, New York, 1992), 51, 5806 (1995). p. 111. [12] V. Cuesta, M. Montesinos, and J. D. Vergara, arXiv:hep-th/ [5] M. Henneaux, C. Teitelboim, and J. D. Vergara, Nucl. 0611333. Phys. B387, 391 (1992). [13] M. Montesinos, C. Rovelli, and T. Thiemann, Phys. Rev. D [6] C. Teitelboim, in Physical Origins of the Time Asymmetry, 60, 044009 (1999). ´ edited by J. J. Halliwell, J. Perez-Mercader, and W. H. [14] V. Cuesta, Ph.D. thesis, Cinvestav, 2007. Zurek (Cambridge University Press, Cambridge, [15] C. Rovelli, Phys. Rev. D 65, 124013 (2002). England, 1994), p. 442. [16] P. Singh and K. Vandersloot, Phys. Rev. D 72, 084004 [7] M. Montesinos and J. D. Vergara, Phys. Rev. D 65, 064002 (2005). (2002). [17] B. Dittrich and T. Thiemann, Classical Quantum Gravity [8] M. Montesinos and J. D. Vergara, Gen. Relativ. Gravit. 33, 23, 1089 (2006). 921 (2001). 025025-10

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