The Hamiltonian formulation of tetrad gravity: Three-dimensional case

The Hamiltonian formulation of tetrad gravity in any dimension higher than two, using its first-order form where tetrads and spin connections are treated as independent variables, is discussed, and the complete solution of the three-dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of Poisson brackets among all constraints is calculated. The algebra of Poisson brackets, among first-class secondary constraints, locally coincides with Lie algebra of the ISO(2,1) Poincar, group. All the first-class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same as found by Witten without recourse to the Hamiltonian methods [Nucl. Phys. B 311, 46 (1988)]. The gauge symmetry of the tetrad gravity generated by the Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is not derivable as a gauge symmetry from the Hamiltonian formulation of tetrad gravity in any dimension where tetrads and spin connections are used as independent variables.