The canonical analysis of the lambda R model extended with the term due to Blas Pujolas and Sibiryakov BPS is performed. The analysis is developed for any value of lambda but particular attention is paid to the point lambdafrac13 because of the closeness with linearized General Relativity GR. Then we add the higherorder conformal term the socalled Cottonsquare term to study the constraint structure of what constitutes an example of kineticconformal Horavas gravity. At the conformal point an extra secondclass constraint appears this does not arise at other values of lambda. Then the Dirac brackets are constructed and we will observe that the lambda RCottonsquare model shares the same number of degrees of freedom with linearized GR.
['General Relativity and Quantum Cosmology', 'Mathematical Physics']
New canonical analysis for consistent extension of lambda R gravity
{'New canonical analysis for consistent extension of λR gravity': "Alberto Escalante ∗ and P. Fernando Oca˜na-Garc'ıa † Instituto de F'ısica, Benem'erita Universidad Aut'onoma de Puebla. Apartado Postal J-48 72570, Puebla Pue., M'exico, (Dated: September 25, 2024)  \nThe canonical analysis of the λR model extended with the term due to Blas, Pujolas, and Sibiryakov [ BPS ] is performed. The analysis is developed for any value of λ , but particular attention is paid to the point λ = 1 3 because of the closeness with linearized General Relativity [GR]. Then, we add the higher-order conformal term, the so-called Cotton-square term, to study the constraint structure of what constitutes an example of kinetic-conformal Horava's gravity. At the conformal point, an extra second-class constraint appears; this does not arise at other values of λ . Then, the Dirac brackets are constructed, and we will observe that the λR -Cotton-square model shares the same number of degrees of freedom with linearized GR .  \nPACS numbers: 98.80.-k,98.80.Cq", 'I. INTRODUCTION': "Hoˇrava gravity is a higher-order theory that stands out as a serious candidate for generating a complete quantum gravity theory by following the schemes of perturbative quantum field theory [13]. The remarkable features of Hoˇrava gravity are a preferred foliation of spacetime and invariance under the group of diffeomorphisms preserving this structure (FDiff), which conveniently allows us to consider anisotropy. The proposal of breaking the local Lorentz symmetry is intended to avoid the ghosts due to higher-time derivatives [4], representing an alternative path to solve the problems of unitarity and renormalizability [5, 6]. In fact, the renormalizability of the projectable version has been proven in [7], whereas the quantization of the non-projectable case has been hampered due to the difficulty posed by the presence of second-class constraints [8-10]. Remarkably, a proof has recently been presented in which the quantization is performed through the Batalin-FradkinVilkovisky (BFV) formalism, and the renormalization is achieved by using the approach of Barvinsky et al. based on the background field formalism [11].  \nRegarding the underlying structure, the preservation of the foliation provides an absolute distinction between time and space similar to the Newtonian one, which allows anisotropy by assuming a different scaling between space and time according to  \nt → b -z t, x i → b -1 x i , (1)  \nwhere z is the so-called critical exponent. On this basis, the theory is constructed with a potential containing terms with spatial derivatives of different orders and a kinetic part that employs solely time derivatives of order two. In order to ensure power-counting renormalizability, at least six-order terms must be considered in 3 + 1 dimensions [1].  \nConcerning the kinetic part, a central aspect in setting up the Hoˇrava action is the introduction of the λ parameter that determines the separate compatibility of the kinetic terms with FDiff. An outstanding feature due to its dynamic implications is that, in the realm of non-projectability, the kinetic part acquires an anisotropic conformal symmetry at λ = 1 3 [1]. The Weyl transformations are anisotropic in the sense that the lapse function scales with a weight different from the one of the spatial metric and the shift vector  \n˜ g ij = Ω 2 g ij , ˜ N = Ω 3 N, ˜ N i = Ω 2 N i , (2)  \nwhere the dependence of Ω = Ω( x, t ) is consistent only with non-projectability. Thus, by including the extension of the non-projectable potential provided by Blas, Pujol'as, and Sibiryakov (BPS) [12], we can generate a full anisotropic conformal theory if the potential is chosen to be conformal. Conversely, if the potential is not conformal, we obtain a not-conformal gravitational theory, which is called the kinetic-conformal Hoˇrava theory [13]. In both cases, the conformal symmetry of the kinetic part gives rise to a primary constraint that decreases the degrees of freedom of the theory, propagating two as in GR . However, this constraint changes from second-class in the kineticconformal case to a gauge symmetry associated with infinitesimal conformal transformations in the anisotropic-conformal case [13]. An excellent analysis of the dynamics of both versions is performed in [14].  \nOn the other hand, the physical feasibility of Hoˇrava's proposal can be assessed by its behavior to low energies, say, long distances. In this sense, the original non-projectable Hoˇrava gravity leads to the λR model, conformed by the z = 1 compatible terms. It is essential to highlight that although the Einstein-Hilbert action is obtained here identically when λ = 1, i.e., when it is restored the full diffeomorphism symmetry, the compatibility can be achieved regardless λ [15]. Consequently, the non-projectable case has field equations closer to GR . It is worth mentioning that implementing a complete canonical analysis has been fundamental to consolidating these results [15, 16]. Indeed, it is well known that the fundamental aspects of gauge theories can be better handled by employing this formalism [17, 18]. In a previous work [16], the canonical analysis of perturbative λR gravity was performed by implementing a 3 + 1 scheme based on the introduction of an extrinsic curvature type variable. This approach allowed for a closer identification of the constraints, just like in the familiar analysis of perturbative GR is done [19].  \nWith all discussed above, in this paper, by using the framework reported in [16], we perform a perturbative canonical analysis of the λR model extended with its corresponding z = 1 BPS term. The complete set of constraints and their classification into first and second-class for any value of λ are reported; in the analysis, we consider special attention for λ = 1 3 . This extended version gives rise to a prime example of kinetic-conformal Hoˇrava theory at this value. On the other hand, we  \nalso add the square of the Cotton tensor to the potential being a conformal six-order term and the analysis is developed. In this model, we study the structure of the constraints when considering a higher potential in what is regarded as a theory with the soft breaking of conformal symmetry [13]: although the Cotton-square term is conformally invariant, the extended λR is not.  \nThe paper is organized as follows. In section II, we present the perturbative analysis of the system λR plus the BPS term for any value of λ , with an emphasis on the modification to the constraint structure generated at the point λ = 1 3 . In section III the λR plus the Cotton-square term is analyzed. We report the constraint structure, and counting the degrees of freedom is carried out. Finally, the Dirac brackets are constructed for each case.", 'II. THE λR GRAVITY PLUS THE BPS TERM': "As we commented above, Hoˇrava theory is grounded in the group of diffeomorphism that preserves the foliation, given by  \nt → t ' ( t ) , x i → x ' i ( /vectorx, t ) , (3)  \nin coordinates adapted to the foliation. The analysis of the dynamics concerning this particular gauge group has been of great importance at the classical level. Its very structure suggests the presence of a strongly coupled additional degree of freedom to the two of GR [20]. Although it has been possible to find useful cosmological applications for this extra mode [21-23], initially, this put into debate the consistency of Hoˇrava theory, which at the IR regime would differ from GR and its well-tested predictions. Remarkably, the λR theory has been shown to be fully consistent with GR despite the reduced symmetry group [15, 16]. Here, the extra scalar mode that occurs in the full action is suppressed by the emergent constraints structure, thus only propagating two degrees of freedom.  \nOn the other hand, the extended λR model is the lowest-order effective action, up to second order in derivatives, of the complete Hoˇrava theory; it is constructed including the z = 1 BPS term that is symmetry-compatible and depends on the FDiff-covariant vector a i = ∂ i ln N , resulting in the following second-order action written in the Arnowitt-Deser-Misner formalism (ADM) [24].  \nS = ∫ dtd 3 x √ gN ( K ij K ij -λK 2 + R + αa i a i ) , (4)  \nwhere N is the lapse function, and its dependence on x and t characterizes the non-projectability, R is the spatial Ricci scalar, K ij = 1 2 N ( ˙ g ij -2 ∇ ( i N j ) ) is the extrinsic curvature and g ij is the Riemannian spatial metric.  \nOur canonical analysis will focus on the action (4) but it will be carried out in the perturbative sector. Although the action is originally written regarding ADM variables, we will use a different analysis method. Namely, we will use the perturbative 3+1 formalism, which is also compatible with the preferred time direction defined by FDiff, and it is helpful to economize the analysis at the perturbative level [16, 25, 26]. The implementation of this formalism starts by considering the  \nwell-known Fierz-Pauli Lagrangian for massless particles of spin 2 [27].  \nL FP = 1 4 ˙ h ij ˙ h ij -˙ h ij ∂ i h 0 j -˙ h j j ∂ i h 0 i -1 4 ( ˙ h i i ) 2 -1 2 ∂ i h 0 j ∂ i h 0 j + 1 2 ∂ i h j 0 ∂ j h i 0 + 1 2 ∂ i h 00 ∂ j h ij -1 2 ∂ i h k k ∂ j h ij -1 2 ∂ i h 00 ∂ i h k k + 1 4 ∂ i h j j ∂ i h k k + 1 2 ∂ i h jk ∂ j h ik -1 4 ∂ i h jk ∂ i h jk . (5)  \nThis action describes linearized gravity on a Minkowski background and it is written in its 3+1 form; the perturbation is given by g muν = η µν + h µν with η µν = diag( -1 , +1 , +1 , +1). The complete compatibility with FDiff is established by introducing an extrinsic curvature type variable given by K ij = 1 2 ( ˙ h ij -∂ i h 0 j -∂ j h 0 i ) , and expressing the kinetic part of L FP in a new fashion. In fact, the action will be written in a Horava-like form, and it will be in agreement with the action (4). Thus, introducing K ij and adding the linearized BPS term into L FP , we obtain  \nL = G ijkl K ij K kl -1 2 h 00 R -1 2 h ij ( R ij -1 2 δ ij R ) + α∂ i h 00 ∂ i h 00 , (6)  \nwhere  \nG ijkl = 1 2 ( δ ik δ jl + δ il δ jk ) -λδ ij δ kl , (7)  \nand the spatial part has been condensed by using  \nR ij = 1 2 ( ∂ k ∂ i h k j -∂ k ∂ k h ij -∂ j ∂ i h k k + ∂ j ∂ k h ik ) , R = ∂ i ∂ j h ij -∇ 2 h i i . (8)  \nThe λ parameter occurs in (6) using (7), and the introduction of G ijkl is relevant to simplify the calculation of the canonical momenta as we will see below.  \n/negationslash", 'A. Canonical analysis for λ = 1 3': 'Since the primary constraints depend only on the kinetic part, these are the same as the respective previous case. Thus, the primary Hamiltonian takes the form  \nH = G ijkl π kl π ij -2 h j 0 ∂ i π ij + 1 2 h 00 R + 1 2 h ij ( R ij -1 2 δ ij R ) -α∂ i h 00 ∂ i h 00 + w∂ i R j k ∂ i R k j -w∂ i R j i ∂ k R k j -3 w 8 ∂ i R∂ i R -w 2 ∂ i R i j ∂ j R + uπ 00 + u i π 0 i . (33)  \nSimilarly, since the high-order potential does not involve h 00 or h 0 i , the consistency of primary constraints results in the following secondary constraints  \nH : 1 2 R +2 α ∇ 2 h 00 ≈ 0 , H i : ∂ j π ji ≈ 0 . (34)  \nThe preservation in time of these constraints does not lead us to more constraints. Note that, at the perturbative level, the added potential does not affect the sets of first and second-class constraints. Thus, there are 3 degrees of freedom and the Dirac brackets are those found in (19) and (20). The same would be true for any higher-order potential that does not involve terms that depend on h 00 , such as the BPS terms.', 'B. Canonical analysis for λ = 1 3': "Following the same above consideration about the primary constraints, now the primary Hamiltonian takes the form  \nH = π ij π ij -2 ∂ i π ij h 0 j + 1 2 h 00 R + 1 2 h ij ( R ij -1 2 δ ij R ) -α∂ i h 00 ∂ i h 00 + w∂ i R j k ∂ i R k j -w∂ i R j i ∂ k R k j -3 w 8 ∂ i R∂ i R -w 2 ∂ i R i j ∂ j R + uπ 00 + u i π 0 i + vπ. (35)  \nIn addition to the set (34), the presence of π adds a secondary constraint whose structure is determined by the dependence on h ij of the high-order terms, so that it differs from its counterpart in (25). Thus, from consistency of π we obtain the following constraint  \nγ : ∇ 2 h 00 + 1 2 R +2 w ∇ 2 ∂ i ∂ j R ij + 1 2 w ∇ 2 ∇ 2 R ≈ 0 . (36)  \nThe presence of ∇ 2 h 00 in H and γ leads to expressions containing the Lagrange multiplier u when the consistency condition is applied, and as in the previous cases, the preservation of H i is identically satisfied. Thus we have obtained the complete set of constraints. To make the separation into first and second-class, let us note that the non-zero Poisson brackets between them are  \n{ γ, π 00 } = ∇ 2 δ 3 ( x -y ) , { π, γ } = ( ∇ 2 + w ∇ 2 ∇ 2 ∇ 2 ) δ 3 ( x -y ) , {H , π } = -∇ 2 δ 3 ( x -y ) , { H , π 00 } = 2 α ∇ 2 δ 3 ( x -y ) . (37)  \nThus, we obtain six first-class constraints given by  \nΓ i 1 : π 0 i ≈ 0 , Γ i 2 : ∂ j π ji ≈ 0 , (38)  \nand the following four second-class constraints  \nχ 1 : 1 2 R +2 α ∇ 2 h 00 ≈ 0 , χ 2 : π ≈ 0 , χ 3 : π 00 ≈ 0 , χ 4 : ∇ 2 h 00 + 1 2 R +2 w ∇ 2 ∂ i ∂ j R ij + 1 2 w ∇ 2 ∇ 2 R ≈ 0 . (39)  \nAlthough there is an obvious modification in the second-class constraints compared to (29), the gauge symmetries associated with the first-class constraints prevail, as well as the propagation of two degrees of freedom. We will now calculate the Dirac brackets that arise from the set (39). The matrix of the Poisson brackets between the second-class constraints is  \nC ab = χ 1 χ 2 χ 3 χ 4           χ 1 0 -∇ 2 2 α ∇ 2 0 χ 2 ∇ 2 0 0 ∇ 2 + w ∇ 2 ∇ 2 ∇ 2 χ 3 -2 α ∇ 2 0 0 -∇ 2 χ 4 0 -∇ 2 -w ∇ 2 ∇ 2 ∇ 2 ∇ 2 0 δ 3 ( x -y ) , (40)  \nand its inverse is given by  \nC ab = χ 1 χ 2 χ 3 χ 4           χ 1 0 1 1 + w ∇ 4 0 χ 2 -1 0 0 2 α χ 3 -1 -w ∇ 4 0 0 1 χ 4 0 -2 α -1 0 1 (1 -2 α (1 + w ∇ 4 )) ∇ 2 δ 3 ( x -y ) . (41)  \nThe Dirac brackets that can be built with this matrix are shown below  \n{ h 00 , π 00 } D = 0 , (42)  \n{ h ij , π 00 } D = 0 , (43)  \n{ h ij , π lm } D = 1 2 ( δ l i δ m j + δ m i δ l j ) δ 3 ( x -y ) + δ ij 2Ξ ( ∂ l ∂ m -δ lm ∇ 2 ) δ 3 ( x -y ) -αδ ij Ξ ( ∂ l ∂ m -δ lm ∇ 2 ) ( 1 + 3 w ∇ 4 ) . (44)  \nwhere Ξ = (1 -2 α (1 + w ∇ 4 )) ∇ 2 . Now, to observe the IR effective action, we take w → 0, and the Dirac brackets are reduced to  \n{ h ij , π lm } D = 1 2 ( δ l i δ m j + δ m i δ l j -δ ij δ lm ) δ 3 ( x -y ) + δ ij ∂ l ∂ m 2 ∇ 2 δ 3 ( x -y ) . (45)  \nIt is worth mentioning that in [31], a different higher-order model was studied; it was constructed in terms of linearized ADM variables and considered the dynamical part and a quadratic term R ij R ij , this theory propagates two degrees of freedom, and its fundamental brackets were given by (45). Furthermore, the first class constraints remain and we can fix the gauge. In fact, by fixing the gauge the following constraints arrive  \nχ 1 : 1 2 R +2 α ∇ 2 h 00 ≈ 0 , χ 2 : π ≈ 0 , χ 3 : π 00 ≈ 0 , χ 4 : ∇ 2 h 00 + 1 2 R +2 w ∇ 2 ∂ i ∂ j R ij + 1 2 w ∇ 2 ∇ 2 R ≈ 0 , χ 5 : π 0 i ≈ 0 , χ 6 : h 0 i ≈ 0 , χ 7 : ∂ j π ji ≈ 0 , χ 8 : ∂ j h ji ≈ 0 . (46)  \nThe matrix whose entries are the Poisson brackets between these constraints is given by  \n(  \n)  \nand after long calculations, it's inverse takes the form  \n(  \n)  \n<!-- image -->  \nwhere Γ = ( -1 + 2 α ( 1 + w ∇ 4 )) ∇ 4 and β = -1 + 2 α ( 1 + w ∇ 4 ) . Thus, the final Dirac's brackets  \n,  \nare given by  \n{ h ij , π lm } D = 1 2 ( δ l i δ m j + δ m i δ l j ) δ 3 ( x -y ) + 1 2Γ ( ∂ i ∂ j -δ ij ∇ 2 ) ( ∂ l ∂ m -δ lm ∇ 2 ) ( 1 -2 α ( 1 + 3 w ∇ 4 )) δ 3 ( x -y ) -1 2 ∇ 4 (( ∂ i ∂ l δ m j + ∂ i ∂ m δ l j + ∂ j ∂ l δ m i + ∂ j ∂ m δ l i ) ∇ 2 -2 ∂ i ∂ j ∂ l ∂ m ) δ 3 ( x -y ) , (49)  \nIn the IR limit, say w → 0, these brackets are reduced to those of linearized GR (30). If w = 0 and α = 1 2 , then these brackets are w independent.", 'III. LINEARIZED λR GRAVITY PLUS A COTTON-SQUARE TERM': 'Now we consider a Hoˇrava theory with soft breaking of conformal symmetry [13]. This is composed by the previous extended λR model that is not conformal and a square term of the Cotton tensor that is conformally invariant. The Cotton term that we shall add is C ij C ij , where  \nC ij = /epsilon1 ikl ∇ k ( R j l -1 4 Rδ j l ) . (31)  \nOn a Minkowski background, the linearized Lagrangian now is written as  \nL = G ijkl K ij K kl -1 2 h 00 R -1 2 h ij ( R ij -1 2 δ ij R ) + α∂ i h 00 ∂ i h 00 -w∂ i R j k ∂ i R k j + w∂ i R j i ∂ k R k j + 3 w 8 ∂ i R∂ i R + w 2 ∂ i R i j ∂ j R, (32)  \nwhere w is an arbitrary constant. Now we proceed to perform the canonical analysis.  \n/negationslash', 'IV. CONCLUSIONS': "/negationslash  \nThe Hamiltonian analysis for the extended λR model and the Hoˇrava theory with smooth breaking of conformal symmetry for any value of λ were reported. The constraints and the fundamental Dirac's brackets were found for the extended model at λ = 1 3 . We observed that the theory propagates three degrees of freedom at this value, one more than linearized GR . In addition, the BPS constant can not be zero, indicating a strong coupling of the field h 00 . Furthermore, at the critical point, the constraints were found and classified, then we observed that the model propagates two degrees of freedom, just like linearized GR . The fundamental Dirac's brackets were constructed, and we obtained those reported for λR gravity and linearized GR ; the BPS constant does not appear in the fundamental brackets.  \n/negationslash  \nOn the other hand, at the value λ = 3 , the λR -Cotton-square action presents three degrees of freedom, and the canonical structure was similar to that for the extended model at the same value. However, at the critical point, the action shares a canonical structure with linearized GR . In fact, the theory propagates two degrees of freedom, if we take the limit in the IR, say w → 0, then the fundamental brackets are reduced to those reported for linearized GR and λR gravity. In addition, the BSP constant is not restricted; if α = 1 2 , then the brackets are w independent, this is a difference between the extended λR and λR -Cotton-square gravity at the critical point. 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