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Canonical analysis for Chern-Simons modification of general relativity

By using the Gitman-Lyakhovich-Tyutin canonical analysis for higher-order theories a four-dimensional Chern-Simons modification of general relativity is analyzed. The counting of physical degrees of freedom, the symmetries, and the fundamental Dirac brackets are reported. Additionally, we report the complete structure of the constraints and its Dirac algebra is developed.


I. INTRODUCTION
It is well known that the addition of topological terms to physical theories presents contributions that could change the original physical description of the base theory. In fact, topological terms can transform the canonical structure as well as increase the number of physical degrees of freedom.
For instance, the addition of the topological Chern-Simons [CS] term to three-dimensional Maxwell theory turns the gauge fields into massive ones, and the dynamics are modified because there is an extra degree of freedom [1][2][3]. In the four-dimensional case, the coupling of the Pontryagin invariant to Maxwell's theory modifies the photon physics [4]. In fact, the dynamics are modified, now the photon retains two degrees of freedom, but in vacuum Lorentz invariance and parity are lost. In the case of theories describing gravity, there are also interesting scenarios in which topological theories or topological terms play an interesting role in the physical degrees of freedom of the gravitational field. In the tree dimensional case, the coupling of the CS invariant to Einstein Hilbert [EH] action provides mass to the graviton and modifies the symplectic structure of theory [5]. Moreover, in the four-dimensional stage, the Macdowell-Mansouri formulation is worth of mention. In fact, that formulation breaks down the symmetry of a BF theory from SO (5) to SO(4) obtaining the Palatini action plus the sum of topological invariants [6][7][8]; thus, the braking symmetry of a topological theory leads to new physical degrees of freedom. On the other hand, in the canonical gravity context, there is also the well-known Holst action [9]. The Holst action is given by the Palatini action plus a topological term, it depends on the so-called Barbero-Immirizi [BI] parameter and provides a set of actions classically equivalent to Einstein's theory. In fact, both Palatini and Holst actions share the same equations of motion, however, the topological term through the BI parameter contributes at the classical level in the symplectic structure of the theory. Furthermore, by coupling of the Holst action with fermionic mater, the BI parameter determines the coupling constant interaction between fermions [10]. From the quantum point of view, this parameter gives a contribution in the quantum spectra of the area and volume operators in the context of loop quantum gravity [11][12][13]. Moreover, the Holst term facilitates the canonical description of gravity, and depending on the values of the BI parameter one can reproduce the different scenarios found in canonical gravity.
On the other hand, there is a model found in the literature proposed by R. Jackiw and Y. Pi [JY] where the Einstein-Hilbert [EH] action is coupled through a auxiliary field to the topological Pontryagin term [14]. Under a particular configuration of the external field, this theory presents some interesting features: the Schwarzschild metric is also a solution of the modified theory, thus, there are no changes in the basic predictions of general relativity. Moreover, the theory describes the propagation of gravitational waves with velocity c, but these carry different intensities and violate spatial reflection symmetry. However, the analysis reported in [14] was performed at Lagrangian level and we have already mentioned that the presence of any modification could change the canonical structure of the base theory or the number of physical degrees of freedom. Thus, it is mandatory to perform a canonical analysis in order to determine if there are any changes in the canonical structure of the constraints in the modified theory. Keeping in mind that the constraints are the best guideline to perform a canonical quantization.
With all this being said, the cornerstone of this paper is to present the canonical analysis of the theory reported in [14]. Our study will be developed by using a perturbation around the Minkowski spacetime, this particular choice will show that the JY theory is a higher-order singular theory in the temporal derivatives. Furtheremore, it is worth commenting that the canonical analysis of gauge higher-order theories is a difficult task to develop and modern tools for performing the analysis are needed. In this respect, there are two approaches for performing the analysis of higher-order gauge theories the so-called Ostrogradski-Dirac [OD] and Gitman-Lyakhovich-Tyutin [GLT] methods [15][16][17]. The former is based on the extension of the phase space, where the choice of the fields and their temporal derivatives become the canonical variables, thus, a generalization of the canonical momenta for the higher-temporal derivative of the fields is introduced. However, it is claimed that in some cases the usual OD framework does not allow an easy identification of the complete structure of the constraints, then the constraints are fixed by hand in order to achieve consistency [18]. On the other hand, the GLT framework is a generalization of Ostrogradski's method, based on the introduction of extra fields reducing a problem with higher temporal derivatives to ones with firstorder time derivatives. Then by using either the definition of the momenta for all the fields or the introduction of the Dirac brackets, the second class constraints, and non-physical degrees of freedom (extra fields) can be removed [19,20]. In this manner, in this paper, we will use the GLT formalism to perform our analysis due to it is sturdiness at handling the constraints and their consistency.
In fact, we will report that there exists a contribution to the canonical structure of the EH theory due to the CS modification. We will report the complete structure of the constraints and we will perform the counting of degrees of freedom, then we will remove the non-physical degrees of freedom by introducing the fundamental Dirac-brackets and the Dirac algebra between all constraints will be reported.
The paper is organized as follows. In Section II we present a brief description of the JY theory, then the GLT analysis is performed. We will report the complete canonical structure of the theory, namely, the complete structure of the constraints, the counting of physical degrees of freedom is developed and the fundamental Dirac brackets are constructed. In Section III we present the conclusions.

II. THE GITMAN-LYAKHOVICH-TYUTIN ANALYSIS.
We start with the following action composed by the EH action plus the Pontryagin invariant [14], this is where g µν is the metric tensor, θ is a coupling field which in general is not constant and * R σ τ µν = 1 2 ε µναβ R σ τ αβ . Along this paper we will use Greek letters for labeling the spacetime indices µ, ν, .., α = 0, ..., 3 and latin ones i, j.., k = 1, 2, 3 for labeling space indices. Furthermore, we can rewrite the action by noticing that Then, up to a boundary term the action takes the form where v µ ≡ ∂ µ θ. The variation of the action (3) yields the following equations of motion here G µν is the Einstein tensor and is a four dimensional Cotton-type tensor. We can fix a particular form of the coupling field, namely thus, the Schwarzchild metric is also a solution of (4) [14,21] and the classical test of GR are considered in the modification. In this manner, we are interested in performing a canonical analysis of the action (3) in order to find any contribution of the CS term to the well-known canonical structure of GR. For our aims, we shall find the linearized action around the Minkowski spacetime by using the usual perturbation hence, by considering (5) and (6) into (1) we get variation of (7) yields where here is the D'Alambertian operator. Furthermore, we can observe that the action (7) can be written as hence, this new fashion of the action (7) will be analyzed bellow. As far as we know, the canonical analysis of the action (10) has not been reported in the literature.
We will remove the overall factor of − 1 2 from (10), which of course, does not change the dynamics. Moreover, we perform the 3 + 1 decomposition of action (10), thus, the action takes the following where we have defined µ ≡ 2Ω. We can observe that the action is a higher order theory and we will use the GLT method for performing the canonical analysis. By following the GLT framework, we introduce the following change of variables [19,20] G µν ≡ḣ µν , hence, by introducing these new variables through Lagrange multipliers, the action (11) will take the form where We now introduce the momenta (π µν , p µν ,v µν , Λ 1 µν , Λ 2 µν ) canonically conjugate to (h µν , G µν , v µν , λ µν 1 , λ µν 2 ), this is We observe that the equations (15)-(16) allows us to identify the Lagrange multipliers (λ µν 1 , λ µν 2 ) as canonical variables through the momenta (π µν , p µν ) respectively. Aditionally, the primary constraints will be given by [19] in this manner, we identify the following primary constraints ϕ 00 ≡ p 00 ≈ 0, and the fundamental Poisson brackets of the theory will be expressed as With all momenta identified we define the canonical Hamiltonian given by thus, the primary Hamiltonian will be expressed in the following form where ∆ µν are Lagrange multipliers enforcing the primary constraints. In order to identify further constraints, we will calculate the consistency conditions of the primary constraints. From the consistency conditions we obtain the following secondary constraintṡ Thus, from consistency of the secondary constraints (26), (27) and (28) we obtaiṅ where we identify the following tertiary constraints and the following relations for the multipliers ∆ ij Now, we calculate consistency on the tertiary constraints (32) and (33), we finḋ where we observe that there are no more constraints because (36) gives relations for the Lagrange multipliers. On the other hand, we can observe from the relations of the Lagrange multipliers that ∂ j (34) = (36), thus, the process for identifying further constraints is finished. Therefore, the full set of constraints is given by With all constraints at hand, we need to classify them into first class and second class. For this aim we form the matrix W IJ , whose entries are the Poisson brackets between all constraints, this is where W IJ is a 24×24 matrix. The non-zero Poisson brackets in W IJ are given by Hence, after a long algebraic work, we find that the matrix W IJ has 12 null vectors, therefore, this implies that there are 12 first class constraints [20], which are given by and the matrix has a rank=12, then the rank allows us to identify the following 12 second class constraints The Poisson brackets between those constraints are given by In this manner, with all constraints classified into first and second class, we can perform the counting of physical degrees of freedom as follows: there are 40 canonical variables (h µν , G µν ), the number of first class constraints is 12 and there are 12 second class constraints, therefore the number of physical degrees of freedom is DOF = 1 2 [(40) − 2(12) − 12] = 2, just like the EH action. It is important to comment that those degrees of freedom are associated to two linearly independent polarizations of gravitational waves.
On the other hand, we can remove the second class constraints by introducing the Dirac brackets.
In fact, the Dirac brackets are defined by where (C ab ) −1 is the inverse of the matrix C ab whose entries are the Poisson brackets between the second class constraints, this is the explicit form of C ab is given by we can observe that this matrix is not singular, therefore it has an inverse. Hence, after long algebraic manipulations, we find the following non-trivial Dirac brackets where Ξ = −µ 2 (∇ 2 + µ 2 )(∇ 4 + µ 2 4 ). We can see a contribution due to the CS term which could be important in the analysis of quantization of the theory. Furthermore, with this new set of brackets one can calculate the Dirac algebra between the first class constraints and the second class constraints, the Dirac brackets are given by Once we have introduced the Dirac brackets we proceed to identify the symmetries of the theory, for instance, the gauge transformations. For this part we will use the first class constraints and the Dirac brackets calculated above. Hence, the local gauge transformations are given by where ω α are gauge parameters. From the calculation of (53) for the perturbation h αβ and the G αβ variables, we obtain δG 00 = ω 1 , and G µν in the following way moreover, we can express (59)-(64) in a covariant way by introducing the following ω 6 = −2Λ 0 and ω 4i = −2Λ i , thus, the gauge transformations reads The action (10) is invariant under the transformations (65). In fact, we observe that (9) transforms like δG lin µν = 0 and δC lin µν = 0. Furthermore, we observe that ∂ µ G lin µν = 0 and ∂ µ C lin µν = 0, therefore

III. CONCLUSIONS
In this paper, a detailed GLT analysis for JY theory has been performed. Using the null vectors, we have reported the complete structure of the constraints. Furthermore, with the correct constraints at hand, the counting of physical degrees of freedom was performed; we identified two physical degrees of freedom, thus, the EH and JY theories share either the same classical test of gravity or the number of physical degrees of freedom. Despite this similarity, both theories present significant differences at the fundamental Dirac brackets level due to the modification, of course, if we take the standard limit on µ → ∞, then the usual Dirac brackets of EH theory are recovered [18,25]. In this respect, the Dirac brackets were useful to improve Ostrogradski's instability. In fact, with the Dirac brackets at hand, we used the second class constraints for eliminating the linear momenta terms in the Hamiltonian. Furthermore, Dirac's brackets were also used to find the involution relations between the constraints , as well as the unfree gauge symmetry. It is important to comment that the Dirac brackets could be important in the analysis of quantum aspects, like the calculation of the propagators of the fields. However, the calculations of the propagatos is a dificult task when higher derivatives are present [26]. In any case, these ideas are being analyzed in the non-perturbative and perturbative sector, they are still in progress and this work will be the base for the subject of forthcoming works [27].