Topological Field Theories In N Dimensional Spacetimes And Cartans Equations
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Topological Field Theories In N Dimensional Spacetimes And Cartans Equations Topological Field Theories In N Dimensional Spacetimes And Cartans Equations Document Transcript

  • PHYSICAL REVIEW D 78, 064046 (2008) Topological field theories in n-dimensional spacetimes and Cartan’s equations Vladimir Cuesta,1,* Merced Montesinos,2,+ Mercedes Velazquez,2,‡ and Jose David Vergara1,x ´ ´ 1 ´ ´ ´ Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, 70-543, Ciudad de Mexico, Mexico´ 2 ´ ´ ´ Departamento de Fısica, Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional, ´ ´ Instituto Politecnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de Mexico, Mexico´ (Received 23 June 2008; revised manuscript received 11 August 2008; published 16 September 2008) Action principles of the BF type for diffeomorphism invariant topological field theories living in n-dimensional spacetime manifolds are presented. Their construction is inspired by Cuesta and Montesinos’ recent paper where Cartan’s first and second structure equations together with first and second Bianchi identities are treated as the equations of motion for a field theory. In opposition to that paper, the current approach involves also auxiliary fields and holds for arbitrary n-dimensional space- times. Dirac’s canonical analysis for the actions is detailedly carried out in the generic case and it is shown that these action principles define topological field theories, as mentioned. The current formalism is a generic framework to construct geometric theories with local degrees of freedom by introducing addi- tional constraints on the various fields involved that destroy the topological character of the original theory. The latter idea is implemented in two-dimensional spacetimes where gravity coupled to matter fields is constructed out, which has indeed local excitations. DOI: 10.1103/PhysRevD.78.064046 PACS numbers: 04.60.Ds, 04.20.Cv, 04.20.Fy I. INTRODUCTION With this in mind, it is natural to ask if the theoretical framework developed in Ref. [3] can be naturally extended There is a renewed interest in the study of BF theory and to arbitrary finite-dimensional spacetimes. The answer is in general relativity written as a constrained BF theory moti- the affirmative, this being one of the two main results vated by the progress of the spin foam [1] and loop quan- tum gravity [2] approaches to the nonperturbative and reported in this paper. In fact, inspired by those results, background-independent quantization of gravity. On one Cartan’s equations are supplemented with auxiliary fields hand, the topological nature and the diffeomorphism in- I , IJ , I , and IJ in such a way that the largest set variance of BF theory makes it a suitable laboratory to test of equations of motion define diffeomorphism invariant technical as well as conceptual issues related with classical topological field theories. The new theories living on and quantum gravity. On the other, there are still several n-dimensional spacetime manifolds contain pure BF the- issues connecting pure BF theory and BF gravity that de- ory having SOðmÞ or SOðm À 1; 1Þ structure groups as serve to be explored deeply. particular cases. It is important to emphasize that the In this context, it was recently proposed in Ref. [3] that auxiliary fields ’s and ’s are not involved in the frame- Cartan’s first and second structure equations together with work [3], this is a major difference between these first and second Bianchi identities can be interpreted as approaches. equations of motion for the tetrad, the connection, and a set Following the viewpoint of Ref. [3], it is also natural to of two-form fields T I and RI J . It was shown that these ask whether or not it is also possible to relate the topologi- equations define a topological field theory, which can be cal theories mentioned in the previous paragraph to known obtained from an action principle of the BF type. More- or new geometric theories having a nonvanishing number over, four-dimensional general relativity was obtained of local excitations by means of the introduction of geo- there by doing a suitable modification of the original action metric or algebraic relationships among the field variables principle that destroyed its topological character and at the involved. Once again the answer is in the affirmative, and same time allowed the degrees of freedom for gravity to the idea is explicitly implemented by building up a model arise. In this way, the results and the philosophy of the with local degrees of freedom in two-dimensional space- paper [3] is that Cartan’s equations encode a topological times starting from a topological field theory. This is the field theory and that the topological property of the theory second result of this paper. It is very interesting concep- disappears once Einstein’s equations are brought into the tually because it opens the possibility of applying the same framework. idea in n-dimensional spacetimes to try to get new formu- lations for gravity or to build suitable modifications of it that might be also worthwhile and interesting enough. *vladimir.cuesta@nucleares.unam.mx This paper is organized as follows: action principles + merced@fis.cinvestav.mx for diffeomorphism invariant field theories living in ‡ mquesada@fis.cinvestav.mx n-dimensional spacetime manifolds are given in Sec. II, x vergara@nucleares.unam.mx their canonical analyses are carried out and it is explicitly 1550-7998= 2008=78(6)=064046(10) 064046-1 Ó 2008 The American Physical Society
  • ´ CUESTA, MONTESINOS, VELAZQUEZ, AND VERGARA PHYSICAL REVIEW D 78, 064046 (2008) shown that the theories are topological; the two- nection ! . The other two auxiliary fields I and IJ ¼ IJ dimensional and three-dimensional cases are reported in À JI are (n À 3)-forms that will be associated to the the Appendices A and B, respectively. In Sec. III A it is ‘‘torsion’’ T I and the ‘‘curvature’’ RIJ , respectively. Note shown that by imposing suitable restrictions on the fields it that RI J is not the same as RI J ½!Š: RIJ is a set of two-forms is possible to build additional topological field theories while RI J ½!Š is the curvature of !I J (an analog comment from the original action principles, and in Sec. III B it is applies to T I , see Ref. [3] for more details). In this way, for shown that the theories introduced in Secs. II and III A n-dimensional spacetimes Mn with n ! 3, the field theo- are just particular members of the largest class of diffeo- ries studied in this paper are defined by the action principle morphism invariant topological field theories whose gauge group need not be the orthogonal group of the vielbeins. S½!I J ; eI ; T I ; RI J ; I ; IJ ; I ; IJ Š Finally, in Sec. IV, using the results of the previous Z ¼ ½I ^ ðdeI þ !I J ^ eJ À T I Þ þ IJ ^ ðd!IJ sections, a theory with local degrees of freedom which Mn lives on two-dimensional spacetimes and resembles two- þ !I K ^ !KJ À RIJ Þ þ ^ ðdT I þ !I J ^ T J I dimensional gravity plus matter fields is constructed out. This model can also be written as two interacting BF À RI J ^ eJ Þ þ IJ ^ ðdRIJ þ !I K ^ RKJ theories. The conclusions and perspectives are collected þ !J K ^ RIK ފ; (1) in Sec. V. where !IJ ¼ À!JI is a Lorentz (or Euclidean) connec- tion valued in the soðn À 1; 1Þ or soðnÞ Lie algebra, eI is a basis of one-forms, T I is a set of n two-forms, RIJ ¼ II. ACTION PRINCIPLES OF THE BF TYPE ÀRJI is a set of nðn À 1Þ=2 two-forms. The indices In the first part of this paper topological field theories of I; J; K; . . . , are raised and lowered with the Minkowski the BF type will be studied. The main goal is to build an ( ¼ À1) or Euclidean ( ¼ þ1) metric ðIJ Þ ¼ action that reproduces as field equations in n dimensions diagð; þ1; þ1; . . . ; þ1Þ (see Ref. [4] for the canonical Cartan’s first and second structure equations. To that end, analysis of BF theory with structure group SOð3; 1Þ, four different types of auxiliary fields are introduced. The Refs. [5,6] for alternative action principles for SOð3; 1Þ first one I is a (n À 2)-form that will be associated to the BF theory, and Ref. [7] for the study of its symmetries). basis of vielbeins eI ; the second one IJ ¼ ÀJI is also a The equations of motion that follow from the variation (n À 2)-form that will be associated to the Lorentz con- of the action (1) with respect to the independent fields are I : deI þ !I J ^ eJ À T I ¼ 0; IJ : d!IJ þ !I K ^ !KJ À RIJ ¼ 0;  I : dT I þ !I J ^ T J À RI J ^ eJ ¼ 0;  IJ : dRIJ þ !I K ^ RKJ þ !J K ^ RIK ¼ 0; T I : I þ ðÀ1ÞnÀ3 D I ¼ 0; RIJ : IJ þ ðÀ1ÞnÀ3 D IJ þ 1ð 2 I ^ eJ À J ^ eI Þ ¼ 0; ! : ðÀ1Þ IJ nÀ2 DIJ þ 2ðI 1 ^ eJ À J ^ eI Þ À 2ð I ^ 1 TJ À J ^ TI Þ À ð IK ^ RJ K À JK ^ RI K Þ ¼ 0; eI : ðÀ1Þ nÀ2 DI þ J ^ RJ I ¼ 0; (2) where D is the covariant derivative computed with respect with the corresponding interpretation for the fields in- to the connection !I J . volved: !I J is an soð2Þ or an soð1; 1Þ connection one- In two-dimensional spacetimes M2 , on the other hand, form, I and IJ are two and one 0-forms, respectively, only Cartan’s first and second structure equations are al- etc. Note that connection !I J involved in the action prin- lowed because there is no room for the first and second ciples (1) and (3) is not flat, its curvature is equal to the Bianchi identities, i.e., the terms involving the ’s in two-form field RIJ as it follows from the variation of the Eq. (1) are not allowed. Consequently, the natural action actions (1) and (3) with respect to IJ . principle is given by In order to count the number of degrees of freedom, the canonical analyses of the theories (1) and (3) are per- Z S½!I J ; eI ; T I ; RI J ; I ; IJ Š ¼ ½I ðdeI þ !I J ^ eJ formed. Let ðx Þ ¼ ðx0 ; xa Þ ¼ ðx0 ; x1 ; . . . ; xnÀ1 Þ be local M2 coordinates on Mn , which is assumed to be of the form À T Þ þ IJ ðd!IJ I Mn ¼ S  R; the coordinate time x0 labels the points along R and the space coordinates xa label the points on þ !I K ^ !KJ À RIJ ފ; (3) S, which is assumed to have the topology of SnÀ1 . The 064046-2
  • TOPOLOGICAL FIELD THEORIES IN n-DIMENSIONAL . . . PHYSICAL REVIEW D 78, 064046 (2008) canonical analyses of the field theories on M and M are 2 3 ordinate the extended phase space. The expressions for the explicitly carried out in the Appendices A and B, respec- momenta in terms of the original Lagrangian variables are tively. Even though the case on M2 is relevant because it does not involve the fields I and IJ , it is much more 1 a I :¼ "0ab1 ...bnÀ2 Ib1 ...bnÀ2 ; interesting to see the changes in the canonical analysis in ðn À 2Þ! the chain M2 ! M3 ! M4 ! M5 ! Á Á Á . As will be 1 clear in the lines below, the canonical analyses of the a IJ :¼ "0ab1 ...bnÀ2 IJb1 ...bnÀ2 ; ðn À 2Þ! theory for n ! 5 are very similar to the structure in n ¼ (5) 4, with minor changes. However, the Hamiltonian descrip- 1 ðÀ1ÞnÀ3 0abc1 ...cnÀ3 Åab I :¼ " Ic1 ...cnÀ3 ; tions in n ¼ 2 and in n ¼ 3 are very different from the 2 ðn À 3Þ! cases n ! 4, that is why theses cases are explicitly reported 1 ðÀ1ÞnÀ3 0abc1 ...cnÀ3 in the appendices. Moreover, the two-dimensional case Åab IJ :¼ " IJc1 ...cnÀ3 ; 2 ðn À 3Þ! will be very useful to introduce local degrees of freedom (see Sec. IV). Thus, the canonical analysis for the action while the ones for the Lagrange multipliers are: I ¼ (1) on M4 and higher-dimensional spacetimes is given in ÀeI 0 , IJ ¼ À!IJ 0 , ÃI a ¼ T I 0a , ÃIJ a ¼ RIJ 0a , and what follows. The Hamiltonian form of the action (1) is obtained through Dirac’s method [8] 1 1 Z uI ab ¼ À "0abc1 ...cnÀ3 I0c1 ...cnÀ3 ; 2 ðn À 3Þ! S ¼ ½a I eI a þ a IJ !IJ a þ Åab I T I ab þ Åab IJ RIJ ab _ _ _ _ 1 1 uIJ ab ¼ À "0abc1 ...cnÀ3 IJ0c1 ...cnÀ3 ; À H Šdn x; 2 ðn À 3Þ!  1 0abc H ¼ I gI þ IJ GIJ þ ÃI a da I þ ÃIJ a Da IJ þ uI ab CI ab " if n ¼ 4; vI abc ¼ 2 1 I0 1 ...dnÀ4 1 "0abcd if n ! 5; þ uIJ ab IJ ab þ vI abc CI abc þ vIJ abc IJ abc ; (4) 2 ðnÀ4Þ! I0d1 ...dnÀ4  1 0abc À2" if n ¼ 4; where H is the extended Hamiltonian [9]. From this ex- vIJ abc ¼ IJ0 À 1 ðnÀ4Þ! "0abcd1 ...dnÀ4 IJ0d1 ...dnÀ4 if n ! 5; 1 pression it follows that the canonical pairs are: ðeI a ; b J Þ, 2 ð!IJ a ; b KL Þ, ðT I ab ; Åcd J Þ, and ðRIJ ab ; Åcd KL Þ, which co- (6) which impose the constraints gI :¼ Da a I À Åab J RJ Iab % 0; GIJ :¼ Da a IJ þ 1ða I eJa À a J eIa Þ þ 1ðÅab I TJab À Åab J TIab Þ þ Åab IK RJ K ab À Åab JK RI K ab % 0; 2 2 da I :¼ a I þ 2Db Åab I % 0; Da IJ :¼ a IJ þ 2Db Åab IJ þ Åab I eJb À Åab J eIb % 0; (7) CI ab :¼ Da eI b À Db eI a À T I ab % 0; IJ ab :¼ @a !IJ b À @b !IJ a þ !I Ka !KJ b À !I Kb !KJ a À RIJ ab % 0; CI abc :¼ RI J½ab eJ cŠ À D½a T I bcŠ % 0; IJ abc :¼ D½a RIJ bcŠ % 0; where D is the covariant derivative compute with respect to !I Ja . A straightforward computation shows that the evolution of the constraints (7) gives no additional ones. The constraints are smeared with test fields Z Z Z Z Z gðaÞ ¼ aI gI ; GðUÞ ¼ UIJ GIJ ; dðbÞ ¼ baI daI ; DðVÞ ¼ V aIJ DaIJ ; fðcÞ ¼ cIab CIab ; Z Z Z (8) hðdÞ ¼ d IJab IJab ; kðfÞ ¼ f Iabc CIabc ; lðgÞ ¼ g IJabc IJabc ; to compute their Poisson algebra 064046-3
  • ´ CUESTA, MONTESINOS, VELAZQUEZ, AND VERGARA PHYSICAL REVIEW D 78, 064046 (2008) fDðVÞ; gðaÞg ¼ dðV Á aÞ; ðV Á aÞaK ¼ VaKL aL ; fGðUÞ; gðaÞg ¼ gðU Á aÞ; ðU Á aÞK ¼ UKL aL ; fGðUÞ; dðbÞg ¼ dðU Á bÞ; ðU Á bÞaJ ¼ UJK baK ; fGðUÞ; DðVÞg ¼ Dð½U; VŠÞ; ½U; VŠaKJ ¼ UK L VaLJ À UJ L VaLK ; fGðU1 Þ; GðU2 Þg ¼ Gð½U1 ; U2 ŠÞ; ½U1 ; U2 ŠIK ¼ U1I L U2LK À U1K L U2LI ; fgðaÞ; fðcÞg ¼ hðc Á aÞ; ðc Á aÞIJab ¼ 1ðcIab aJ À cJab aI Þ; 2 fGðUÞ; fðcÞg ¼ fðU Á cÞ; ðU Á cÞJab ¼ UJ K cKab ; fGðUÞ; hðdÞg ¼ hð½U; dŠÞ; ½U; dŠJLab ¼ UJ K dKLab À UL K dKJab ; fgðaÞ; kðfÞg ¼ lð½a; fŠÞ; ½a; fŠIJabc ¼ 1½aI fJabc À aJ fIabc Š; 2 fGðUÞ;kðfÞg ¼ kðU Á fÞ; ðU Á fÞJabc ¼ UJ K fKabc ; fGðUÞ; lðgÞg ¼ lð½U; gŠÞ; ½U; gŠJLabc ¼ UJ K gKLabc À UL K gKJabc ; fDðVÞ; kðfÞg ¼ fðV Á fÞ; ðV Á fÞIcd ¼ VbI J fJbcd ; flðgÞ; DðVÞg ¼ hð½V; gŠÞ; ½V; gŠILcd ¼ VbI K gKLbcd À VbL K gKIbcd ; fkðfÞ; dðbÞg ¼ hð½b; fŠÞ; ½b;fŠIJcd ¼ 1½bb I fJbcd À bb J fIbcd Š; 2 (9) and the Poisson brackets that are not listed vanish strongly. ia1 a2 ...anÀ3 % 0; Therefore, all the constraints in Eq. (7) are first class. If a # naive counting of the number of local degrees of freedom ranÀ3 ia1 a2 ...anÀ3 ¼ 0; were made without taking into account the reducibility of # (12) the constraints (7), the outcome would be a negative num- . . ber. In fact, as it is explained below, the reducibility pat- . tern for the constraints (7) is essentially the same as the # one for the reducible constraints for pure BF theory in ra1 Á Á Á ranÀ3 ia1 a2 ...anÀ3 ¼ 0; n-dimensional spacetimes. Therefore, before making the which are totally antisymmetric in the free indices. With analysis of the reducibility of the first-class constraints the help of the diagram (12), the counting of the number for the current theory, it will be convenient to recall of local degrees of freedom for pure BF theory is straight- the corresponding analysis for pure BF theory living on forward. The key point to do that is to realize that the n-dimensional spacetimes Mn for n ! 4 [10]. In that case, equations in the first row in Eq. (12) correspond to the first- the reducible first-class constraints are class constraints, the equations in the second row corre- spond to reducibility equations for the equations in the first row, however, these reducibility equations are also not "0a1 a2 ...anÀ3 anÀ2 anÀ1 Fi anÀ2 anÀ1 ðAÞ % 0; (10) independent among themselves, they are linked by the i ¼ 1; . . . ; dimðgÞ; equations in the third row, and the equations in the third row are also not independent among themselves because they are linked by the ones in the fourth row, and so on until where ð1 Fi ab ðAÞdxa ^ dxb Þ  Ji is the curvature of the 2 reaching the last row. Therefore, the number of indepen- connection one-form ðAi a dxa Þ  Ji , Ji are the generators dent first-class constraints in Eq. (10) is equal to the num- of the Lie algebra g and satisfy ½Ji ; Jj Š ¼ ck ij Jk . Nev- ber of equations in the first row in Eq. (12) minus the ertheless, the constraints (10) are not independent among number of equations in the second row plus the number themselves because the following chain of equations of equations in the third row minus the number of equations in the fourth row plus . . ., etc., alternating the sign in the terms of the series until counting the number of equations "0a1 a2 ...anÀ3 anÀ2 anÀ1 Fi anÀ2 anÀ1 ðAÞ % 0; in the last row with its corresponding sign. The result is # that the number of independent first-class constraints in "0a1 a2 ...anÀ3 anÀ2 anÀ1 ranÀ3 Fi anÀ2 anÀ1 ðAÞ ¼ 0; Eq. (10) is ðn À 2Þ Â dimðgÞ which must be added to # dimðgÞ equations contained in the Gauss law. Conse- (11) . quently, the total number of independent first-class con- . . straints is ðn À 1Þ Â dimðgÞ so that the number of local # degrees of freedom in the configuration space is 1 f2½ðn À 2 "0a1 a2 ...anÀ3 anÀ2 anÀ1 ra1 Á Á Á ranÀ3 Fi anÀ2 anÀ1 ðAÞ ¼ 0; 1Þ Â dimðgފ À 2½ðn À 1Þ Â dimðgފg ¼ 0, which means that the theory is topological. So much for pure BF theory. Coming back to the theory studied in this paper, here it is can be obtained from them through the application of the also required to know the number of independent first-class internal covariant derivative ra or, equivalently, renaming constraints in Eq. (7). This number can be obtained by first last expressions noting that the number of independent first-class con- 064046-4
  • TOPOLOGICAL FIELD THEORIES IN n-DIMENSIONAL . . . PHYSICAL REVIEW D 78, 064046 (2008) a a straints in gI , GIJ , d I , and D IJ is simply equal to the "0a1 ...anÀ2 anÀ1 CI anÀ2 anÀ1 % 0; number of these constraints minus their number of reduc- ibility equations NðgI Þ þ NðGIJ Þ, "0a1 ...anÀ2 anÀ1 I JanÀ2 anÀ1 % 0; (15) NðgI Þ þ NðGIJ Þ þ Nðda I Þ þ NðDa IJ Þ À NðgI Þ À NðGIJ Þ "0a1 ...anÀ3 anÀ2 anÀ1 CI anÀ3 anÀ2 anÀ1 % 0; ¼ Nðda I Þ þ NðDa IJ Þ; (13) "0a1 ...anÀ3 anÀ2 anÀ1 I JanÀ3 anÀ2 anÀ1 % 0; where NðgI Þ denotes the number of equations in gI and so or, renaming (15) on; the reducibility equations among gI , GIJ , da I , and Da IJ are obtained by applying the operator Da to the constraints ÈIa1 ...anÀ3 % 0; ÈI J a1 ...anÀ3 % 0; da I and Da IJ . What remains is to know the number of in- (16) dependent first-class constraints among the remaining ones ÉIa1 ...anÀ4 % 0; ÉI J a1 ...anÀ4 % 0; CI ab % 0; I Jab % 0; which are totally antisymmetric in the free indices. The (14) constraints (16) behave as constraints (10) do for BF the- C I % 0; I % 0; abc Jabc ory, i.e., repeatedly applying the operator Da to Eqs. (16), which are equivalent to a chain of reducibility equations arises ÈIa1 ...anÀ3 % 0; ÈI J a1 ...anÀ3 % 0; ÉIa1 ...anÀ4 % 0; ÉI J a1 ...anÀ4 % 0; ÈIa1 ...anÀ4 ¼ 0; ÈI J a1 ...anÀ4 ¼ 0; ÉIa1 ...anÀ5 ¼ 0; ÉI J a1 ...anÀ5 ¼ 0; . . . . . . . . . . . . ÈIa1 ...anÀ3Àk ¼ 0; ÈI J a1 ...anÀ3Àk ¼ 0; ÉIa1 ...anÀ4Àk ¼ 0; ÉI J a1 ...anÀ4Àk ¼ 0; . . . . . . . . . . . . ÈIa1 a2 ¼ 0; ÈI J a1 a2 ¼ 0; ÉIa1 ¼ 0; ÉI J a1 ¼ 0; ÈIa1 ¼ 0; ÈI J a1 ¼ 0; ÉI ¼ 0; ÉI J ¼ 0; ÈI ¼ 0; ÈI J ¼ 0: By using this chain, it is possible to make the counting of independent first-class constraints in Eq. (14) by adding (with the corresponding sign) the numbers contained in each row of the following diagram: ðþÞ0 NðÈIa1 ...anÀ3 Þ; NðÈI J a1 ...anÀ3 Þ; NðÉIa1 ...anÀ4 Þ; NðÉI J a1 ...anÀ4 Þ; ðÀÞ1 NðÈIa1 ...anÀ4 Þ; NðÈI J a1 ...inÀ4 Þ; NðÉIa1 ...anÀ5 Þ; NðÉI J a1 ...anÀ5 Þ; . . . . . . . . . . . . . . . ðÀÞk NðÈIa1 ...anÀ3Àk Þ; NðÈI J a1 ...anÀ3Àk Þ; NðÉIa1 ...anÀ4Àk Þ; NðÉI J a1 ...anÀ4Àk Þ; . . . . . . . . . . . . . . . ðÀÞnÀ5 NðÈIa1 a2 Þ; NðÈI J a1 a2 Þ; NðÉIa1 Þ; NðÉI J a1 Þ; ðÀÞnÀ4 NðÈIa1 Þ; NðÈI J a1 Þ; NðÉI Þ; NðÉI J Þ; ðÀÞnÀ3 NðÈI Þ; NðÈI J Þ: Taking into account the signs, it is clear that the third and of independent first-class constraints in Eq. (7) is Nðda I Þ þ fourth numbers of the first row are canceled by the first and NðDa IJ Þ þ NðCI ab Þ þ Nð IJ ab Þ. Therefore, the number of second numbers of the second row, the third and fourth local degrees of freedom in the configuration space is numbers of the second row are canceled by the first and 2 f2½Nðe a Þ þ Nð!IJ a Þ þ NðT I ab Þ þ NðRIJ ab ފ second terms of the third row, etc.; in such a way that the 1 I two numbers of the last row are canceled by the third and fourth terms of the penultimate row. Therefore, the num- À 2½Nðda I Þ þ NðDa IJ Þ þ NðCI ab Þ þ Nð IJ ab ފg ¼ 0; ber of independent first-class constraints in the set (14) (17) is simply NðÈIa1 ...anÀ3 Þ þ NðÈI J a1 ...anÀ3 Þ ¼ NðCI ab Þ þ Nð I Jab Þ, i.e., the information needed to get the right so the theory defined by the action (1) is topological. Note number of independent constraints is encoded in the num- that the number of variables in eI a , denoted by NðeI a Þ, is ber of the constraints CI ab and I Jab only. Using this result equal to the number of equations in dI a , denoted by NðdI a Þ, and the previous one it is concluded that the total number and so on. 064046-5
  • ´ CUESTA, MONTESINOS, VELAZQUEZ, AND VERGARA PHYSICAL REVIEW D 78, 064046 (2008) Finally, for spacetimes M with n ! 3, it is possible to n gI :¼ Da a I % 0; use the equations of motion contained in the fifth and sixth rows of Eq. (2) and insert back into the action (1) the GIJ :¼ Da a IJ þ 1ða I eJa À a J eIa Þ % 0; 2 expressions for I and IJ in terms of the fields !I J , I , CI ab :¼ Da eI b À Db eI a % 0; I IJ , and e to seek for an equivalent form for the action principle. However, this leads to the following result: IJ ab :¼ @a !IJ b À @b !IJ a þ !I Ka !KJ b À !I Kb !KJ a Z % 0: (21) ðÀ1ÞnÀ3 d½ I ^ ðT I À DeI Þ Mn A straightforward computation shows that the algebra of þ IJ ^ ðR À R ð!Þފ: IJ IJ (18) constraints closes so that the constraints (21) are first class. Alternatively, from the constraint algebra of the Hamilto- The fact that the Lagrangian n-form of the action (1) can be nian analyses for the two-dimensional, three-dimensional, written as the differential of a (n À 1)-form is, in a certain and the generic theory developed in Appendices A and B, sense, analog to the fact that products of curvature of type and in Sec. II, respectively, it follows that the constraints FðAÞ wedge FðAÞ are equal to the differential of the Chern- da I , Da IJ , CI abc , and IJ abc can be dropped from the Ham- Simons Lagrangians. The difference between them and the iltonian form (4) in such a way that the constraints of the current theory lies in the fact that in those cases the result is smaller set also close, so they are first-class constraints obtained without using any equations of motion while here too. The constraints of Eq. (21) are irreducible for two- some equations of motion were used to get (18). dimensional and three-dimensional spacetime manifolds while the constraints CI ab and IJ ab become reducible for n ! 4. Following the same procedure made for the III. PARTICULAR TOPOLOGICAL FIELD analysis of the reducibility of the constraints performed THEORIES AND GENERALIZATIONS in Sec. II it follows that the number of independent first- A. Particular theories class constraints among CI ab and IJ ab is ðn À 2Þ Â By plugging T I ¼ 0 and RIJ ¼ 0 into the Lagrangian NðgI Þ þ ðn À 2Þ Â NðGIJ Þ. On the other hand, the con- action (1) leads to the field theory straints gI and GIJ are always irreducible. Thus, the total number of independent first-class constraints in (21) is Z S½!I J ; eI ; I ; IJ Š ¼ ½I ^ ðdeI þ !I J ^ eJ Þ ðn À 2Þ Â NðgI Þ þ ðn À 2Þ Â NðGIJ Þ þ NðgI Þ þ NðGIJ Þ Mn þ IJ ^ ðd!IJ þ !I K ^ !KJ ފ; (19) ¼ ðn À 1Þ Â NðgI Þ þ ðn À 1Þ Â NðGIJ Þ; (22) which is well defined for spacetime manifolds Mn with which implies that the number of local degrees of freedom n ! 2. Note that this action principle contains also BF the- in the configuration space is ory with SOðnÞ or SOðn À 1; 1Þ structure groups as par- 2f2½Nðe a Þ þ Nð!IJ a ފ À 2½ðn À 1Þ Â NðgI Þ 1 I ticular cases. Using the results of Sec. II, it is easy to show that this action defines a topological field theory too.1 The þ ðn À 1Þ Â NðGIJ ފg ¼ 0; (23) Hamiltonian form for the action (19), obtained through Dirac’s method, is which means that the theory (19) is topological. Z S¼ ½a I eI a þ a IJ !IJ a À H Šdn x; _ _ B. Generalizations (20) In the action principles (1), (3), and (19), studied in H ¼ I gI þ IJ GIJ þ uI ab CI ab þ uIJ ab IJ ab ; Secs. II and III A, the group of local orthogonal rotations was taken to be SOðnÞ or SOðn À 1; 1Þ, i.e., the one that where now corresponds naturally to n-dimensional spacetime mani- folds. Nevertheless, it follows immediately from their ca- 1 As far as we know, this is the first time the theory (19) is nonical analyses that there is no need of restricting the reported. With respect to this, in Ref. [11] an action principle for analysis to that group in spite of the fact that the theories a two-dimensional theory was reported in Eq. (A1), which contains the action (19) as part of the action reported there. are defined on n-dimensional spacetimes. In fact, the group Nevertheless, there the idea was to report the action (A1) as a can be SOðmÞ or SOðm À 1; 1Þ with m Þ n. If this were whole thing defining a two-dimensional topological field theory allowed, the counting of the degrees of freedom would be and an analysis of their parts was not carried out. More precisely, left unaltered and the various theories would remain topo- the action (19) was not studied by itself (alone) in Ref. [11]. logical, the challenge would be the interpretation of the Furthermore, it was not realized there that the action (19) was topological as is done in this paper. Also, action (19) is defined various fields involved only (see Sec. IV). This feature is for spacetimes with dimension 2 and higher, not just two- indeed very interesting and has implications on at least the dimensional ones. following two issues: 064046-6
  • TOPOLOGICAL FIELD THEORIES IN n-DIMENSIONAL . . . PHYSICAL REVIEW D 78, 064046 (2008) (1) First of all, it is well known that the freedom in this way of generating local degrees of freedom is right the choice of the dimension of the group SOðmÞ from the Hamiltonian viewpoint, the challenge is that this or SOðm À 1; 1Þ can be used to introduce ‘‘matter procedure is covariant in the sense that it is associated to a fields’’ that will interact with those degrees of free- Lagrangian action principle. dom naturally living on n-dimensional spacetime In what follows it is shown that this is indeed possible manifolds (see Sec. IV for a concrete implementa- and it is illustrated in two-dimensional spacetimes using tion of this idea, for instance). the results contained in Appendix A. More precisely, this (2) On the other hand, if m > n then the theories studied idea is implemented through the following steps: along this paper can be naturally coupled to other (1) Take the Hamiltonian action principle given in theories living on higher-dimensional spacetime Eq. (A2). manifolds Mm having SOðmÞ or SOðm À 1; 1Þ (2) Take SOð2; 1Þ or SOð3Þ as the internal gauge group, as the group of local orthogonal rotations. From i.e., even though the spacetime is two-dimensional, this point of view, the former theories might be the gauge group is not SOð1; 1Þ or SOð2Þ but interpreted as extended objects (strings, membranes, SOð2; 1Þ or SOð3Þ. It has been shown in Sec. III etc.) that can be naturally coupled and acting as that this is indeed allowed without destroying the sources for the fields of geometric theories living topological character of the theory. on higher-dimensional spacetimes Mm . In fact, (3) Add the equation among the momenta following this way of thinking, in the context of nonperturbative quantum gravity, various two- IJ À a"IJK K ¼ 0; (24) dimensional topological field theories allowing the where a is a constant. existence of ‘‘tetrad fields’’ (and therefore allowing either SOð3; 1Þ or SOð4Þ groups) living on a two- The key point is to realize that this relationship corre- dimensional spacetime have been constructed out sponds to a reducibility equation and coupled to four-dimensional BF theories [11]. The last models are conceptually different from and DIJ À a"IJK dK ¼ 0; (25) technically equal to those analyzed in Ref. [12] among the constraints dI and DIJ . In this way, the new where the coupling of (n À 3)-dimensional mem- theory has more reducibility equations than the original branes to n-dimensional BF theory defined for a one (A2), the counting is not balanced, and the new theory large class of structure groups was studied. On this is not topological anymore, it has matter see also Refs. [13,14] as well as the previous results contained in Refs. [15,16]. 1=2½2ð3 þ 3Þ À 2ð3 þ 3 þ 3 þ 3 À 3 À 3 À 3ފ ¼ 1=2½2ð6Þ À 2ð3ފ ¼ 3 (26) IV. ADDING LOCAL DEGREES OF FREEDOM local degrees of freedom. Even though this Hamiltonian Up to now, the paper has been focused in the analysis of way of generating degrees of freedom is correct, it remains various diffeomorphism invariant topological field theories to see that it corresponds to a Lagrangian action. This is that are per se interesting enough. Nevertheless, one might really so, the corresponding Lagrangian action is wonder about the relationship between them and theories Z with local degrees of freedom, such as general relativity or S½!I J ; eI ; T I ; RI J ; I Š ¼ ½I ðdeI þ !I J ^ eJ À T I Þ modifications of it, for instance. To be precise, the question M2 is how to build theories with local degrees of freedom from þ a"IJK K ðd!IJ þ !I M ^ !MJ the topological theories discussed in Secs. II and III. This is the issue studied in this section. À RIJ ފ: (27) The easiest way of modifying the action principles dis- The field theory of the action principle (27) is two- cussed up to now, in order to build field theories with local dimensional gravity coupled to additional fields. The degrees of freedom, is to use the canonical analyses per- contact with two-dimensional gravity is made through formed in Secs. II and III, and in the appendices. From the usual identification I; J ¼ a; 2 with a ¼ 0, 1 (see ^ ^ them it follows that there are, essentially, three parameters Refs. [17,18]) at hand to generate local degrees of freedom: the number of ^ ^ phase space variables, the number of constraints, and the !a b ¼ a b ; ^ ^ !a2 ¼ # a ; ^ ^ ðI Þ ¼ ða ; Þ; (28) ^ number of reducibility equations. The main idea developed ^ ^ here is that by means of an appropriate handling of these where a b is a two-dimensional Lorentz connection a^ parameters it is possible to get theories with a nonvanishing and # is a two-dimensional local Lorentz frame; two- number of physical degrees of freedom. This process im- ^ ^ dimensional Lorentz indices a, b are raised and lowered plies to get rid of the topological nature of the original with the metric ða;b Þ ¼ diagð; þ1Þ. Therefore, action ^ ^ theory to allow local excitations to emerge. Even though (27) becomes 064046-7
  • ´ CUESTA, MONTESINOS, VELAZQUEZ, AND VERGARA PHYSICAL REVIEW D 78, 064046 (2008) Z In fact, plugging the equation of the second row into the S¼ ½a d ea þ a # a ^ e2 À a T a ^ ^ ^ ^ ^ ^ M2 one of the first row of (30) leads to ^ þ ðde2 À #a ^ ea À T 2 Þ þ a"a b Ra b ðÞ ^ ^ ^ ^ ^ g
  • @ @
  •  þ #a @ #
  • a @
  •  þ 2 ^ ^ ^ ^ ^ ^ À a"a b ^ ^ ^ #a ^ #b À a"a b ^ ^ Ra b À ^ 2a"a b b d # a ^ ^ ^ þ #a a b #
  • b @
  •  ¼ 0; (31) ^ ^ ^ ^ ^ þ 2a"a b  R Š; ^ ^ b a2 (29) where g
  • ¼ #a # a
  • is the inverse of the induced two- ^ ^ ^ ^ ^ ^ ^ ^ with "a b 2 ¼ "a b and ^ ^ ^ ^ Ra b ðÞ ¼ þ c^ da b ^ ^ a c b . ^ ^ dimensional metric g
  • ¼ # a # b
  • a b . The expression ^ ^ From this expression it is clearly observed that the first, ^ ^ for a b  can be obtained from the variation of the action second, and fourth terms in the second row correspond to ^ ^ ^ ^ (27) with respect to I , it is of the form a b  ¼ Àa b  þ two-dimensional gravity (see Eq. (2.12) of Ref. [18]). The ^ ^ ^ ^ ^ ^ additional terms in (29) give the explicit nonminimal cou- Sa b  where Àa b  is the spin connection and Sa b  includes plings of the matter fields to gravity. Even though the terms the contribution of matter fields. Note that Eq. (31) is an in the action (29) show the nature of the dynamical fields extension of Eq. (2.28) of Ref. [19] because in that case ^ ^ ^ ^ through their couplings, the particular dynamics of one of a b  does not include Sa b  . the matter fields involved can be illustrated even more from Furthermore, by defining the fields the equations of motion that follow from the variation of the action (27) with respect to eI IJ A :¼ !IJ þ "IJ K eK ; A :¼ !IJ þ
  • "IJ K eK ;
  • IJ ^ (32) da þ # a  þ a b b ¼ 0; ^ ^ ^ ^ a ¼ #a @ : (30) ^ ^ with À
  • Þ 0, the meaning of the action (27) becomes clearer due to the fact it can be cast in the equivalent form Z  
  • 1 1 þ
  • S½ AIJ ;
  • AIJ ; T I ; RI J ; I Š ¼ "IJK K FIJ ð AÞ À "IJK K FIJ ð
  • AÞ À "IJK K RIJ À I T I ; M2 2
  • À 2
  • À 2
  • (33) that involves two interacting BF theories sharing the ‘‘B V. CONCLUDING REMARKS field’’; the constant a in (27) is related to the constants A generalization of part of the results contained in and
  • through a ¼ þ
  • , which has been chosen in order to 2
  • Ref. [3] concerning Cartan’s equations and Bianchi iden- eliminate the quadratic terms in both connections A and tities was presented. Contrary to the previous work, the
  • A that appear when the quadratics terms are recollected. current action principles for diffeomorphism invariant to- Of course, it would be nice to try to get an equivalent form pological field theories hold for arbitrary n-dimensional for the action (27), (29), and (33) of the model containing spacetime manifolds and involve auxiliary fields. It is the true (physical) degrees of freedom only. However, that worthwhile to mention that the connection !I J involved is not the point here; the point is to show that idea putting in the action principles (1) and (3) is not flat; its curvature is forward in this paper works, namely, that it is possible to equal to the two-form field RIJ . This is a major difference build theories with local degrees of freedom from the between theories (1) and (3) of this paper and pure BF original topological theory by means of a suitable modifi- theories. It was also shown that it is possible to use this cation that destroys its topological nature and that allows theoretical framework to build a two-dimensional field the emerging of local excitations. theory with local degrees of freedom by imposing addi- It is worth noting that the local excitations have tional restrictions on the fields involved, destroying the arisen essentially by establishing some additional rela- topological nature of the original theory. It would be tions among the variables involved that were not present interesting to apply the same strategy in four-dimensional in the original topological field theory. Note, however, that spacetimes to find alternative formulations for general the constraints were directly imposed on the fields involved relativity or modifications (generalizations) of it, just start- in the model (I and IJ , in this case). Of course it is ing from the action principle (1) and constraining the fields also possible to incorporate the constraints on the fields suitably. This is left for future work. Because of the fact by introducing more auxiliary fields that impose these that the framework developed in Secs. II and III is quite constraints, which would be much more in the spirit of generic, it is natural to expect that it can also be applied to the relationship between BF gravity and pure BF theory analyze general relativity in arbitrary finite-dimensional [20–26]. spacetime manifolds, in the sense of [27]. 064046-8
  • TOPOLOGICAL FIELD THEORIES IN n-DIMENSIONAL . . . PHYSICAL REVIEW D 78, 064046 (2008) There are various topics that were not touched in the and computing their Poisson brackets gives the nonvanish- paper, but they deserve also to be explored, among these: ing ones (1) the interpretation of the auxiliary fields involved, (2) the relationship between this approach and the one of Ref. [3] if the analysis of this paper is restricted to four- fGðuÞ; gðaÞg ¼ gðu Á aÞ; fDðUÞ; gðaÞg ¼ dðU Á aÞ; dimensional spacetimes, (3) the possible relationship be- fGðuÞ; GðvÞg ¼ Gð½u; vŠÞ; fGðuÞ; dð Þg ¼ dðu Á Þ; tween the various topological field theories reported in the paper with other topological theories, and (4) the inclusion fGðuÞ; DðUÞg ¼ Dð½u; UŠÞ; (A5) of fermion fields in the current theoretical framework. where ðu Á aÞI :¼ uIJ aJ , ðU Á aÞI :¼ UIJ aJ , ½u; vŠIJ :¼ ACKNOWLEDGMENTS uI K vKJ À uJ K vKI , ðu Á ÞI :¼ uIJ J , and ½u; UŠIJ :¼ This work was supported in part by CONACYT, uI K UKJ À uJ K UKI . So, the 2 þ 1 þ 2 þ 1 ¼ 6 constraints ´ Mexico, Grants No. 56159-F and No. 47211-F. J. D. in Eq. (A3) are first class. However, they are reducible Vergara acknowledges the support from DGAPA-UNAM because of the 2 þ 1 ¼ 3 reducibility equations ´ Grant No. IN109107. V. Cuesta and M. Velazquez ac- knowledge the financial support from CONACYT. gI À @x1 dI þ !K I1 dK ¼ 0; GIJ À @x1 DIJ þ !K I1 DKJ þ !K J1 DIK (A6) APPENDIX A: TWO-DIMENSIONAL THEORY À 1ðdI eJ1 À dJ eI1 Þ ¼ 0: 2 The equations of motion that follow from the variation of the action (3) with respect to the independent fields are Therefore, there are just 6 À 3 ¼ 3 independent first-class I :¼ deI þ !I J ^ eJ À T I ¼ 0; constraints in Eq. (A3). Because of the fact that there are IJ :¼ d!IJ þ !I K ^ !KJ À RIJ ¼ 0; 2 þ 1 ¼ 3 configuration variables, the number of local degrees of freedom is 1 ½2ð3Þ À 2ð6 À 3ފ ¼ 0 and so the T I :¼ I ¼ 0; RIJ :¼ IJ ¼ 0; (A1) 2 action (3) defines a topological field theory. ! :¼ DIJ þ IJ 2ðI eJ 1 À J eI Þ ¼ 0; In this appendix the structure group was taken to be SOð2Þ or SOð1; 1Þ. Nevertheless, as it was explained in e :¼ DI ¼ 0: I Sec. III B, the computation can be performed generically The Hamiltonian form of the action (3) is and it turns out that the theory remains topological for the group SOðmÞ or SOðm À 1; 1Þ in spite of the fact that the S½eI 1 ; !IJ 1 ; I ; IJ ; I ; IJ ; ÃI ; ÃIJ Š theory lives in a two-dimensional spacetime. This com- Z ment about the structure group also applies to the analysis ¼ ½I eI 1 þ IJ !IJ 1 À I gI À IJ GIJ À ÃI dI _ _ carried out in Appendix B where the structure group was taken to be SOð3Þ or SOð2; 1Þ. À ÃIJ DIJ Šdx0 ^ dx1 ; (A2) where APPENDIX B: THREE-DIMENSIONAL THEORY The theory is defined by the action (1) where the fields gI :¼ @x1 I À !K I1 K % 0; I are three one-forms and IJ are three one-forms while GIJ :¼ @x1 IJ À !K I1 KJ À !K J1 IK þ 1ðI eJ1 À J eI1 Þ 2 I are three 0-forms, and IJ are three 0-forms, respec- tively. The Hamiltonian form of the action (1) is % 0; dI :¼ I % 0; DIJ :¼ IJ % 0; (A3) Z ½a I eI a þ a IJ !IJ a þ Åab I T I ab þ Åab IJ RIJ ab À I gI _ _ _ _ and the following definition of variables has been made: I :¼ I , IJ :¼ IJ , I :¼ ÀeI 0 , IJ :¼ À!IJ 0 , ÃI :¼ À IJ GIJ À ÃI a da I À ÃIJ a Da IJ À uI hI À uIJ H IJ Šd3 x; T I 01 , and ÃIJ :¼ RIJ 01 . Smearing the constraints (A3) with (B1) test fields whose indices have the corresponding symme- tries of the constraints where the definitions (in terms of the original variables) of Z Z the momenta are: a I :¼ "ab Ib , a IJ :¼ "ab IJb , gðaÞ :¼ dx1 aI gI ; GðuÞ :¼ dx1 uIJ GIJ ; Åab I :¼ 1 "ab I , Åab IJ :¼ 1 "ab IJ while the Lagrange 2 2 (A4) multipliers I :¼ ÀeI 0 , IJ :¼ À!IJ 0 , ÃI a :¼ T I 0a , Z Z dð Þ :¼ dx1 I I ; DðUÞ :¼ dx1 UIJ IJ ; ÃIJ a :¼ RIJ 0a , uI :¼ À 1 I0 , and uIJ :¼ À 1 IJ0 impose 2 2 the constraints 064046-9
  • ´ CUESTA, MONTESINOS, VELAZQUEZ, AND VERGARA PHYSICAL REVIEW D 78, 064046 (2008) gI :¼ Da a I À Åab J RJ Iab % 0; GIJ :¼ Da a IJ þ 1ða I eJa À a J eIa Þ þ 1ðÅab I TJab À Åab J TIab Þ þ Åab IK RJ K ab À Åab JK RI K ab % 0; 2 2 da I :¼ a I þ 2Db Åab I % 0; Da IJ :¼ a IJ þ 2Db Åab IJ þ Åab I eJb À Åab J eIb % 0; hI :¼ "ab ðDa eI b À Db eI a À T I ab Þ % 0; H IJ :¼ "ab ð@a !IJ b À @b !IJ a þ !I Ka !KJ b À !I Kb !KJ a À RIJ ab Þ % 0: (B2) Some of the differences with the theory in two dimensions Because of the fact that there are 6 þ 6 þ 3 þ 3 ¼ 18 are the following: (1) there are now velocities of the fields configuration variables, the number of local degrees of T I 0a and RIJ 0a , (2) the new constraints hI and H IJ come freedom is 1 ½2ð18Þ À 2ð24 À 6ފ ¼ 0 and so the action 2 from the fact that now I and IJ are one-forms. The 3 þ (1) defines a topological field theory. 3 þ 6 þ 6 þ 3 þ 3 ¼ 24 constraints in Eq. (B2) are first class and reducible because of the 3 þ 3 ¼ 6 reducibility equations D a da I À gI þ 1"ab Åab J H J I ¼ 0; 2 Da Da IJ À GIJ þ 1ðda I eJa À da J eIa Þ 2 (B3) À 1"ab ðÅab I hJ À Åab J hI Þ 4 þ 1"ab ðH K I Åab KJ À H K J Åab KI Þ ¼ 0: 2 [1] A. Perez, Classical Quantum Gravity 20, R43 (2003); D. [12] J. C. Baez and A. Perez, Adv. Theor. Math. Phys. 11, 451 Oriti, Rep. Prog. Phys. 64, 1703 (2001); J. C. Baez, Lect. (2007). Notes Phys. 543, 25 (2000); Classical Quantum Gravity [13] W. J. Fairbairn and A. Perez, Phys. Rev. D 78, 024013 15, 1827 (1998); J. Engle, R. Pereira, and C. Rovelli, Phys. (2008). Rev. Lett. 99, 161301 (2007); E. R. Livine and S. Speziale, [14] W. J. Fairbairn, R. Brasselet, and A. Perez, ‘‘Quantization Europhys. Lett. 81, 50004 (2008). of string-like sources coupled to BF theory: physical [2] T. Thiemann, Modern Canonical Quantum General scalar product and spinfoam models’’ (to appear). Relativity (Cambridge University Press, Cambridge, [15] K. Noui and A. Perez, Classical Quantum Gravity 22, England, 2007); C. Rovelli, Quantum Gravity 4489 (2005); 22, 1739 (2005); L. Freidel and D. Louapre, (Cambridge University Press, Cambridge, England, Classical Quantum Gravity 21, 5685 (2004). 2004); A. Ashtekar and J. Lewandowski, Classical [16] S. Carlip, Nucl. Phys. B324, 106 (1989); P. de Sousa Quantum Gravity 21, R53 (2004); A. Perez, arXiv: Gerbert, Nucl. Phys. B346, 440 (1990). gr-qc/0409061. [17] A. H. Chamseddine and W. Wyler, Phys. Lett. B 228, 75 [3] V. Cuesta and M. Montesinos, Phys. Rev. D 76, 104004 (1989). (2007). [18] A. H. Chamseddine, Nucl. Phys. B346, 213 (1990). ´ [4] M. Mondragon and M. Montesinos, J. Math. Phys. (N.Y.) [19] A. H. Chamseddine and D. Wyler, Nucl. Phys. B340, 595 47, 022301 (2006). (1990). [5] M. Montesinos, J. Phys. Conf. Ser. 24, 44 (2005). ´ [20] J. F. Plebanski, J. Math. Phys. (N.Y.) 18, 2511 (1977). [6] M. Montesinos, Classical Quantum Gravity 23, 2267 [21] M. P. Reisenberger, Nucl. Phys. B457, 643 (1995). (2006). [22] R. Capovilla, J. Dell, and T. Jacobson, Classical Quantum [7] M. Montesinos, Classical Quantum Gravity 20, 3569 Gravity 8, 59 (1991). (2003). [23] M. P. Reisenberger, Classical Quantum Gravity 16, 1357 [8] P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer (1999). Graduate School of Science, New York, 1964). [24] R. De Pietri and L. Freidel, Classical Quantum Gravity 16, [9] M. Henneaux and C. Teitelboim, Quantization of Gauge 2187 (1999). Systems (Princeton University, Princeton, NJ, 1992). [25] D. C. Robinson, J. Math. Phys. (N.Y.) 36, 3733 (1995). [10] M. I. Caicedo, R. Gianvittorio, A. Restuccia, and J. [26] R. Capovilla, M. Montesinos, V. A. Prieto, and E. Rojas, Stephany, Phys. Lett. B 354, 292 (1995). Classical Quantum Gravity 18, L49 (2001). [11] M. Montesinos and A. Perez, Phys. Rev. D 77, 104020 [27] L. Freidel, K. Krasnov, and R. Puzio, Adv. Theor. Math. (2008). Phys. 3, 1289 (1999). 064046-10