Hamiltonian dynamics and Faddeev–Jackiw formulation of 3D gravity with a Barbero–Immirzi like parameter

Abstract

A detailed Dirac and Faddeev-Jackiw formulation of Bonzom-Livine model describing gravity in three dimensions is performed. The full structure of the constraints, the gauge transformations and the generalized FaddeevJackiw brackets are found. In addition, we show that the Faddeev-Jackiw and Dirac brackets coincide.

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

“…However, this is not a serious restriction because even if the original Lagrangian is not of first-order, it is p ossible to introduce auxiliary fields in order to obtain a first-order one (usually, the canonical momenta are chosen as auxiliary fields see for instance the cites [53][54][55]). In this manner, we can construct a first-order Lagrangian for a physical system as follows…”

“…The term V (ξ ), which is called symplectic potential, is assumed to be free of time derivatives of ξ I , and it is easy to see that in comparison with the Dirac method, the potential is the negative of the canonical Hamiltonian. Furthermore, the functions a I (ξ ) are the canonical one-forms and they are of interest because they can be identified with either the original dynamical variables or the canonical momenta; in [FJ] framework there is a freedom for choosing the symplectic variables [53][54][55]. However, there are comments in this respect.…”

“…In the Faddeev-Jackiw formalism the momenta are part of the symplectic one-forms as auxiliar variables. Second, in order to obtain a symplectic tensor, the presence of the momenta force to fixing the gauge in a nontrivial form [54,55], however by using a suitable gauge, the Dirac and the generalized FJ brackets coincide to each other. Furthermore, the Euler-Lagrange equations of motion for Lagrangian (A1) can be written as…”

Symplectic analysis for the Holst action with Dirac fields

“…The symplectic analysis of gravitation theories with Immirzi parameters has been made in the context of Faddeev-Jackiw formalism, with the focus on obtaining the generalized brackets [13,14], with results generally coinciding with those obtained in analysis via Dirac's formalism [15], although some differences arise in regards to the set of constraints of the theory [14]. There are also symplectic analysis of theories with half-integer spin variables not coupled to gravity [16].…”

Hamilton–Jacobi analysis for three dimensional gravity without dynamics

“…Furthermore, the relation between CS and 3dg theories can be extended in order to obtain models with more general structure than Palatini's theory, for instance the Bonzom Livine model [BL] [5,6] and the models reported by V. Hussain, in which there is not generator of the dynamics [7,8]. The BL model describes a set of actions sharing the equations of motion with Palatini's theory; however, the symplectic structure in the BL model depends on a Barbero-Immirizi-like parameter, which may represent a difference at dynamical level [5,6]. In contrast to real gravity theory, on the other hand, in the Hussain theories have not a Hamiltonian constraint, but the vector and the Gauss constraints are present and this fact facilitates the study of quantum aspects of gravity being it a difficult task to perform, thus, the study of toy models brings us insights for study the symmetries of gravity.…”