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

Alberto Escalante,Prihel Cavildo-Sánchez

doi:10.1155/2018/3474760

Alberto Escalante, Prihel Cavildo-Sánchez

Open Access

https://doi.org/10.1155/2018/3474760

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Abstract

Using the symplectic framework of Faddeev-Jackiw, the three-dimensional Palatini theory plus a Chern-Simons term [P-CS] is analyzed. We report the complete set of Faddeev-Jackiw constraints and we identify the corresponding generalized Faddeev-Jackiw brackets. With these results, we show that although P-CS produces Einstein’s equations, the generalized brackets depend on a Barbero-Immirzi-like parameter. In addition, we compare our results with those found in the canonical analysis showing that both formalisms lead to the same results.

Highlights

It is well known that the core of canonical gravity is based on the Hamiltonian formalism for singular systems developed by Dirac [1, 2]
We reported the complete set of FJ constraints, the gauge symmetry, and the FJ generalized brackets which are not reported in the literature
We observed that the generalized brackets depend on γ, and this fact makes the P-CS theory different from Palatini theory

Summary

Introduction

It is well known that the core of canonical gravity is based on the Hamiltonian formalism for singular systems developed by Dirac [1, 2]. The BL theory turns out to be classically equivalent to threedimensional Palatini theory It provides a set of actions sharing the classical equations of motion with the Palatini theory, and its symplectic structure depends on a γ-like parameter. The P-CS theory provides a classical set of actions sharing the equations of motion with Einstein’s theory and their symplectic structure depends on the γ parameter. Hamiltonian points of view [10]; all the fundamental steps were ignored; that is, the complete structure of the constraints and the symmetries of the theory were not reported In this manner, with the antecedents mentioned above, in this paper, we will study the P-CS theory from a symplectic point of view.

Symplectic Analysis

Conclusions