Faddeev–Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions

Abstract

A detailed Faddeev-Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions is performed. We obtain for the theories under study the constraints, the gauge transformations, the generalized Faddeev-Jackiw brackets and we perform the counting of physical degrees of freedom. In addition, we compare our results with those found in the literature where the canonical analysis is developed, in particular, we show that both the generalized Faddeev-Jackiw brackets and Dirac's brackets coincide to each other. Finally we discuss some remarks and prospects.

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

“…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.…”

“…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…”

“…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…”

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

“…It is important to comment that in the FJ framework we are free to choose the symplectic variables; we can choose the field configuration variables or the phase space variables. In previous sections, we have constructed the Dirac brackets by eliminating the second class constraints, hence, in order to obtain these results by means of the FJ method we will use the configuration space as symplectic variables [16]. To this aim, we choose from the symplectic Lagrangian (35) the symplectic variables ξ (0)a (x) = {e i a , e i 0 , A i a , A i 0 }, and the components of the symplectic 1-forms are a (0) a (x) = { 0ab e bi , 0, 2 0ab e bi + β 0ab A bi , 0}.…”

“…In fact, in order to compare the two approaches, it is necessary to work out a complete Dirac analysis. Hence, we need to know the complete structure of the constraints over the full phase space for constructing the Dirac brackets and to compare these brackets with the generalized FJ ones [16]. Furthermore, we shall prove that the FJ approach is more economic than Dirac's one.…”