arXiv:1912.01019v1 [gr-qc] 2 Dec 2019 Canonical analysis involving first-class constraints only of the n-dimensional Palatini action Merced Montesinos, * Ricardo Escobedo, † and Jorge Romero ‡ Departamento de Física, Cinvestav, Avenida Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360 Gustavo A. Madero, Ciudad de México, Mexico Mariano Celada § Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, Apartado Postal 61-3, Morelia, Michoacán 58090, Mexico (Dated: December 4, 2019) We carry out the canonical analysis of the n-dimensional Palatini action with or without a cos- mological constant (n ≥ 3) introducing neither second-class constraints nor resorting to any gauge fixing. This is accomplished by providing an expression for the spatial components of the con- nection 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. I. 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 relativ- ity [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 met- ric g µν in terms of the lapse function N , the shift vec- tor 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 impos- ing the scalar (or Hamiltonian) and diffeomorphism con- straints, respectively, whereas q ab and its canonically con- jugate momentum ˜ p ab –an object related to the extrinsic curvature–constitute the canonical variables that label the points of the phase space. Even though the canon- ical quantization program emerging from this approach has failed [4], the ADM variables have been extensively used in other instances of general relativity such as ini- tial 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 fundamen- tal variables are an orthonormal frame of 1-forms e I * merced@fis.cinvestav.mx † rescobedo@fis.cinvestav.mx ‡ ljromero@fis.cinvestav.mx § mcelada@matmor.unam.mx (vielbein) and a SO(n − 1, 1) or SO(n) connection 1- form ω I J depending on whether the spacetime metric 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 tech- nique 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 intro- ducing second-class constraints whatsoever in the Hamil- tonian formalism. It is clear from that approach that the second-class constraints are unnecessary and superfluos for doing the canonical analysis of the Holst action, and thus they are also unnecessary for doing the Hamilto- nian 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