Hamiltonian analysis for new massive gravity

A detailed canonical analysis for three- dimensional massive gravity is performed. The construction of the fundamental Dirac brackets, the complete structure of the constraints and the counting of the physical degrees of freedom are reported. In addition, it is shown that the extended Hamiltonian is healed from Orstrogradki’s instabilities.


I. INTRODUCTION
It is well-known that topological massive gravity [T M G] is a toy model, however, it is a great laboratory for testing new ideas which could be useful for the understanding of either the classical or quantum structure of gravity.T M G is constructed by the Einstein-Hilbert [EH] action coupled with a Chern-Simons term; at the linearized regime it describes the propagation of a single massive state of helicity ±2 on a Minkowski background [1][2][3].The theory breaks parity and it is not-unitary.
In fact, it is claimed that if Newton's constant is negative, then black hole states lead to non-unitary theory.Moreover, if Newton's constant is positive and the Chern-Simons coupling is not tuned to the chiral point, then massive excitation lead also to non-unitarity.However, T M G theory is a dynamical model that is naively power counting renormalizable; all these features make the theory so interesting [4][5][6][7].
On the other hand, there are alternative models for describing the propagation of physical degrees of freedom in three dimensions, the so-called Minimal Massive Gravity [M M G] and New Massive Gravity [N M G] [8][9][10].M M G is an extension of T M G that includes curvature-squared symmetric tensors and describes the propagation of a single local degree of freedom about an anti-de Sitter (AdS) vacuum, this propagation mode is physical; it is neither a tachyon nor a ghost.For this reason, M M G is an attractive model due to it can be explored in the (AdS) and quantum field holographic correspondence.On the other hand, N M G is a parity-conserving theory and describes the propagation of two massive degrees of freedom of helicity ±2.These properties make the theory very interesting because it shares the same number of degrees of freedom with general relativity, considering the clear exception that in three dimensions the physical degrees of freedom are massive.The theory is constructed by employing the EH action together with squared terms of the theory that could suffer from Ostrogradski's instabilities.In this respect, the theory was analyzed at Lagrangian level in [9], however, a detailed Hamiltonian analysis has not been reported in the literature.A Hamiltonian description of the theory would allow us to understand how to exorcise the apparently ghosts in N M G, and the canonical structure of the theory would help to make any progress in the quantization program.The usual method used for analyzing higher-order theories is by employing the Ostrogradski-Dirac [OD] method [11][12][13][14][15].This is based on the extension of the phase space, where the fields and their temporal derivatives become the configuration variables, thus, it is introduced a generalization of the canonical momenta for the higher-temporal derivatives of the fields.However, it is claimed that the OD method does not allow direct identification of the complete structure of the constraints, so the constraints are fixed by hand to achieve consistency [3].With all commented above, in this paper we will perform the canonical analysis of N M G by following a different scheme from that established in the OD framework.To this end, a set of variables will be introduced to reduce the second-order time derivatives to first-order.Then, by using the null vectors of the theory, the correct structure of the constraints is obtained without the need to fix them by hand as in other approaches is done [16][17][18].Furthermore, we will introduce the Dirac brackets, thus the second-class constraints will be useful for either to build the extended Hamiltonian or healing the Ostrogradski sickness [19].It is worth commenting, that usually the extended Hamiltonian is not built on a standard canonical analysis.In fact, if there are non-trivial second-class constraints, then its construction is a difficult task.For constructing the extended Hamiltonian, we need to calculate the Dirac brackets and identify all Lagrange multipliers associated with the second-class constraints, then, the second class constraints are used for obtaining the new structure of the Hamiltonian.Thus, the structure of the extended Hamiltonian will allow us to observe if the Ostrogradski instability could be healed.To complete the analysis, we will study a close model to N M G.This model will present a ghost and the trivial structure of the second-class constraints will implies that the Ostrogradski instability will not be healed, so the extended Hamiltonian will be unstable.
The paper is organized as follows: in Section II, through the implementation of a set of auxiliary variables and their respective Lagrange multipliers, the canonical analysis of N M G will be performed.The correct structure of the constraints will be reported; with all constraints at hand, the Dirac brackets will be calculated and the extended Hamiltonian will be constructed.In addition, the counting of the physical degrees of freedom will be done and the Dirac algebra between the extended Hamiltonian and the first-class constraints is reported.To highlight the role of the extended Hamiltonian in the identification of ghosts, a close model to N M G in Appendix A is studied.The same auxiliary variables of Section II are introduced and the extended Hamiltonian is also constructed, then the differences between the theories are discussed.We finish the paper with the conclusions.
We start our analysis with the following action [9,10] where g µν is the metric tensor, α, β, ... = 0, 1, 2; κ is a constant with dimension of mass in fundamental units [κ] = −1/2, m is a "relative" mass parameter and J is a higher-order term given by From the action, the following equations of motion arise where G µν = R µν − 1 2 Rg µν is the Einstein tensor and J µν is given by here is the D'Alambertian operator.We observe that the trace of (3) implies For our aims, we will work by considering the standard perturbation of the metric around a Minkowski background, g µν = η µν + h µν .Hence, the linearized versions of the Ricci tensor and scalar curvature takes the form In this manner, by using (6) into (3) and ( 5), the following linearized equations of motion arise We observe that the first equation of motion is a Klein-Gordon equation for the linerarized Einstein tensor.The second one, is an implicit constraint that at Hamiltonian level enforces the presence of second-class constraints, this will be shown below.
Furthermore, another way for obtaining the linearized equations ( 7) is by the variation of the linearized version of (1), which gives the following Lagrangian The Lagrangian (8) will be our subject of study.In fact, by performing the 2 + 1 decomposition, we obtain the following form of the Lagrangian where As far as we know, a complete canonical analysis of the Lagrangian ( 9) has not been reported in the literature.In this respect, we have commented above that the standard way for analyzing a higher-order theory is by using the OD framework [11,12], however, this approach is long and the constraints are not under control.So, we will perform our analysis by introducing the following variables [20,21] these variables are an extrinsic curvature type.In this way, the Lagrangian is rewritten in the following new fashion where α ij are Lagrange multipliers enforcing the relation (10) and The new canonical variables of the system are given by h µν , K ij , α ij and their corresponding canonical momenta given by π µν , P ij and τ ij .From the definition of the momenta, we identify the following relations and the fundamental Poisson brackets between the canonical variables will be From the Lagrangian (11) and the canonical momenta (12), we can construct the canonical Hamiltonian, it is given by we can observe the presence of linear terms in the conjugate momenta π ij , which could be related to an Ostrogradski's instability.However, in further lines we will remove this apparently instability by means the Dirac brackets and the second-class constraints.
On the other hand, from the definition of the momenta, we identify the following primary constraints φ 00 : π 00 ≈ 0, we observe from the definition of the momenta (12) that the velocity K can not be expressed in terms of the canonical momenta, then the constraint ψ arise.The constraint ψ is not trivial and it will be of second-class, this constraint will be useful at the end of the analysis for removing the apparent Ostrogradski's instability.On the other hand, we calculate the non-zero Poisson brackets between primary constraints which are given by Thus, the primary Hamiltonian reads where u µν , ζ ij and ξ are Lagrange multipliers enforcing the primary constraints.With the primary Hamiltonian, we calculate the consistency conditions on the primary constraints, thus, we obtain from these equations we can observe that S, S i , W are secondary constraints while S ij , Q ij only give relations between the Lagrange multipliers.From consistency of the secondary constraints we find that we can observe that not further constraints arise.Hence, the complete set of constraints is given by φ 00 : π 00 ≈ 0, With all constraints at hand, we can perform their classification into first and second-class.For this step we calculate the following matrix, whose entries are the Poisson brackets between all constraints where the non-zero Poisson brackets between the constraints are given by After a long calculation, we can see that the matrix (20) has a rank= 8 and 6 null vectors.From the null vectors we find the following 6 first-class constraints which allow us to identify the following 8 second-class constraints where Γ i 3 is the so-called Gauss constraint.With the classification of the constraints, we carry out the counting of physical degrees of freedom as follows: there are 24 canonical variables, eight second-class constraints and six first-class constraints, thus these degrees of freedom corresponds to massive modes of helicities ±2 [9].
We shall introduce the Dirac brackets.To achieve this, we first calculate the algebra between the second-class constraints, it is given by and we express it in matrix form as follows It is well known that the Dirac brackets play an important role in the quantization of any theory with second-class constraints.In fact, they are promoted to commutators and they can be used for the identification of observables.On the other hand, at classical level either the Dirac brackets or the second class constraints can be used for constructing the extended Hamiltonian.It is worth commenting, that the equations of motion obtained by means of the extended Hamiltonian are mathematically different from the Euler-Lagrange equations, but the difference is unphysical, thus, the construction of the extended Hamiltonian is important.In this manner, the Dirac brackets are defined as where C αβ is the inverse of (26) given by Hence, we obtain the following non-trivial Dirac brackets Now, we shall construct the extended Hamiltonian, which is a fundamental element in the canonical formulation.In fact, the extended Hamiltonian is a first-class function and it is used in the quantization program due to it contains all relevant information of the theory.The extended Hamiltonian is defined by where w α are the Lagrange multipliers associated with the second-class constraints, these multipliers can be determined through [23,24] In this manner, by using the second-class constraints (23) and the matrix (28) the following expressions for the Lagrange multipliers are obtained hence, by using the second-class constraints (23) and the Lagrange multipliers (32) into the extended Hamiltonian, it will take the following form we can observe a squared term in the momenta π which implies that the Hamiltonian is bounded from below.This means that the two degrees of freedom are physical rather than ghosts, in agreement with the Lagrangian analysis reported in [9].Furthermore, we can see that the inclusion of the second-class constraints fixes the Ostrogradski instability, hence, for obtaining detailed results in the canonical formalism of higher-order theories, it is mandatory to develop an analysis as has been done in this work.In this respect, we have added the appendix A where the system Klein-Gordon-Einstein (see 7) without the constraint R Lin = 0 has been analyzed.This theory is described by a higher-order Lagrangian and we will observe that its second-class constraints will have a trivial structure and the Ostrogradski instability will not be removed.
We complete our canonical analysis with the Dirac algebra between the first-class constraints and the extended Hamiltonian, this is given by where we observe that the algebra is closed and the extended Hamiltonian is of first-class as expected.
Furthermore, by using the Dirac brackets and the extended Hamiltonian it is possible to find the the following equations of motion which is the relation between K ij and h ij given in (10), and namely that corresponds to the equation of motion R Lin = 0. Hence, our analysis is complete and extends those results reported in literature.
Furthermore, if we perform the linearization of the action (39) we can identify the following linearized We observe that the system corresponds to a higher-order theory as expected.Moreover, if we perform the 2+1 decomposition and the variables (10) are introduced, then the Lagrangian is rewritten in the following new fashion The canonical variables of the system are given by h µν , α ij and K ij , and its corresponding canonical momenta π µν , τ ij and P ij .Hence, by performing the canonical analysis, the main results are the following: the canonical Hamiltonian is given by where we can observe that there is only a linear term in the canonical momenta π ij and this fact will be associated to the presence of ghosts degrees of freedom.
On the other hand, the complete set of constraints is given by the following six first-class constraints and the following six second-class constraints We can observe that the second-class constraints have a trivial structure, and it is easy to observe that the Dirac and Poisson brackets coincide to each other, thus we expect that the instability of the Hamiltonian (44) will be present.In fact, we can use the Dirac brackets and the second-class constraints for calculating the extended Hamiltonian, we obtain In this case, we can see that second class constraints do not heal the instability and there will be ghosts.In fact, the counting of physical degrees of freedom is performed as follows: there are 24 canonical variables, six first-class and six second-class constraints, so now there are two modes of helicity ±2 and one mode of zero helicity being a ghost.Our results complete and extend those reported in [9].

V. APPENDIX B
In this appendix we will resume the canonical analysis of the higher-order term given in the action (1), this term is expressed by By performing the linearization and the change of variables given in (10), we find the following Lagrangian Hence, the results from the canonical analysis are the following: the canonical Hamiltonian is given by where we observe again the linear term in the π ij momenta.Furthermore, the complete set of the constraints is given by the following eight first-class constraints Γ 1 : π 00 ≈ 0, Γ i 2 : π 0i ≈ 0, Γ i 3 : ∂ j π ij ≈ 0, Γ 4 : ∂ i ∂ j P ij ≈ 0, and the following six second-class constraints It is worth commenting, that the second-class constraints have a trivial form just like the system analyzed in appendix A, therefore, the Dirac brackets will be trivial.In this manner, with the results obtained in previous sections we expect that the system (48) will present the Ostrogradski sickness.
In this respect, we can carry out the counting of physical degrees of freedom as follows: there are 24 canonical variables, eight first-class constraints and six second-class constraints, thus this degree of freedom corresponds to a ghost.
We would like to thank Daniel Grumiller for the comments and cites recommended.