Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter

A detailed analysis of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transfor...

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Дата:	2011
Автори:	Montesinos, M., Velázquez, M.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147658
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Цитувати:	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter / M. Montesinos, M. Velázquez // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 26 назв. — англ.

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spelling	irk-123456789-1476582019-02-16T01:25:36Z Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter Montesinos, M. Velázquez, M. A detailed analysis of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transformed action principle involves two BF terms and the two Lorentz invariants that appear in the original action principle generically. As an application of this formalism, the action principle used by Engle, Pereira, and Rovelli in their spin foam model for gravity is recovered and the coupling of the cosmological constant in such a formulation is obtained. 2011 Article Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter / M. Montesinos, M. Velázquez // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 83C05; 83C45 http://dspace.nbuv.gov.ua/handle/123456789/147658 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A detailed analysis of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transformed action principle involves two BF terms and the two Lorentz invariants that appear in the original action principle generically. As an application of this formalism, the action principle used by Engle, Pereira, and Rovelli in their spin foam model for gravity is recovered and the coupling of the cosmological constant in such a formulation is obtained.
format	Article
author	Montesinos, M. Velázquez, M.
spellingShingle	Montesinos, M. Velázquez, M. Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Montesinos, M. Velázquez, M.
author_sort	Montesinos, M.
title	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter
title_short	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter
title_full	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter
title_fullStr	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter
title_full_unstemmed	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter
title_sort	equivalent and alternative forms for bf gravity with immirzi parameter
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147658
citation_txt	Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter / M. Montesinos, M. Velázquez // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 26 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT montesinosm equivalentandalternativeformsforbfgravitywithimmirziparameter AT velazquezm equivalentandalternativeformsforbfgravitywithimmirziparameter
first_indexed	2025-07-11T02:34:48Z
last_indexed	2025-07-11T02:34:48Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 103, 13 pages Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter? Merced MONTESINOS and Mercedes VELÁZQUEZ Departamento de F́ısica, Cinvestav, Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de México, México E-mail: merced@fis.cinvestav.mx, mquesada@fis.cinvestav.mx URL: http://www.fis.cinvestav.mx/~merced/ Received August 31, 2011, in final form November 07, 2011; Published online November 11, 2011 http://dx.doi.org/10.3842/SIGMA.2011.103 Abstract. A detailed analysis of the BF formulation for general relativity given by Capo- villa, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transformed action principle involves two BF terms and the two Lorentz invariants that appear in the original action principle generically. As an appli- cation of this formalism, the action principle used by Engle, Pereira, and Rovelli in their spin foam model for gravity is recovered and the coupling of the cosmological constant in such a formulation is obtained. Key words: BF theory; BF gravity; Immirzi parameter; Holst action 2010 Mathematics Subject Classification: 83C05; 83C45 1 Introduction One of the main challenges nowadays is to establish links between loop quantum gravity (LQG) [1] and spin foam models [2], which are the main approaches to the nonperturbative and background-independent quantizations of general relativity. Whether or not the two quan- tization schemes yield two different quantum theories is still an open problem (see [3]). The search for the links between these approaches lies mostly in the quantum realm, but there are still some aspects of this correspondence that are unclear classically. It is possible to say that the connection between the two frameworks at the classical level is the relationship between the Holst’s action and the BF formulations for general relativity because LQG is based at the classical level on Holst’s action but spin foam models for gravity are related to constrained BF theories. This is the issue studied in this paper. General relativity expressed as a constrained BF theory was given by Plebański many years ago [4]. The basic idea behind the Plebański formulation is that the fundamental variables for describing the gravitational field (general relativity) are neither a metric (as it is in the Einstein–Hilbert action) nor a tetrad together with a Lorentz connection (as it is in the Palatini action), but rather two-form fields, a connection one-form, and some Lagrange multipliers. The geometry of spacetime is built up from these fundamental blocks. In order to bring tetrads into the formalism, the two-forms are eliminated by solving an equation among them, which implies that the two-forms can be expressed in terms of tetrad fields, and by inserting back this expression for the two-forms into the Plebański action, it becomes the self-dual action for general relativity [5, 6]. ?This paper is a contribution to the Special Issue “Loop Quantum Gravity and Cosmology”. The full collection is available at http://www.emis.de/journals/SIGMA/LQGC.html mailto:merced@fis.cinvestav.mx mailto:mquesada@fis.cinvestav.mx http://www.fis.cinvestav.mx/~merced/ http://dx.doi.org/10.3842/SIGMA.2011.103 http://www.emis.de/journals/SIGMA/LQGC.html 2 M. Montesinos and M. Velázquez This view point has been adopted in the construction of other action principles, which also express general relativity as a constrained BF theory [7, 8, 9, 10, 11, 12]. The link with tetrad gravity is again made by solving the constraints for the two-form fields which amounts to express them in terms of tetrad fields. In particular, a formulation for real general relativity expressed as a constrained BF theory that involves the Immirzi parameter [13, 14, 15] was given in [12] by Capovilla, Montesinos, Prieto, and Rojas (hereafter CMPR formulation). It is well-known that the Immirzi parameter in such a formulation appears naturally when the two Lorentz invariants ηIKηJL−ηILηJK and εIJKL are introduced in the constraint on the Lagrange multipliers φIJKL. This action principle involves just one BF term. In this work it is shown that by performing a suitable transformation on the fields involved in the theory, the original action principle can be written in a form that involves two BF terms, one of them containing a parameter that will be identified a posteriori with the Immirzi parameter (see also [16]). This allows us to relate the CMPR formulation with different real BF formulations of gravity currently employed in the literature (see e.g. [17]). Furthermore, the same transformation is applied to the action principle studied in [18], which includes the cosmological constant, and we obtain the coupling of the cosmological constant in the framework of [17]. The material reported in this paper is part of the work presented in [19]. 2 CMPR action for gravity The action principle for pure gravity introduced by Capovilla, Montesinos, Prieto, and Rojas in [12] is given by S[Q,A,ψ, µ] = ∫ M4 [ QIJ ∧ FIJ [A]− 1 2 ψIJKLQ IJ ∧QKL − µ ( a1ψIJ IJ + a2ψIJKLε IJKL ) ] , (1) where AIJ is an Euclidean or Lorentz connection one-form, depending on whether SO(4) or SO(3, 1) is taken as the internal gauge group, and F IJ [A] = dAIJ +AIK ∧AKJ is its curvature; the Q’s are a set of six two-forms on account of their antisymmetry QIJ = −QJI ; the Lag- range multiplier ψIJKL has 21 independent components due to the properties ψIJKL = ψKLIJ , ψIJKL = −ψJIKL, and ψIJKL = −ψIJLK ; the Lagrange multiplier µ implies the additional restriction a1ψIJ IJ + a2ψIJKLε IJKL = 0 on the Lagrange multiplier ψIJKL. The Lorentz (Eu- clidean) indices I, J,K, . . . = 0, 1, 2, 3 are raised and lowered with the Minkowski (Euclidean) metric (ηIJ) = diag(σ,+1,+1,+1) where σ = +1 for Euclidean and σ = −1 for Lorentzian signatures, respectively. The variation of the action (1) with respect to the independent fields gives the equations of motion δQ : FIJ [A]− ψIJKLQKL = 0, δA : DQIJ = 0, δψ : QIJ ∧QKL + 2a1µη [I\|K\|ηJ ]L + 2a2µε IJKL = 0, (2) δµ : a1ψIJ IJ + a2ψIJKLε IJKL = 0. By contracting equation (2) with the Killing–Cartan metric η[I\|K\|ηJ ]L = 1 2(ηIKηJL − ηILηJK) and εIJKL, one gets a1µ = − 1 12Q IJ ∧ QIJ and a2µ = − σ 4!Q IJ ∧ ∗QIJ respectively, where ∗QIJ := 1 2ε IJ KLQ KL. The non-degenerate case corresponds to µ 6= 0 whereas the degenerate case corresponds to µ = 0. Let us restrict the analysis to the non-degenerate case. Inserting back a1µ and a2µ into (2), it is obtained QIJ ∧QKL − 1 6 ( QMN ∧QMN ) η[I\|K\|ηJ ]L − 2σ 4! ( QMN ∧ ∗QMN ) εIJKL = 0, Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter 3 together with 2a2Q IJ ∧QIJ − σa1Q IJ ∧ ∗QIJ = 0, that follows from the equality of the two expressions for µ and the fact that a1 6= 0 and a2 6= 0. It is shown in [12] that QIJ = α ∗ ( eI ∧ eJ ) + βeI ∧ eJ , (3) is the general solution for the Q’s provided that the constants α and β satisfy a2 a1 = α2 + σβ2 4αβ . (4) By inserting the solution (3) into the action principle (1), we get S[e,A] = ∫ M4 [ ∗ (eI ∧ eJ)+ β α eI ∧ eJ ] ∧ FIJ [A]. Notice that, as remarked in [12], the Immirzi parameter appears naturally in equation (3) because the two invariants ψIJ IJ and ψIJKLε IJKL are present in the action. 2.1 CMPR formulation with a1 = 0 or a2 = 0 The cases when a1 or a2 are equal to zero have been analyzed separately [8, 10, 11]. In particular, if a1 = 0 and a2 6= 0 the Lorentz invariant ψIJ IJ is not present in the action, which reduces to S[Q,A,ψ, µ] = ∫ M4 [ QIJ ∧ FIJ [A]− 1 2 ψIJKLQ IJ ∧QKL − µψIJKLεIJKL ] . (5) After solving the constraint on the Q’s, they can be written in terms of the tetrad eI as (i) QIJ = κ1 ∗ (eI ∧ eJ) , (ii) QIJ = κ2e I ∧ eJ , (6) where κ1, κ2 are constants. By inserting these expressions for the Q’s into (5), we get action principles for two different theories, one of which is general relativity [10, 11] (i) S1[e,A] = κ1 ∫ M4 ∗ (eI ∧ eJ) ∧ FIJ [A], (ii) S2[e,A] = κ2 ∫ M4 ( eI ∧ eJ ) ∧ FIJ [A]. (7) In [20] was proposed to consider the action S2[e,A] as a genuine field theory in its own right be- cause it is background-independent and diffeomorphism-invariant. Recently, it has been shown that S2[e,A] is indeed topological if the spacetime M4 has no boundary. The proof is given by performing the covariant canonical analysis to this action as well as by doing the Dirac’s canonical analysis with and without breaking local Lorentz invariance [21, 22, 23]. The rele- vance of the action S2[e,A] is not academic, this field theory is a topological limit of general relativity obtained by taking the Newton constant G → ∞ and the Immirzi parameter γ → 0 while keeping the product Gγ constant [21]. On the other hand, if a1 6= 0 and a2 = 0, the action takes the form [8] S[Q,A,ψ, µ] = ∫ M4 [ QIJ ∧ FIJ [A]− 1 2 ψIJKLQ IJ ∧QKL − µψIJ IJ ] . (8) 4 M. Montesinos and M. Velázquez The solutions for the two-forms in terms of the tetrad eI are given by QIJ = κ [∗ (eI ∧ eJ)±√−σeI ∧ eJ] . Therefore, the action principle in terms of the tetrad takes the form SE [e,A] = κ ∫ M4 [∗ (eI ∧ eJ)± ieI ∧ eJ] ∧ FIJ [A] in the Euclidean case whereas in the Lorentzian case it becomes SL[e,A] = κ ∫ M4 [∗ (eI ∧ eJ)± eI ∧ eJ] ∧ FIJ [A]. Notice that last form of the action includes the coupling of the term eI ∧ eJ ∧ FIJ [A] added by Holst to the Palatini action with (what it is called now) Immirzi parameter equal to ±1, but this form of the action was reported in [8] several years before Holst, Immirzi, and Barbero’s papers. 3 Transformation of the CMPR action for gravity The goal of this section is to study the CMPR formulation for general relativity by performing a linear transformation from the original variables the action principle depends functionally on to a new set of two-forms and Lagrange multipliers. It will be shown that the resulting action principle involves the two possible BF terms, BIJ ∧ FIJ [A] and ∗BIJ ∧ FIJ [A] that can be built when the internal gauge group is SO(3, 1) or SO(4), with the corresponding change in the symplectic structure as it was pointed out in [24] and [25]. In order to do what we have explained, i.e., the alternative writing of the CMPR action, we define QIJ := b1B IJ + b2 ∗BIJ , (9) with b1 and b2 constants, from which it follows the inverse transformation BIJ = 1 b21 − σb22 ( b1Q IJ − b2∗QIJ ) , (10) provided that b21 − σb22 6= 0, (11) holds. Using (9), the Lagrangian of action principle (1) acquires the form( b1B IJ + b2 ∗BIJ ) ∧ FIJ [A]− 1 2 ψIJKL ( b1B IJ + b2 ∗BIJ) ∧ (b1BKL − b2 ∗BKL) −µ ( a1ψIJ IJ + a2ψIJKLε IJKL ) , (12) that can further be rewritten by defining φIJKL := b21ψIJKL + b1b2 ∗ψIJKL + b1b2ψ ∗ IJKL + b22 ∗ψ∗IJKL, (13) where ∗ψIJKL := 1 2ε MN IJψMNKL and ψ∗IJKL := 1 2ε MN KLψIJMN are the dual on the first and on the second pair of Lorentz indices, respectively. From (13) it follows that ψIJKL = 1 (b21 − σb22)2 ( b21φIJKL − b1b2 ∗φIJKL − b1b2φ∗IJKL + b22 ∗φ∗IJKL ) . (14) Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter 5 Therefore, the second term of (12) takes the form ψIJKL ( b1B IJ + b2 ∗BIJ) ∧ (b1BKL − b2 ∗BKL) = φIJKLB IJ ∧BKL. Furthermore, using (14), the two invariants ψIJ IJ and ψIJKLε IJKL can be written in terms of the two invariants φIJ IJ and εIJKLφ IJKL as ψIJ IJ = 1( b21 − σb22 )2 [(b21 + σb22 ) φIJ IJ − b1b2φIJKLεIJKL ] , (15) and ψIJKLε IJKL = 1( b21 − σb22 )2 [(b21 + σb22 ) φIJKLε IJKL − 4σb1b2φIJ IJ ] . (16) Thus, using (15) and (16), the last term of (12) acquires the form a1ψIJ IJ + a2ψIJKLε IJKL = A1φIJ IJ +A2φIJKLε IJKL, (17) with A1 = 1( b21 − σb22 )2 [a1 ( b21 + σb22 ) − 4σa2b1b2 ] , A2 = 1( b21 − σb22 )2 [a2 ( b21 + σb22 ) − a1b1b2 ] . (18) By using the previous steps, the form that the CMPR action (1) acquires once the transfor- mation defined in equations (9), (10) and in equations (13), (14) has been done is S[B,A, φ, µ] = ∫ M4 [ ( b1B IJ + b2 ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL − µ ( A1φIJ IJ +A2φIJKLε IJKL ) ] . (19) Due to the fact that the transformation is invertible, both actions (1) and (19) are equivalent. The action principle (19) can still be written in terms of tetrads and a Lorentz connection by solving the constraint on the B’s coming from it. Alternatively, the expression for the two- forms BIJ in terms of the tetrad field can be obtained from the expression for the Q’s given in (3) and from the use of the equation (10). The relationship between the action principle (1) and (19) will be analyzed in detail in an example given in Section 3.1. Some remarks follow: (a) It can be observed from equations (15), (16), (17), and (18) that even though we had started from action (1) with either a1 = 0 or a2 = 0, it might be possible to obtain generically the two Lorentz invariants in the transformed action (19). (b) It is possible to get just one of the invariants in the transformed action (19) by imposing either A1 = 0 or A2 = 0. For instance, the case A1 = 0 can be achieved by solving for the ratio b2/b1 in terms of the ratio a2/a1, i.e. by choosing a particular transformation (encoded in b1 and b2) and leaving a1 and a2 arbitrary. Alternatively, A1 = 0 can be achieved by solving for the ratio a2/a1 in terms of the ratio b2/b1, i.e. by choosing a particular form for the ratio a2/a1 and leaving the transformation arbitrary. Similarly, the case A2 = 0 can also be handled in two analogous ways. 6 M. Montesinos and M. Velázquez The previous analysis points out that is not correct to refer to the term ∗BIJ ∧ FIJ as “Holst’s term” simply because the term added by Holst to the Palatini action and given by eI ∧ eJ ∧ FIJ [15] (see also [8]) is at the level of tetrads eI and not at the level of BF theories. Even though they might be related, they are not exactly the same thing. In particular, ∗BIJ∧FIJ could be proportional to ∗ ( eI ∧ eJ ) ∧ FIJ , or to eI ∧ eJ ∧ FIJ , or to something else depending on the expression for the B’s that solves the constraint among them, i.e., a priori there is not guarantee that ∗BIJ ∧ FIJ would lead to the term added by Holst, because this will ultimately depend on the expression for the B’s in terms of the tetrads. 3.1 A particular transformation Let us now study a particular case of the transformation (9). Taking b1 = 1 and b2 = 1 γ the transformation is invertible for γ2 6= σ. In this case, the action (1) takes the form S[B,A, φ, µ] = ∫ M4 [( BIJ + 1 γ ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL − µ ( A1φIJ IJ +A2φIJKLε IJKL ) ] , (20) where now A1 = γ2 (γ2 − σ)2 [ a1 ( γ2 + σ ) − 4σa2γ ] , A2 = γ2 (γ2 − σ)2 [ a2 ( γ2 + σ ) − a1γ ] . (21) It is important to notice that, at this stage, the action principle (20) is completely equivalent to action (1) because the coefficients of the transformation satisfy the condition (11). As pointed out in the previous remark (b), it is possible to obtain only one of the invariants in the action (20) by imposing, additionally, either A1 = 0 or A2 = 0. 3.1.1 Case A1 = 0: action with the invariant φIJKLε IJKL only In order to eliminate the term with φIJ IJ in the action (20), A1 must vanish; this is only possible if a2 and a1 satisfy the condition a2 a1 = γ2 + σ 4γσ . (22) Using (22), the CMPR action principle (1) takes the form S[Q,A,ψ, µ] = ∫ M4 [ QIJ ∧ FIJ [A]− 1 2 ψIJKLQ IJ ∧QKL − µa1 ( ψIJ IJ + γ2 + σ 4γσ ψIJKLε IJKL )] , (23) whereas the transformed action principle (20) becomes S[B,A, φ, µ] = ∫ M4 [( BIJ + 1 γ ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL − µa1σγ 4 φIJKLε IJKL ] . (24) This is the form of the action principle used in [17] (see also [16]). Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter 7 Remarks: (c) In the Lorentzian case σ = −1, the particular values γ = ±1 imply γ2 + σ = 0 and thus the Lorentz invariant φIJKLε IJKL is not present in the action (23), which then reduces to the form given in (8). This means that the action (24) can be written as the action studied in [8] by taking γ = ±1 in the Lorentzian case. For any other arbitrary real value of the Immirzi parameter the two invariants ψIJ IJ and ψIJKLε IJKL are present in the action (23) [12]. (d) In the Euclidean case, σ = 1, for real values of γ it follows that γ2 + σ 6= 0 and therefore the two Lorentz invariants are always present in the action (23). This means that the two invariants ψIJ IJ and ψIJKLε IJKL must be involved in order to include arbitrary real values of γ, as it was recognized in [12]. Nevertheless, it is important to notice that γ2 +σ can vanish if complex values of γ are allowed, γ = ±i. For these values the invariant ψIJKLε IJKL is missing in the action (23), which becomes also the one given in (8). Continuing with the analysis, the expression for the B’s can be directly obtained from action principle (24). However, it can be alternatively obtained from the Q’s given in (3) and (4) (and supplemented with (22)) and from the use of the inverse transformation (10) with b1 = 1 and b2 = 1 γ . We are going to follow this last approach. Therefore, from the equality of equations (4) and (22) it follows that in order for the Q’s in (3) to be solutions for the action principle (23), α/β must satisfy the quadratic equation( α β )2 − ( 1 γ + σγ ) α β + σ = 0, whose solutions are (i) α/β = σγ, and (ii) α/β = 1 γ . (25) The first root was explicitly mentioned in [12], but the second one was not recognized there as a possibility to include the Immirzi parameter. Inserting the two roots given in (25) into (3), we get the corresponding expression for the Q’s (i) QIJ = α [ ∗ (eI ∧ eJ)+ σ γ eI ∧ eJ ] , (ii) QIJ = α [ ∗ (eI ∧ eJ)+ γeI ∧ eJ ] , (26) and by plugging them into (10) with the restrictions b1 = 1 and b2 = 1 γ we get the corresponding expressions for the B’s (i) BIJ = α ∗ ( eI ∧ eJ ) , (ii) BIJ = αγeI ∧ eJ , (27) which are precisely the ones given in (6). Furthermore, by comparing (6) and (27), we conclude that κ1 = α for (i) (and thus κ1 = σγβ) whereas κ2 = αγ for (ii) (and thus κ2 = β). By plugging (27) into (24) or, equivalently, by plugging (26) into (23), we get (i) S1[e,A] = κ1 ∫ M4 [ ∗ (eI ∧ eJ)+ σ γ eI ∧ eJ ] ∧ FIJ [A], (ii) S2[e,A] = κ2 γ ∫ M4 [ ∗ (eI ∧ eJ)+ γeI ∧ eJ ] ∧ FIJ [A], which is exactly the same result that we had obtained if we had directly solved the constraint on the B’s that comes from (24) [17]. It is common to take κ1 = ±1 and κ2 = ±1. Nevertheless, it must be stressed that these values do not come out from the sole handling of the equations of motion. 8 M. Montesinos and M. Velázquez 3.1.2 Case A2 = 0: action with the invariant φIJ IJ only In order to obtain the action (24) from (20), a particular function for the ratio a2/a1 has been taken such that A1 in equation (21) vanishes once the transformation is performed. So, it is natural to ask what happens if, instead of A1, it is the coefficient A2 which is forced to vanish in such a way that the invariant φIJKLε IJKL is not present in the action (20). From (21) this condition is equivalent to a1 a2 = γ2 + σ γ . (28) This means that starting from the CMPR action principle in the form S[Q,A,ψ, µ] = ∫ M4 [ QIJ ∧ FIJ [A]− 1 2 ψIJKLQ IJ ∧QKL − µa2 ( γ2 + σ γ ψIJ IJ + ψIJKLε IJKL )] , (29) and using the transformation (10) with b1 = 1 and b2 = 1/γ, this action principle acquires the form S[B,A, φ, µ] = ∫ M4 [( BIJ + 1 γ ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL − µa2γφIJ IJ ] . (30) Remarks: (e) In the case γ2+σ 6= 0 it follows from (29) that the two Lorentz invariants are present in the action and therefore the expression for the two-forms Q’s in terms of the tetrad field, and thus the form of the B’s in (30), can be obtained from (3) following the same procedure carried out in the previous section. In this case the value of a2/a1 comes from (28) and the final form of the actions (29) and (30) in terms of the tetrad field and the connection are (i) S1[e,A] = α ∫ M4 [ ∗ (eI ∧ eJ)+ √ −σγ − √ −σ γ + √ −σ eI ∧ eJ ] ∧ FIJ [A], (ii) S2[e,A] = α ∫ M4 [ ∗ (eI ∧ eJ)−√−σγ + √ −σ γ − √ −σ eI ∧ eJ ] ∧ FIJ [A], corresponding to the two solutions for the B’s. (f) In the case γ2 + σ = 0 the Lorentz invariant ψIJ IJ is missing in the action (29) and it becomes the one given in (5). In that analysis the action principles (7) were obtained by plugging in (5) the solution for the Q’s. In a similar way, the action principle (30) restricted to γ = ± √ −σ can be written in terms of the tetrad field by solving for the B’s and plugging the solutions into (30). The resulting action principles obtained following one or the other procedure are different by a global factor. 4 Application of the transformation to the coupling of the cosmological constant The coupling of the cosmological constant to the CMPR action principle was done in [18] (see also [26]). However, taking into account the previous analysis, it is natural to wonder how the coupling looks like in the transformed CMPR action principles discussed in Section 3. In Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter 9 particular we are interested in the action principle analyzed in Section 3.1.1 and used in [17]. To do this task, the starting point is the CMPR action principle coupled with the cosmological constant given in [18]. This action principle can be transformed to one with two BF terms by applying the general transformation encoded in equations (9) and (14). This will give us an equivalent action principle for gravity with cosmological constant. Nevertheless, as we are interested in the coupling of the cosmological constant to gravity in an action principle of the form given in (24), i.e. with the linear combination BIJ + 1 γ ∗BIJ and using only the Lorentz invariant φIJKLε IJKL, it is necessary to apply the particular transformation studied in Section 3 (defined by b1 = 1 and b2 = 1 γ ), and to restrict the constants included in the CMPR action principle to those which lead us to (24), i.e. to impose on a1 and a2 the condition (22). The action principle introduced in [18] is given by S[Q,A,ψ, µ] = ∫ M4 [ QIJ ∧ FIJ [A]− 1 2 ψIJKLQ IJ ∧QKL − µ ( a1ψIJ IJ + a2ψIJKLε IJKL − λ ) + l1QIJ ∧QIJ + l2QIJ ∧∗ QIJ ] , (31) with λ = a1(4!l2σ a2 a1 + 12l1 + Λ β ). It is important to mention that this expression for λ comes from the fact that Λ is identified with the cosmological constant in the tetrad formalism (see [18] for the details). By applying the transformation given in equation (9), the action principle (31) takes the form S[B,A, φ, µ] = ∫ M4 [ ( b1B IJ + b2 ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL (32) − µ ( A1 φIJ IJ +A2φIJKLε IJKL − λ ) +K1BIJ ∧BIJ +K2BIJ ∧∗ BIJ ] , where the constants A1 and A2 are given by equation (18) and K1 = l1 ( b21 + σb22 ) + 2l2b1b2σ, K2 = l2 ( b21 + σb22 ) + 2l1b1b2. (33) Note that the last two terms of equation (32) come from the analogous terms that appear in equation (31). These terms could still be removed by doing a redefinition of the Lagrange multiplier φIJKL → ϕIJKL but this would imply a transformation ψIJKL → ϕIJKL different from the one given in (13). Such a transformation would be an example of how we can choose the transformation depending on the action principles we want to relate. Nevertheless, in this section we are only interested in an application of the transformation introduced in Section 3, therefore we will continue with the original transformation (13). Notice that λ can be written in terms of K1 and K2 as λ = 12A1K1 + 4!σA2K2 + Λa1 β . Let us now take the particular transformation b1 = 1 and b2 = 1 γ used in Section 3.1.1. In this case, the action (32) takes on the form S[B,A, φ, µ] = ∫ M4 [( BIJ + 1 γ ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL (34) − µ ( A1φIJ IJ +A2φIJKLε IJKL − λ ) +K1BIJ ∧BIJ +K2BIJ ∧∗ BIJ ] , where now A1 and A2 are given by (21), and (33) reduces to K1 = l1 (γ2 + σ)2 γ2 + 2l2 σ γ , K2 = l2 (γ2 + σ)2 γ2 + 2l1 1 γ . 10 M. Montesinos and M. Velázquez It is important to note that, at this stage, the action principle (34) is completely equivalent to action (31) because the coefficients of the transformation b1 and b2 satisfy the condition (11). In order to eliminate the term with φIJ IJ in the action (34), a1 and a2 must satisfy the condition (22) which fixes the constants A1 and A2 to A1 = 0, and A2 = a1γσ 4 , while the constants K1 and K2 do not get modified. Using the values for A1 and A2 the action principle (34) takes the form S[B,A, φ, µ] = ∫ M4 [( BIJ + 1 γ ∗BIJ ) ∧ FIJ [A]− 1 2 φIJKLB IJ ∧BKL − µ (a1γσ 4 φIJKLε IJKL − λ ) +K1BIJ ∧BIJ +K2BIJ ∧∗ BIJ ] , (35) and λ becomes λ = a1 [ 3!γK2 + Λ β ] . (36) Because of the restriction (22), the action principle (35) is a particular case of (34) and thus a particular case of (31). In order to write the action principle (35) in terms of the tetrad and a connection we can follow two approaches. In the first, we simply use the form for the B’s obtained in (27) and the value of λ given in (36) and insert them into (35) to get (i) S[e,A] = α ∫ M4 [( ∗ (eI ∧ eJ)+ σ γ eI ∧ eJ ) ∧ FIJ [A]− Λ 12 εIJKLe I ∧ eJ ∧ eK ∧ eL ] , (ii) S[e,A] = α ∫ M4 [( ∗ (eI ∧ eJ)+ γeI ∧ eJ ) ∧ FIJ [A]− Λ 12 εIJKLe I ∧ eJ ∧ eK ∧ eL ] , which is the Holst action principle with cosmological constant. Note that each solution involves a different Immirzi parameter. In the second approach, we start from the action principle (35) where now λ is not fixed and we simply solve the constraint on the B’s that comes from (35). The relation of λ with the cosmological constant will be obtained at the end of the procedure. The equation of motion that comes from the variation of (35) with respect to the field φIJKL implies 1 2 BIJ ∧BKL = −µa1γσ 4 εIJKL, and therefore BIJ ∧BIJ = 0, BIJ ∧∗ BIJ = −3!µa1γ. (37) From which we obtain the two-forms B’s given by (i) BIJ = κ1 ∗ (eI ∧ eJ) and (ii) BIJ = κ2 ( eI ∧ eJ ) . (38) Using equations (37) and (38) we obtain (i) S[e,A] = κ1 ∫ M4 [( ∗ (eI ∧ eJ)+ σ γ eI ∧ eJ ) ∧ FIJ [A] + κ1σ 2 ( K2 − 1 3!γ λ a1 ) εIJKLe I ∧ eJ ∧ eK ∧ eL ] , Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter 11 (ii) S[e,A] = κ2 γ ∫ M4 [ (∗ (eI ∧ eJ)+ γ eI ∧ eJ ) ∧ FIJ [A] + κ2γ 2 ( K2 − 1 3!γ λ a1 ) εIJKLe I ∧ eJ ∧ eK ∧ eL ] . These actions have the form of the Holst’s action principle with cosmological constant Λ given in each case by (i) Λ 12 = −κ1σ 2 ( K2 − 1 3!γ λ a1 ) , (ii) Λ 12 = −κ2γ 2 ( K2 − 1 3!γ λ a1 ) . They fix the relationship among the constants λ, γ, and a1 that appear in the action princip- le (35), and the coefficients of the solutions given in (38). It is easy to see that, for each case, λ has the form (i) λ = a1 [ 3!γK2 + Λσγ κ1 ] , (ii) λ = a1 [ 3!γK2 + Λ κ2 ] , respectively. Note that in order for these values of λ match the value of λ given in (36) (i.e. the two approaches give the same result), it is required that κ1 = σγβ and κ2 = αγ, respectively. These are the same values obtained for κ1 and κ2 in the analysis after equation (27). 5 Conclusions It has been shown that by performing an invertible transformation of the fields of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas [12], it is possible to obtain the action principle (19) which includes the two BF terms BIJ ∧ FIJ and ∗BIJ ∧FIJ and still involves the two Lorentz invariants φIJKLε IJKL and φIJ IJ generically. One of the results of the analysis is to clearly show the relationship of the two parameters a1 and a2 of the CMPR action principle and of the two parameters b1 and b2 involved in the transformation with the Immirzi parameter. From the analysis is clear that the freedom in the choice of the parameters a1, a2, b1, and b2 can be used to handle the two Lorentz invariants φIJKLε IJKL and φIJ IJ that appear in the transformed action principle. In particular, a suitable combination of these parameters can result in that one of these invariants is missing in the transformed action principle as it is shown in the Section 3.1. As an application of this fact, the action used in [17] is obtained in Section 3.1.1. Finally, and as another application of the transformation discussed in this paper, the coupling of the cosmological constant to the action principle used in [17] is obtained from the coupling of the cosmological constant to the CMPR action principle studied in [18]. Acknowledgements This work was partially supported by Conacyt, grant number 56159-F. References [1] Ashtekar A., Lectures on non-perturbative canonical gravity (Notes prepared in collaboration with Ranjeet S. Tate), Advanced Series in Astrophysics and Cosmology, Vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. 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[26] Smolin L., Speziale S., A note on the Plebanski action with cosmological constant and an Immirzi parameter, Phys. Rev. D 81 (2010), 024032, 6 pages, arXiv:0908.3388. http://dx.doi.org/10.1103/PhysRevD.81.064033 http://arxiv.org/abs/0906.4524 http://dx.doi.org/10.1088/1742-6596/24/1/006 http://arxiv.org/abs/gr-qc/0602072 http://dx.doi.org/10.1088/0264-9381/23/7/004 http://arxiv.org/abs/gr-qc/0603076 http://dx.doi.org/10.1103/PhysRevD.81.024032 http://arxiv.org/abs/0908.3388 1 Introduction 2 CMPR action for gravity 2.1 CMPR formulation with a1=0 or a2=0 3 Transformation of the CMPR action for gravity 3.1 A particular transformation 3.1.1 Case A1=0: action with the invariant IJKL IJKL only 3.1.2 Case A2=0: action with the invariant IJIJ only 4 Application of the transformation to the coupling of the cosmological constant 5 Conclusions References

Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter

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