We carry out the canonical analysis of the n-dimensional Palatini action with or without a cosmological 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 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.I.

We work on a spacetime manifold foliated by timelike leaves. In this setting, we explore the solution of the second-class constraints arising during the canonical analysis of the Holst action with a cosmological constant. The solution is given in a manifestly Lorentz-covariant fashion, and the resulting canonical formulation is expressed using several sets of real variables that are related to one another by canonical transformations. By applying a gauge fixing to this formulation, we obtain a description of gravity as an SU (1, 1) gauge theory that resembles the Ashtekar-Barbero formulation.I. * merced@fis.cinvestav.mx † ljromero@fis.cinvestav.mx ‡ rescobedo@fis.cinvestav.mx § mcelada@fis.cinvestav.mx 1 It however manifests off shell in the classical theory. See Ref. [8].

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In this paper we perform in a manifestly SO(n − 1, 1) [or, alternatively SO(n)] covariant fashion, the canonical analysis of general relativity in n dimensions written as a constrained BF theory. Since the Lagrangian action of the theory can be written in two classically equivalent ways, we analyze each case separately. We show that for either action the canonical analysis can be accomplished without introducing second-class constraints during the whole process. Furthermore, in each case the resulting Hamiltonian formulation is the same as the canonical formulation with only first-class constraints recently obtained in Ref.[1] from the n-dimensional Palatini action.