Canonical analysis of $n$-dimensional Palatini action without second-class constraints

We carry out the canonical analysis of the $n$-dimensional Palatini action with or without a cosmological constant $(n\geq3)$ introducing neither second-class constraints nor resorting to any gauge fixing. This is accomplished by providing an expression for the spatial components of the connection that allows us to isolate the nondynamical variables present among them, which can later be eliminated from the action by using their own equation of motion. As a result, we obtain the description of the phase space of general relativity in terms of manifestly $SO(n-1,1)$ [or $SO(n)$] covariant variables subject to first-class constraints only, with no second-class constraints arising during the process. Afterwards, we perform, at the covariant level, a canonical transformation to a set of variables in terms of which the above constraints take a simpler form. Finally, we impose the time gauge and make contact with the $SO(n-1)$ ADM formalism.


INTRODUCTION
The canonical analysis of general relativity has a very long history starting with attempts by Dirac himself (see for instance Refs. [1,2]). However, it was not until the discovery of the ADM variables for general relativity [3] that the program to canonically quantize gravity acquired a suitable and feasible form. These variables arise from the canonical analysis of the Einstein-Hilbert action through the parametrization of the spacetime metric g µν in terms of the lapse function N , the shift vector N a , and the spatial metric q ab := g ab . It turns out that in the resulting Hamiltonian form of the action both N and N a play the role of Lagrange multipliers imposing the scalar (or Hamiltonian) and diffeomorphism constraints, respectively, whereas q ab and its canonically conjugate momentump ab -an object related to the extrinsic curvature-constitute the canonical variables that label the points of the phase space. Even though the canonical quantization program emerging from this approach has failed [4], the ADM variables have been extensively used in other instances of general relativity such as initial value problems, spacetime symmetries, asymptotic behavior of gravitational fields, numerical relativity, etc.
On the other hand, the metric formulation is not the appropriate theoretical framework to couple fermion fields to general relativity, for which we have to use the first-order formalism of the theory, where the fundamental variables are an orthonormal frame of 1-forms e I (vielbein) and an SO(n − 1, 1) or SO(n) connection 1form ω I J depending on whether the spacetime metric * merced@fis.cinvestav.mx † rescobedo@fis.cinvestav.mx ‡ ljromero@fis.cinvestav.mx § mcelada@matmor.unam.mx has Lorentzian or Euclidean signature. The equations of motion of the theory are then obtained from the Palatini (also called Einstein-Cartan or Hilbert-Palatini) action.
The standard canonical analysis of the Palatini action involves second-class constraints, which must be either handled with the Dirac bracket [5], or explicitly solved. In 4-dimensional spacetimes, the second-class constraints are irreducible [6] and can be explicitly solved in a manifestly SO (3,1) [or SO (4)] covariant fashion [6,7], whereas in dimensions higher than four they are reducible but can be handled using the approach of Refs. [8,9], where the original second-class constraints are replaced with an equivalent (irreducible) set of constraints that can be explicitly solved. The second-class constraints in dimensions equal or higher than four can also be solved using the approach of Ref. [10]-where the second-class constraints emerging from the canonical analysis of the Holst action [11] are explicitly solved in a manifestly SO (3,1) [or SO (4)] covariant fashion-because that technique is generic and is not restricted to 4-dimensional spacetimes. However, it was recently shown in Ref. [12] that it is possible to perform a manifestly SO(3, 1) [or SO(4)] covariant canonical analysis of the Holst action involving first-class constraints only, i.e., without introducing second-class constraints whatsoever in the Hamiltonian formalism. It is clear from that approach that the second-class constraints are unnecessary and superfluous for doing the canonical analysis of the Holst action, and thus they are also unnecessary for doing the Hamiltonian analysis of the 4-dimensional Palatini action as can be seen from taking the limit γ → ∞ in Ref. [12], where γ is the Immirzi parameter [13].
In this paper we extend the theoretical approach of Ref. [12] to higher dimensions and perform from scratch the canonical analysis of the n-dimensional Palatini action with a cosmological constant. In this framework, the original frame variables e µ I are parametrized in terms of the momentum variables, the lapse function, and the shift vector, whereas the original connection variables ω µ I J are expressed in terms of the configuration variables, some auxiliary fields, and some Lagrange multipliers. The outstanding aspect of this parametrization is that it straightforwardly leads to the Hamiltonian form of the n-dimensional Palatini action after getting rid of the auxiliary fields involved in the action. Moreover, the resulting canonical formulation is manifestly SO(n−1, 1) [or SO(n)] covariant and features first-class constraints only.
This paper is organized as follows. In Sec. II we perform the (n − 1) + 1 decomposition of the n-dimensional Palatini action with or without a cosmological constant (n ≥ 3) and provide the appropriate parametrizations of the frame and the connection. We then identify the auxiliary fields present in the action and eliminate them, thus getting the Hamiltonian form of the n-dimensional Palatini action with manifest local SO(n − 1, 1) [or SO(n)] symmetry that involves just first-class constraints. In Sec. III we perform a canonical transformation to new SO(n − 1, 1) [or SO(n)] variables that simplify the expressions of the constraints. In Sec. IV we impose the time gauge and obtain the SO(n − 1) ADM formulation of general relativity. In Sec. V we give some conclusions. In addition, in Appendix A we discuss in detail the 3dimensional Palatini action (for which the auxiliary fields are absent from the very beginning), and in Appendix B we depict an alternative approach for the 4-dimensional case.
In the first-order formalism, general relativity with a vanishing or nonvanishing cosmological constant Λ is described by the Palatini (or Einstein-Cartan) action 1 where F I J := dω I J + ω I K ∧ ω K J is the curvature of ω I J , ρ := (1/n!) I1···In e I1 ∧ · · · ∧ e In is the volume form of M , κ is a constant related to Newton's constant, and " " is the Hodge dual map given by (2) To perform the canonical analysis of the action (1), we first make the (n−1)+1 decomposition of it by expressing the frame and the connection respectively as e I = e t I dt+ e a I dx a and ω I J = ω t I J dt+ω a I J dx a . It is also convenient to introduce the unit normal to each leaf Σ, n := n I e I , that fulfills n I n I = σ and n(∂ a ) = 0 (or, equivalently, e a I n I = 0), which has the following explicit form: with q := det(q ab ) > 0 (of weight +2), q ab := e aI e b I being the induced metric on Σ, whose inverse is denoted by q ab . This object allows us to introduce the projector on the orthogonal plane to n I as q I J := q ab e a I e bJ = δ I J − σn I n J .
Therefore, the (n − 1) + 1 decomposition of the action (1) is given by (we recall that all spatial boundary terms will be neglected because Σ has no boundary) where we have defined being the curvature of ω a I J and where we have also suppressed a wedge product between dt and d n−1 x := dx 1 ∧ · · · ∧ dx n−1 in (5) to simplify notation.
To continue our analysis, we express e t I in terms of the lapse function N and the shift vector N a [3] as and compute the inverse of the expression (6a) whereh ab is the inverse ofh ab :=Π aIΠb I and h := det(h ab ) has weight 2(n − 2). Notice that the right-hand side of (8) is a function ofΠ aI only. As a consequence of this, n I in (3) can also be expressed in terms ofΠ aI as (9) Substituting (8) and (9) into the right-hand side of (7) we can reinterpret e t I as a function of the n 2 variables N , N a , andΠ aI . With this in mind, relations (7) and (8) define a one-to-one map from the n 2 variables N , N a , andΠ aI to the original n 2 frame components e α I . The inverse map that sends e α I to N , N a , andΠ aI is given by (6a) together with where n I must be understood as that given by (3). Therefore, using (7), (8), and (9), the action (5) acquires the form 2 For future purposes, we introduce the covariant derivative ∇ a defined on each leaf Σ that annihilates e a I through with Γ aIJ = −Γ aJI and Γ a bc = Γ a cb . These are n(n−1) 2 inhomogeneous linear equations for n(n−1) 2 /2 unknowns 2 From (8) we get h = q n−2 , and thus √ q = h 1 2(n−2) . Γ aIJ and n(n − 1) 2 /2 unknowns Γ a bc , so that the solution is unique. It turns out that Γ a bc are the Christoffel symbols associated with the induced metric q ab on Σ, whereas the explicit solution for Γ aIJ is given by Furthermore, from (6a) and (13), we find that the operator ∇ a annihilatesΠ aI as well Either by solving this equation similarly as we did for (13) or simply by substituting (8) into the right-hand-side of (14), we find Now, following the same approach of Refs. [10,12], we realize that the term involving ∂ t ω aIJ in (11) can be written as It is worthwhile to remark that the equality (17) is exact. That is to say, neither temporal nor spatial boundary terms have been neglected. The relation (17) clearly suggests to define the n(n − 1) configuration variables which thus are canonically conjugate toΠ aI . The variables Q aI embody the combination of the components of the connection ω a IJ contributing to the dynamical variables of the theory; those variables are precisely singled out by the object W a b IJK . We can interpret (19) as n(n − 1) linear equations for n(n − 1) 2 /2 unknowns ω aIJ . In consequence, the solution for ω aIJ must involve n(n − 1) 2 /2 − n(n − 1) = n(n − 1)(n − 3)/2 free variables. Let us call these variablesλ abc , which satisfỹ λ abc = −λ acb and the traceless conditionλ abch ab = 0; both conditions guarantee the right amount of independent variables thatλ abc must contain. The solution for ω aIJ can be expressed as with The objects (18), (21), (22) and (23) all together fulfill the orthogonality relations U cde gIJÑ g f ab The presence of the second term on the right-hand side of (25b) is a consequence of both traceless conditions h bcÑa bcd IJ = 0 andh abŨ abc dIJ = 0. Using (20) together with the relations (25a) and (25b), we get (19) as well as which shows that Q aI andλ abc are independent variables among themselves. Furthermore, we have the completeness relation with where R ab It is remarkable thatG IJ -given by (29a)-involves nõ λ abc . It is also surprising thatṼ a andC-given correspondingly by (29b) and (29c)-contain no spatial derivatives ofλ abc , because (12a) and (12b) contain spatial derivatives of ω a I J . By inspection, it is pretty obvious that the variablesλ abc are auxiliary fields [14]. At this point, there are two, equivalent, ways to continue. The first way consists in to first fix the variablesλ abc by using their equation of motion and then to substitute them back into the action (28). Next, a redefinition of the La-grange multiplier in front of the Gauss constraintG IJ is required (this way was followed in Ref [12]). The second way consists in first to redefine the Lagrange multiplier in front ofG IJ and then to get rid of the auxiliary fields λ abc . We will follow the second way. Then, factoring out all terms inṼ a andC involvingG IJ , we get with whereD a andS are the diffeomorphism and Hamiltonian constraints, respectively. Also, as promised, we have replaced ω tIJ with Λ IJ via the field redefinition Therefore, the original connection variables ω α IJ have been replaced with the independent variables Q a I ,λ abc (satisfying the properties already mentioned for them), and Λ IJ . It is clear by now thatλ abc are auxiliary fields that can be eliminated by using their own equation of motion. In fact, by making the variation of the action (30) with respect toλ abc (taking into account the properties for them), we havẽ which impliesλ Substituting backλ abc into (30), we arrive at the Hamiltonian form of the n-dimensional Palatini action with a cosmological constant Λ: with the Gauss, diffeomorphism and scalar constraints given bỹ respectively. It is worth mentioning that, although the spacetime dimension n shows up in the term involving the (n − 2)-th root of h in (36c), the constraints (36a)-(36c) take exactly the same form in all spacetime dimensions. For Λ = 0, the form of the constraints is actually independent of the spacetime dimension. Therefore, we have obtained a manifestly Lorentzcovariant Hamiltonian formulation (35) for the Palatini action (1). This Hamiltonian form of the action emerged from parametrizing the original frame variables e α I in terms of the momentum variablesΠ aI , the lapse N , and the shift N a as given by (7)- (8), whereas the original connection variables ω α I J have been parametrized in terms of the configuration variables Q a I , the auxiliary fields λ abc , and the Lagrange multipliers Λ IJ as depicted in (20) and (32).
Notice that the map from ω a I J to Q a I andλ abc through (19) and (26), with inverse map given by (20), can be seen as a change of variables. Nevertheless, as is clear from (17) and (19), the presymplectic structure present in (11) becomes the canonical symplectic structure present in (28) when such a map is used. Therefore, we reach a smaller phase-space and simultaneously parametrize it with manifestly Lorentz-covariant canonical variables (Q a I ,Π a I ). The reduction map is given by (ω a I J ,Π a I ) −→ (Q a I ,Π a I ) using (19). This reduction process leaves the null directions of the presymplectic structure (11) out of the canonical symplectic structure present in (28). The null directions are clearly alongλ abc , which turn out to be auxiliary fields that can be eliminated from the action by using their own equation of motion. The variables Λ IJ , N a , andÑ are Lagrange multipliers imposing the SO(n − 1, 1) [or SO(n)] Gauss, diffeomorphism, and scalar constraints; respectively. These constraints depend on the phase space variables (Q a I ,Π We close this section with two remarks: (i) For 4-dimensional spacetimes, the canonical description of general relativity with a cosmological constant given in (35) is the same as the one obtained from the canonical variables for the Holst action through a canonical transformation (see Sec. IV of Ref. [12]).
(ii) As shown in Appendix A, for 3-dimensional spacetimes there are no auxiliary fieldsλ abc (notice that U abc dIJ identically vanishes for n = 3, as for any object with the same symmetries ofλ abc in three of its spatial indices). In spite of this, the resulting Hamiltonian form of the theory has exactly the same structure given by (35).

III. OTHER MANIFESTLY LORENTZ-COVARIANT PHASE-SPACE VARIABLES
It is important to emphasize that the manifestly Lorentz-covariant canonical analysis of general relativity with a cosmological constant embodied in the action (35) is not the canonical description of the Palatini action given in Refs. [8,9]. We show in what follows that the latter can be obtained from our Hamiltonian formulation through a very simple canonical transformation leaving the momentumΠ aI unchanged: (Q aI ,Π aI ) −→ (Q aI ,Π aI ). Both configuration variables are related to each other by This transformation is indeed canonical because and since Σ has no boundary, the last term of the equality (39) does not contribute to the Hamiltonian action. More precisely, using (38), the action (35) acquires the form withG This is the formulation obtained in Ref. [8,9] through a lengthy process of solving the second-class constraints involved there. Notice also that the canonical variables (Q aI ,Π aI ) are SO(n − 1, 1) [or SO(n)] vectors. Alternatively, the manifestly Lorentz-covariant Hamiltonian formulation (40) can also be directly obtained from (11) by following an analogous procedure to that developed in Sec. II. To achieve this, we have to handle the equality (17) as follows: The reason to keep Γ aIJ with the minus sign is because vector. The next step is to define the expression inside the brackets as the configuration variables and so −2Π aI n J ∂ t ω aIJ = 2Π aI ∂ t Q aI − 2∂ a n I ∂ tΠ aI .(44) The following step is to solve (43) for ω aIJ , which gives with M a b IJK andÑ a bcd IJ given by (21) and (22), respectively; and the variablesũ abc satisfyũ abc = −ũ acb and the traceless conditionũ abch ab = 0. The cases n = 3 (that does not involveũ abc ) and n ≥ 4 must be analyzed separately as we already explained. The next step is to substitute (45) into the action (11) and then redo the analysis performed in Sec. II to eliminate the auxiliary fieldsũ abc and thus obtain (40). This is done as follows. Substituting (45) into (11), we get with Factoring outG IJ inṼ a andC, we obtain withG and where we have also replaced ω tIJ with λ IJ through The action (48) depends on the phase space variables (Q aI ,Π aI ), the Lagrange multipliers (λ IJ , N a ,Ñ ), and the auxiliary fieldsũ abc . Now, we can get rid of the variablesũ abc by using their own equation of motion, which is given byÑh Given thatÑ = 0, its solution forũ abc is Substituting this into the constraints of (48) we get precisely the Hamiltonian formulation (40).

IV. TIME GAUGE
We shall fix the boost freedom to reduce the gauge group SO(n − 1, 1) [or SO(n)] to the rotation group SO(n − 1). This is achieved by imposing by hand the gauge conditionΠ a0 ≈ 0, which forms a second-class set [5] with the boost constraintG 0i ≈ 0 because defines an invertible (n−1)×(n−1) matrix for nondegen-erateΠ ai , something that we assume. This assumption combined withΠ a0 ≈ 0 in turn implies n i ≈ 0. So, making the second-class constraints strongly equal to zero, we get from (29a) whereΠ ai denotes the inverse ofΠ ai [we also recall that (16) implies Γ a0i = 0, whereas Γ aij is a function of Π ai and their derivatives]. So, the action (35) becomes In analogy with the 4-dimensional case [6], this formulation could be called the SO(n − 1) ADM formulation of general relativity [2]. On the other hand, if the gauge fixing is imposed directly in the action (40), we have Q a0 = 0 and we get exactly the action (55) with Q ai taking the place of Q ai . The fact that Q ai = Q ai can be easily seen from the relation (38). Therefore, in the time gauge, the same formulation (55) arises from both (35) and (40).

V. CONCLUSIONS
In this paper we performed, in an SO(n − 1, 1) [or SO(n)] covariant fashion, the canonical analysis of the n-dimensional Palatini action with or without a cosmological constant (1). We followed an strategy akin to that used in Ref. [12], where the introduction of secondclass constraints in the canonical analysis of the Holst action was entirely avoided. To that end, we expressed the components of the connection ω aIJ in terms of the variables Q aI andλ abc as shown in the relation (20). The construction underlying these variables is laid out in Sec. II, which entails a reduction of the presymplectic structure of the theory to a canonical symplectic structure. It turns out that the variables Q aI play the role of the configuration variables of the resulting theory, whereas the variablesλ abc are auxiliary fields that can be eliminated from the action by using their own dynamics. The final phase space is thus parametrized by the canonical pair (Q aI ,Π aI ), whereΠ aI is related to the spatial components of the orthonormal frame by the expression (6a), subject to the Gauss, diffeomorphism, and scalar constraints (36a)-(36c), which are first-class and make up the full set of constraints of the theory. Therefore, the introduction of second-class constraints and the subsequent elimination of them is completely bypassed in our approach.
In addition, we have also performed the canonical transformation (38), which maps (Q aI ,Π aI ) into (Q aI ,Π aI ); in terms of these variables, the diffeomorphism constraint remains the same, whereas the Gauss and scalar constraints get much simpler [see the expressions (41a)-(41c)]. The ensuing canonical formulation (40) is actually the one obtained in Refs. [8,9] for the higher-dimensional Palatini action after eliminating the second-class constraints arising in the canonical analysis carried out by the authors. This procedure is long and highly nontrivial, since the resulting second-class constraints are not independent (and thus reducible) for n > 4. In contrast, our approach is quite straightforward and leads to the Hamiltonian action (40) in no time. For the sake of completeness, we detail the case n = 3 (where there are no variablesλ abc ) in Appendix A, and also present an alternative approach for the case n = 4 in Appendix B. Finally, we imposed the time gauge on both actions (35) and (40), and obtained as a result the SO(n − 1) ADM formulation of general relativity embodied in the action (55).
It is worth stressing the simplicity and tidiness of our approach to arrive at the Hamiltonian action (35). What is really remarkable is that such a decomposition (20) of the connection exists for general relativity in all dimensions n ≥ 3 (recall that in n = 3 there are no variables λ abc ), something that enormously simplifies the canonical analysis of the theory, as we have shown in this paper. This decomposition is not only convenient for pure gravity, but can also be employed to build up the Hamiltonian formulation of general relativity coupled to matter fields. Perhaps the most interesting case would be the coupling of a spin 1/2 field, because given that it couples directly to the SO(n−1, 1) connection, then the variablesλ abc are expected to get nontrivial contributions from this matter field. On the other hand, given that the diffeomorphism and scalar constraints can be combined into a single con-straintH I := h −1/[2(n−2)] 2Π a IDa + σn IH , it would be really interesting to investigate how this covariant constraint is related to the Lagrangian gauge symmetry unveiled in Ref. [15] for the n-dimensional Palatini action. We finally remark that the approach of this paper can also be used to do deal with the so-called "space gauge" following the same ideas of Ref. [16].  To perform the canonical analysis for 3-dimensional general relativity with a cosmological constant, we start from the definition (19), which defines a system of 6 linear equations for the unknowns ω a I J whose solution is Notice that there are noλ abc variables involved. Substituting (A1) into the action (11), we obtain C = −σΠ aIΠbJ R abIJ + 2Π a[IΠ|b|J] (Q aI Q bJ +2Q aI Γ bJK n K + Γ aIL Γ bJK n K n L + 2σΛh +2Π aI n J ∇ aGIJ .
Factoring outG IJ inṼ a andC, we arrive at the Hamiltonian formulation of the 3-dimensional Palatini action with a cosmological constant H := −σΠ aIΠbJ R abIJ + 2Π a[IΠ|b|J] (Q aI Q bJ +2Q aI Γ bJK n K + Γ aIK Γ bJL n K n L + 2σhΛ, are the SO(2, 1) [or SO (3)] Gauss, diffeomorphism and scalar constraints, respectively; and where we have redefined the Lagrange multiplier ω tIJ through It is worth mentioning that the action (A5) is precisely the same Hamiltonian formulation (35) obtained in Sec. II for n > 3 (when the auxiliary fieldsλ abc are present). Therefore, the Hamiltonian formulation (35) holds for n ≥ 3.

Canonical transformations
To close this appendix, we perform a canonical transformation-depending on two real parameters α and β-that leave the momentumΠ aI unchanged. The transformation from (Q aI ,Π aI ) to the phase space variables (Y aI ,Π aI ) is such that the configuration variables Y aI are defined by where W a b IJK has been defined in (18). This transformation is indeed canonical because Hence, in terms of the canonical variables (Y aI ,Π aI ), the action (A5) becomes with where G abcd :=h abhcd −h (a|ch|b)d has weight +4. Now, factoring out the Gauss constraintG IJ inṼ a andC, and redefining the Lagrange multiplier ω tIJ , the action becomes