Cartans Equations Define A Topological Field Theory Of The Bf Type


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Cartans Equations Define A Topological Field Theory Of The Bf Type

  1. 1. PHYSICAL REVIEW D 76, 104004 (2007) Cartan’s equations define a topological field theory of the BF type Vladimir Cuesta* and Merced Montesinos† ´ ´ ´ Departamento de Fısica, Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional, ´ ´ ´ Avenida Instituto Politecnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de Mexico, Mexico (Received 6 August 2007; published 2 November 2007) Cartan’s first and second structure equations together with first and second Bianchi identities can be interpreted as equations of motion for the tetrad, the connection and a set of two-form fields T I and RI . J From this viewpoint, these equations define by themselves a field theory. Restricting the analysis to four- dimensional spacetimes (keeping gravity in mind), it is possible to give an action principle of the BF type from which these equations of motion are obtained. The action turns out to be equivalent to a linear combination of the Nieh-Yan, Pontrjagin, and Euler classes, and so the field theory defined by the action is topological. Once Einstein’s equations are added, the resulting theory is general relativity. Therefore, the current results show that the relationship between general relativity and topological field theories of the BF type is also present in the first-order formalism for general relativity. DOI: 10.1103/PhysRevD.76.104004 PACS numbers: 04.60.Ds, 04.20.Cv, 04.20.Fy graph, it is natural to ask if a relationship between general I. INTRODUCTION relativity and a topological field theory is already present in In the middle of the 1970’s, starting from the first-order the first-order formalism. It will be shown in this paper that formalism for general relativity, Plebanski wrote the equa- this relationship is indeed possible. To be precise, we tions of motion for four-dimensional general relativity in address and give an answer to the following questions: is such a way that there are no tetrad fields anymore in the it possible to see the relationship between general relativity resulting equations of motion and gave an action principle and a topological field theory in the first-order formalism? from which those equations can be obtained [1]. The new If this is possible, what is the topological theory, and how is action principle is a BF theory supplemented with con- general relativity related to it? straints on some of the fields involved. Since then, another The results of this paper follow from the very simple alternative Lagrangian formulations of the BF type for observation that Cartan’s equations and Bianchi identities general relativity have also been proposed, the difference can be interpreted as the equations of motion for the tetrad among them is the type of constraints on the fields [2 –7]. A eI , the Lorentz connection !I J , and the set of two-forms T I peculiar feature of ‘‘pure’’ BF theory (that where the fields and RI J . An action principle from which these equations of B’s are not constrained) is that it is topological in the sense motion are obtained is then given. The resulting theory is that it has no local degrees of freedom [8], while general of the BF type, where the fields RI J play the role of the relativity, of course, has them. Even though the relation- fields B’s in the part of the action that is the BF theory. ship between general relativity and BF theory is not com- Once the fields T I and RI J are eliminated from the action plicated technically, the conceptual link that connects the by means of their equations of motion, the action principle two theories seems to be deep. becomes a linear combination of the Nieh-Yan, Pontrjagin, In fact, this close relationship between these two theo- and Euler classes, from which the topological nature of the ries at Lagrangian level has increased the possibility of theory is clearly appreciated. The action principle is also quantizing general relativity using the techniques em- written in various, equivalent, ways. For the sake of com- ployed in the so-called spin foam models which are applied pleteness, an analysis of the Pontrjagin and Euler classes to quantum BF theory. The main difficulty faced by spin expressed as action principles of the BF type is also given, foam people is that currently there is not a clear under- and a similar analysis for the Nieh-Yan class. Finally, the standing of how (the elements that form) the building relationship of the topological theory with general relativ- blocks for the partition function for quantum BF theory ity is analyzed too. might be modified in order to capture the degrees of free- dom of a particular theory with local degrees of freedom, and particularly, those of quantum gravity [9,10]. II. CARTAN’S EQUATIONS On the other hand, in the first-order formalism for In the theoretical framework of the first-order formalism general relativity, there is a kinematical framework that for general relativity, the kinematical equations (namely, must be set before introducing Einstein’s equations of those equations before Einstein’s equations are introduced) motion [11]. So, from what is mentioned in the first para- are Cartan’s first structure equations [11], deI ‡ !I J ^ eJ ˆ T I ; (1) * † Cartan’s second structure equations, 1550-7998= 2007=76(10)=104004(6) 104004-1 © 2007 The American Physical Society
  2. 2. VLADIMIR CUESTA AND MERCED MONTESINOS PHYSICAL REVIEW D 76, 104004 (2007) d!I J ‡ !I K ^ !K J ˆ RI J ; (2) Similarly, Eq. (2) acquires the form the first Bianchi identities, ÿ @a !I J0 ‡ !I Ja ‡ !I K0 !K Ja ÿ !I Ka !K J0 ˆ RI J0a ; _ dT I ‡ !I J ^ T J ˆ RI J ^ eJ ; (3) @a !I Jb ÿ @b !I Ja ‡ !I Ka !K Jb ÿ !I Kb !K Ja ˆ RI Jab : and the second Bianchi identities, (6) dRI J ‡ !I K ^ RK J ÿ !K J ^ RI K ˆ 0: (4) In the same way, Eq. (3) is written as Even though Eqs. (1)–(4) hold for arbitrary frames in any _ ÿ@‰a T I j0jbŠ ‡ 1T I ab ‡ 1!I J0 T J ab manifold, in what follows the analysis will be restricted to 2 2 the case when !I J and RI J are valued in the Lie algebra of ÿ!I J‰a T J j0jbŠ ˆ RI J0‰a eJ bŠ ‡ 1RI Jab eJ 0 ; 2 SO…3; 1† or SO…4† and the manifold on which the fields live will be taken to be a four-dimensional one. @‰a T I bcŠ ‡ !I J‰a T J bcŠ ˆ RI J‰ab eJ cŠ : (7) As already mentioned, the viewpoint adopted in this paper is very simple but it is nontrivial. So, in order to Finally, Eq. (4) becomes make clear the main idea behind the current approach it _ will be convenient to emphasize that, strictly speaking, ÿ@‰a RI jJ0jbŠ ‡ 1RI Jab ‡ 1!I K0 RK Jab ÿ !I K‰a RK jJ0jbŠ 2 2 Eqs. (1) and (2) are not equations in the usual sense of ÿ1!K J0 RI Kab ‡ !K J‰a RI jK0jbŠ ˆ 0; 2 the word, but definitions for the torsion T I and the curva- ture RI J , i.e., the torsion T I and the curvature RI J are @‰a RI jJjbcŠ ‡ !I K‰a RK jJjbcŠ ÿ !K J‰a RI jKjbcŠ ˆ 0: (8) computed from the fields eI and !I J which in this sense play the role of ‘‘potentials’’ for T I and RI J . A much better Note that in Eqs. (5)–(8) only time derivatives of the fields notation for this fact would be to write T I ‰e; !Š and RI J ‰!Š eI a , !I Ja , T I ab , and RI Jab appear. Moreover, the equations rather than simply T I and RI J , a notation which is not of motion that involve no time derivatives only involve commonly used. Anyhow, to avoid misunderstandings these variables and their space derivatives. This strongly about the ideas developed along this paper, it is required suggests to consider eI a , !I Ja , T I ab , and RI Jab as the labels to clearly distinguish between T I ‰e; !Š; RI J ‰!Š; . . . and T I , for the points of the phase space and eI 0 , !I J0 , T I 0a , and RI J . . . . It is time to set up the main idea put forward in this RI J0a as Lagrange multipliers. More precisely, the equa- paper. It is formed by two parts: tions that involve no time derivatives might be considered (i) The first part of the proposal consists in considering as primary constraints in Dirac’s Hamiltonian formalism: all fields eI , !I J , T I , and RI J as independent coor- dinates that label the points of a certain phase space. CI ab :ˆ @a eI b ÿ @b eI a ‡ !I Ja eJ b ÿ !I Jb eJ a ÿ T I ab 0; (ii) The second part of the proposal consists in interpret- I Jab :ˆ @a !I Jb ÿ @b !I Ja ‡ !I Ka !K Jb ÿ !I Kb !K Ja ing the relations (1)–(4) among these fields as the equations of motion for a dynamical system which ÿ RI Jab 0; will be referred to as Cartan theory, Cartan dynam- CI abc :ˆ RI J‰ab eJ cŠ ÿ @‰a T I bcŠ ÿ !I J‰a T J bcŠ 0; ics, or Cartan fields from now on. The solutions of the equations of motion (1)–(4) define, by construc- I Jabc :ˆ @‰a RI jJjbcŠ ‡ !I K‰a RK jJjbcŠ ÿ !K J‰a RI jKjbcŠ 0: tion, the so-called space of solutions of the theory (9) [12 –14]. In order to explore the nature of the Cartan dynamics, it Note that the constraints are not independent in the sense will be very illuminating to make a 3 ‡ 1 decomposition of that they are linked through the reducibility equations the equations of motion. @‰a CI bcŠ ‡ !I J‰a CJ bcŠ ÿ CI abc ÿ I J‰ab eJ cŠ ˆ 0; 3 ‡ 1 decomposition of Cartan’s equations @‰a I jJjbcŠ ‡ !I K‰a K jJjbcŠ ÿ !K J‰a I jKjbcŠ ‡ I Jabc ˆ 0: To perform the 3 ‡ 1 decomposition of spacetime, it is assumed that spacetime manifold is of the form M4 ˆ (10) R. Let …x † ˆ …x0 ; xa †, a; b; c ˆ 1; 2; 3 be local coor- The next step is to evolve the constraints (9) to see if dinates that label the points of R. By hypothesis, is additional constraints arise. A straightforward computation given as the hypersurface x0 ˆ 0 and xa are local coordi- shows that no more constraints arise and that no one of the nates on . By writing eI ˆ eI 0 dx0 ‡ eI a dxa , !I J ˆ Lagrange multipliers is fixed [15]. This is a strong indica- !I J0 dx0 ‡ !I Ja dxa , Eq. (1) becomes tion of the fact that in a (suitable) Hamiltonian formulation ÿ @a eI 0 ‡ eI a ‡ !I J0 eJ a ÿ !I Ja eJ 0 ˆ T I 0a ; _ of these equations of motion all constraints are going to be (5) first class. With these elements at hand, a first guess is that @a eI b ÿ @b eI a ‡ !I Ja eJ b ÿ !I Jb eJ a ˆ T I ab : there are 104004-2
  3. 3. CARTAN’S EQUATIONS DEFINE A TOPOLOGICAL FIELD . . . PHYSICAL REVIEW D 76, 104004 (2007) eI a T I ab !I Ja RI Jab c1 , and c3 are assumed to be real numbers then these z‚‚‚}|‚‚‚{ z‚‚‚}|‚‚‚{ z‚‚‚}|‚‚‚{ z‚‚‚}|‚‚‚{ equations reduce to those given in (1)–(4). …4 3† ‡ …4 3† ‡ …6 3† ‡ …6 3† ˆ 60 (11) It is possible to go further and to eliminate some of the variables to label each point of the phase space, and that fields in the original action (13) to get alternative actions there are which are classically equivalent to the action (13): (1) If the two-form fields T I are eliminated from the CI ab I Jab CI abc I Jabc z‚‚‚}|‚‚‚{ z‚‚‚}|‚‚‚{ z‚‚‚}|‚‚‚{ z‚‚‚}|‚‚‚{ action (13) by using the equation DeI ˆ T I , then the …4 3† ‡ …6 3† ‡ …4 1† ‡ …6 1† ˆ 40 (12) resulting action is Z constraints, and …4 1† ‡ …6 1† ˆ 10 reducibility equa- S1 ‰e; !; RŠ ˆ c2 ‰…eI ^ eJ † ^ …d!IJ ‡ !IK ^ !K J † tions in Eq. (10). If all the constraints were first class (an hypothesis that is supported from the fact that no one of the Z ÿ DeI ^ DeI Š ‡ …c1 RIJ ‡ c3 RIJ † Lagrange multipliers is fixed during the evolution of the constraints), then the theory would have 1 ‰60 ÿ 2…40 ÿ 2 1 10†Š ˆ 0 local degrees of freedom and so it is topological. ^ d!IJ ‡ !IK ^ !K J ÿ RIJ : (18) 2 In next section, it is shown that this guess is correct. (2) If the two-form fields RI J are eliminated from the III. ACTION PRINCIPLE action (13) by using the equation RI J ˆ The equations of motion (1)–(4) can be obtained from d!I J ‡ !I K ^ !K J , then the resulting action is the action principle1 Z Z S2 ‰e; !; TŠ ˆ c2 ‰…eI ^ eJ † ^ …d!IJ ‡ !IK ^ !K J † S‰e; !; T; RŠ ˆ c2 ‰…eI ^ eJ † ^ …d!IJ ‡ !IK ^ !K J † ÿ 2TI ^ DeI ‡ TI ^ T I Š c Z I I ÿ 2TI ^ De ‡ TI ^ T Š Z ‡ 1 …d!IJ ‡ !I K ^ !KJ † 2 ‡ …c1 RIJ ‡ c3 RIJ † ^ …d!IJ ‡ !IL ^ !L J † ^ d!IJ ‡ !IK ^ ! JK ÿ 1R ; (13) c Z 2 IJ ‡ 3 IJKL …d!IJ ‡ !I K ^ !KJ † 4 with DeI :ˆ deI ‡ !I J ^ eJ and RIJ ˆ 1 IJ KL RKL . In 2 ^ …d!KL ‡ !K M ^ !ML †: (19) fact, the variation of the action (13) with respect to the independent fields yields (3) If both fields RI J and T I are eliminated from the eI : …d!I J ‡ !I K ^ !K J † ^ eJ ÿ DT I ˆ 0; (14) original action (13), then the resulting action is Z !IJ : ÿ c2 D…eI ^ eJ † ‡ c2 …T I ^ eJ ÿ T J ^ eI † S3 ‰e; !Š ˆ c2 ‰…eI ^ eJ † ^ …d!IJ ‡ !IK ^ !K J † ÿ c1 DRIJ ÿ c3 D RIJ ˆ 0; (15) ÿ DeI ^ DeI Š T I : DeI ÿ T I ˆ 0; c Z (16) ‡ 1 …d!IJ ‡ !I K ^ !KJ † 2 RIJ : c1 …d!IJ ‡ !I K ^ !KJ ÿ RIJ † ^ …d!IJ ‡ !IL ^ !L J † ‡ c3 …d!IJ ‡ !I K ^ !KJ ÿ RIJ † ˆ 0; (17) c Z ‡ 3 IJKL …d!IJ ‡ !I M ^ !MJ † 4 with DT I :ˆ dT I ‡ !I J ^ T J , D…eI ^ eJ † ˆ d…eI ^ eJ † ‡ ^ …d!KL ‡ !K P ^ !PL †: (20) !I K ^ …eK ^ eJ † ‡ !J K ^ …eI ^ eK †, and DRIJ ˆ dRIJ ‡ !I K ^ RKJ ‡ !J K ^ RIK , and so on. If the parameters c2 , Thus, the original action (13) is equivalent to a linear 1 Notice that the form of the action (13) is very similar to the combination of the Nieh-Yan [18,19], Euler, and action for the Maxwell theory studied in Ref. [16]. In fact, the Pontrjagin classes [11], and the field theory defined by form of the action for the Maxwell theory  R p analyzed there (13) is topological in the sense that it is a linear combina- is obtained from S‰A ;F Š ˆ M4 d4 x ÿg‰ÿ 1 …@ A ÿ2 tion of topological invariants and, according to the analysis @ A †F ‡ 1 F …F ‡ 82 ? F †Š by setting ˆ 0, g 4 taken to be the Minkowski metric, and M4 ˆ R4 . There are of the Sec. II, has no local excitations per point of space. similar constructions for topologically massive gauge theories Therefore, the field theory defined by the equations (1)–(4) [17]. is topological. Another important point that must be em- 104004-3
  4. 4. VLADIMIR CUESTA AND MERCED MONTESINOS PHYSICAL REVIEW D 76, 104004 (2007) phasized is that the action (13) is of the BF type with terms Z S6 ‰e; !Š ˆ f2 DeI ^ DeI ; (23) quadratic in the fields B’s. The BF part of the action is M4 precisely the second row in the right-hand side of (13), with the fields RIJ playing the role of the fields B’s. Note whose variation with respect to the independent fields also that the two types of BF theories allowed for SO…3; 1† yields the equations of the motion, or SO…4†, and recently studied [20,21], are involved in the eI : D…DeI † ˆ 0; !IJ : eI ^ DeJ ÿ eJ ^ DeI ˆ 0: action (13). For more details on the canonical analysis of (24) BF theories and their symmetries, see Refs. [8,22]. It is also worthwhile to mention that there is an alter- The first equation D…DeI † ˆ 0 is equivalent to …d!I J ‡ native way of showing the topological nature of the action !I K ^ !K J † ^ eJ ˆ 0 while the second is equivalent to (13) which consists in counting directly its number of local DeI ˆ 0 if the tetrad field is nondegenerate. Therefore, degrees of freedom by performing the canonical analysis of both actions (21) and (23) define the same dynamical the action (13). Such an analysis will also provide us as a system, as expected. Finally, notice that the action (23) is bonus the full symmetry of the theory and how it combines also equivalent to the action with local Lorenz invariance and diffeomorphism invari- Z 1 ance. This, however, is out of the scope of this paper. S7 ‰e; !; TŠ ˆ f2 TI ^ DeI ÿ TI ^ T I : (25) M4 2 Before concluding this section, and as an incidental fact, note that the Nieh-Yan class [i.e., the first term in (20)] can Even though both actions (21) and (23) arise from the be fragmented into two pieces which also define by them- Nieh-Yan class which is a topological field theory, there selves two field theories. The first of these theories is is not a priori a guarantee that these two theories are also defined solely by the term topological, i.e., when fragmenting an action for a topo- logical field theory, each of the pieces of the original action Z does not necessarily define a topological field theory. The S5 ‰e; !Š ˆ f1 …eI ^ eJ † ^ …d!IJ ‡ !IK ^ !K J †; M4 first term in (25) might be considered the non-Abelian (21) version of the action analyzed in Ref. [25]; more precisely, the action (25) depends functionally on the connection !I J added by Holst [23] to the Palatini action, whose variation while the action of Ref. [25] does not. with respect to the independent variables yields the equa- tions of motion IV. CHERN AND PONTRJAGIN CLASSES AS BF THEORIES eI : …d!I J ‡ !I K ^ !K J † ^ eJ ˆ 0; (22) The result of the previous section, namely, that Cartan !IJ : D…eI ^ eJ † ˆ 0: dynamics can be obtained from an action principle that is equivalent to a linear combination of the Pontrjagin, Euler, The equation D…eI ^ eJ † ˆ 0 reduces to DeI ˆ 0 if the and Nieh-Yan classes, can also be understood by analyzing fields eI dx are such that the matrix …eI † has a non- separately each one of these terms. The idea is the follow- vanishing determinant. ing. On the first place, it is known that the field theory As far as the authors know, the idea of considering a field defined by the Pontrjagin class, theory defined by the action (21) had not been proposed Z before. Of course that Palatini and Holst actions [Palatini SP ‰!Š ˆ P2 …d!IJ ‡ !I K ^ !KJ † action plus the action (21)] have been studied in detail: 8 they both are actions for general relativity. On the other ^ …d!IJ ‡ !IL ^ !L J †; (26) hand, the field theory defined by the action (21) is not general relativity, but a relative of it and might be consid- is topological. So, it is natural to ask whether or not the ered as uninteresting for this reason. Moreover, people action (26) can be written as a theory of the BF type, and working in loop quantum gravity see the Palatini action the answer is yes. The alternative action of the BF type for as the dish and the term added by Holst to it as the salt. the field theory defined by the action (26) is given by What we are saying here is that it is also allowed to P Z IJ L 1 consider action (21) as the dish itself. The fact that the S‰R; !Š ˆ 2 R ^ d!IJ ‡ !IL ^ ! J ÿ RIJ ; 4 2 Holst action is the starting point for quantizing gravity (27) from the canonical point of view gives us the right to ask for more details about the term added by him to the Palatini whose equations of motion are action. The Hamiltonian analysis of the action (21) is in DRIJ ˆ 0; d!IJ ‡ !I K ^ !KJ ˆ RIJ : (28) progress and will be reported soon [24]. The second theory is defined by the other term involved The first term in the action (27) is precisely a BF theory in the Nieh-Yan class, with RIJ playing the role of the fields B’s. That the action 104004-4
  5. 5. CARTAN’S EQUATIONS DEFINE A TOPOLOGICAL FIELD . . . PHYSICAL REVIEW D 76, 104004 (2007) (27) is equivalent to the action (26) comes from the fact tivity. This is accomplished by adding extra terms to the that the RIJ field can be eliminated from the action (27) by action principle (13), the resulting action, using the second equation of motion given in (28). By doing this, the action (26) is obtained. Z Similarly, the field theory defined by the Euler class, S‰e; !; T; RŠ ˆ c4 …eI ^ eJ † ^ d!IJ ‡ !IK ^ !K J Z Z SE ‰!Š ˆ E 2 IJKL …d!IJ ‡ !I M ^ !MJ † ÿ eI ^ eJ ‡ c2 ‰…eI ^ eJ † 32 6 ^ …d!KL ‡ !K P ^ !PL †; (29) ^ …d!IJ ‡ !IK ^ !K J † ÿ 2TI ^ DeI Z can be formulated as a BF theory too. The corresponding ‡ TI ^ T I Š ‡ …c1 RIJ ‡ c3 RIJ † action principle of the BF type is E Z IJ ^ d!KL ‡ !K K 1 ^ d!IJ ‡ !IK ^ ! J ÿ RIJ ; (33) S‰R; !Š ˆ IJKL R P 2 162 1 ^ !PL ÿ RKL ; (30) is not topological anymore, but rather it describes general 2 relativity, which can be appreciated from the variation of which implies the equations of motion the action (33) with respect to the variables eI , !I J , T I , and RI J . Another way to see that the action principle (33) gives D RIJ ˆ 0; d!IJ ‡ !I K ^ !KJ ˆ RIJ : (31) general relativity is to insert back into the action (33) two The first of these equations reduces to DRIJ ˆ 0. of the equations of motion that follow from its variation, Therefore, both Pontrjagin and Euler classes share the T I ˆ 0 and RI J ˆ d!I J ‡ !I K ^ !K J . By doing this, the same equations of motion. However, the symplectic ge- resulting action S‰e; !Š is the Holst action plus Pontrjagin ometry of the space of solutions induced by the action and Euler classes, and a cosmological constant, which is a principles is very different from one to another [20,21]. well-known action for general relativity [26]. Once again, by eliminating the RIJ field in the action (30) In summary, the results of this paper suggest that the by means of the second equation of motion in (31), the seed for the BF formulations of general relativity in terms action (29) is obtained. of two-forms and connections only, are already contained What about the field theory defined by the Nieh-Yan in the first-order formalism in the sense that gravity arises class? From the discussion of the previous section, it is also from a topological field theory by adding Einstein’s equa- clear that this theory can also be defined by means of the tions for general relativity to the original set of equations of following action principle: motion, very much in the spirit that general relativity can Z be obtained from a BF theory by adding to the BF action S‰e; !; TŠ ˆ NY ‰…eI ^ eJ † ^ …d!IJ ‡ !IK ^ !K J † principle suitable constraints on the fields. Finally, the canonical analysis of the action (33) con- ÿ 2TI ^ DeI ‡ TI ^ T I Š: (32) fronted with the canonical analysis of the action (13) would help us to understand how the large gauge symmetry of the This action is not of the BF type, but it is close to it due to theory (13) breaks down and reduces to the gauge symme- the presence of the term TI ^ DeI . try of general relativity. Now, the reader can clearly appreciate that the actions (27), (30), and (32) are the building blocks used to build the ACKNOWLEDGMENTS action (13). Part of this work was presented in the International Conference on Quantum Gravity LOOP’S 07 held in V. GENERAL RELATIVITY Morelia, Mexico, 2007. We appreciate the questions and Cartan’s equations (1)–(4) are the theoretical framework comments of the people in the parallel session, especially for the first-order formalism of general relativity and, as it those of Alejandro Perez and Carlo Rovelli. We also thank has been shown in this paper, equations of motion (1)–(4) R. Capovilla, G. F. Torres del Castillo, and J. D. Vergara for define a topological field theory whose action principle is very fruitful discussions on the subject. This work was given by (13). Nevertheless, the large symmetry of this BF supported in part by CONACyT, Mexico, Grant ´ theory must be broken down in order to get general rela- No. 56159-F. 104004-5
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