Hamiltonian dynamics and gauge symmetry for three-dimensional Palatini theory with cosmological constant

Abstract

A pure Dirac's framework for 3D Palatini's theory with cosmological constant is performed. By considering the complete phase space, we find out the full structure of the constraints, and their corresponding algebra is computed explicitly. We report that in order to obtain a well defined algebra among the constraints, the internal group corresponds to SO(2, 1). In addition, we obtain the extended action, the extended Hamiltonian, the gauge symmetry, and the Dirac brackets of the theory. Finally, we compare our results with those reported in the literature.

Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

“…and we can observe that these brackets are the same as those obtained using the Dirac method (see the Appendix and [10]). Furthermore, we can observe that there is a -contribution in the fundamental brackets, which makes a difference with respect to Palatini theory [26]. In fact, in Palatini theory, the Dirac brackets between the triad fields are commutative, while in P-CS, they are not; however, we observe that P-CS and Palatini are equivalent in the limit when goes to infinity.…”

The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term

“…These equations for different values of γ represent a set of equations classically equivalent to three dimensional Einstein's theory, however, in spite of this equivalence we will see that the generalized HJ brackets depend on the γ parameter, while in Palatini theory there is not such a dependence [38], in this sense the Palatini theory and PCS are different to each other.…”

“…note that the fields e ′ s are noncommutative due to the presence of the γ parameter. This fact makes PCS theory different to standard Palatini action where the triad is commutative [38]. With the generalized brackets at hand, we introduce the new fundamental HJ differential…”

Gauge symmetry and constraints structure for topologically massive AdS gravity: a symplectic viewpoint

“…This approach, the so-called Faddeev-Jackiw (F-J) symplectic formalism (for a detailed account see [37][38][39][40][41][42][43][44]), is useful to obtain in an elegant way several essential elements of a particular physical theory, such as the physical constraints, the local gauge symmetry, 1 In the presence of a cosmological constant, Minkowski space-time is no longer a vacuum solution and the new maximally symmetric solutions are de Sitter (dS) space-time for positive ( dS has SO(3, 1) isometry) and anti-de Sitter (AdS) space-time for negative (AdS has SO(2, 2) isomety). In this respect, the SO(2, 2) group can be seen as a -deformed Poincaré group [56], if → 0 the AdS algebra contracts to the usual Poincaré algebra. the quantization bracket structure and the number of physical degrees of freedom.…”