Generalized Faddeev-Jackiw Brackets Research Articles

Canonical analysis and modified Faddeev-Jackiw approach for the Jackiw-Teitelboim model in two dimensions

This article presents an analysis of the constraints of the Jackiw-Teitelboim model in two dimensions via the canonical analysis using the Dirac algorithm and modified Faddeev-Jackiw (FJ) approach. The analysis primarily focuses on the identification of constraints, gauge transformations, counting of physical degrees of freedom, and the generalized FJ brackets and Dirac’s brackets. To ensure gauge symmetry within the symplectic formalism and maintain consistency with the Dirac formalism, we employ the Montani-Wotzasek method, which effectively utilizes the zero modes of the symplectic matrix. Additionally, the Poincaré symmetry and diffeomorphisms in the model are identified. Finally, we present the equivalence between the generalized FJ and Dirac brackets when all the second-class constraints are treated as zero equations.

Canonical and symplectic analysis of actions describing linearized gravity

By using the canonical and symplectic approaches an (nonstandard) alternative action describing linearized gravity is studied. We identify the complete set of Dirac's constraints, the counting of physical degrees of freedom is performed and the Dirac brackets are constructed. Furthermore, the symplectic analysis is developed which includes the complete set of Faddeev-Jackiw constraints and a symplectic tensor; from that symplectic matrix we show that the generalized Faddeev-Jackiw brackets and the Dirac ones coincide to each other. With all these results at hand, we prove that the number of physical degrees of freedom are eight, thus, we conclude that the theory does not describe the dynamics of linearized gravity. In addition, we also develop the symplectic analysis of standard linearized gravity and we compare the results for both standard and nonstandard theories. Finally we present some remarks and conclusions.

Faddeev–Jackiw quantization of topological invariants: Euler and Pontryagin classes

The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev–Jackiw context. The Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev–Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions.

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

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.

Hamiltonian dynamics and Faddeev\u2013Jackiw formulation of 3D gravity with a Barbero\u2013Immirzi like parameter

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 Faddeev–Jackiw brackets are found. In addition, we show that the Faddeev–Jackiw and Dirac brackets coincide.

Symplectic analysis of three-dimensional Abelian topological gravity

A detailed Faddeev-Jackiw quantization of an Abelian topological gravity is performed; we show that this formalism is equivalent and more economical than Dirac's method. In particular, we identify the complete set of constraints of the theory, from which the number of physical degrees of freedom is explicitly computed. We prove that the generalized Faddeev-Jackiw brackets and the Dirac ones coincide to each other. Moreover, we perform the Faddeev-Jackiw analysis of the theory at the chiral point, and the full set of constraints and the generalized Faddeev-Jackiw brackets are constructed. Finally we compare our results with those found in the literature and we discuss some remarks and prospects.

Canonical and symplectic analysis for three dimensional gravity without dynamics

In this paper a detailed Hamiltonian analysis of three-dimensional gravity without dynamics proposed by V. Hussain is performed. We report the complete structure of the constraints and the Dirac brackets are explicitly computed. In addition, the Faddeev–Jackiw symplectic approach is developed; we report the complete set of Faddeev–Jackiw constraints and the generalized brackets, then we show that the Dirac and the generalized Faddeev–Jackiw brackets coincide to each other. Finally, the similarities and advantages between Faddeev–Jackiw and Dirac’s formalism are briefly discussed.

On the Faddeev–Jackiw symplectic framework for topologically massive gravity

The dynamical structure of topologically massive gravity in the context of the Faddeev-Jackiw symplectic approach is studied. It is shown that this method allows to avoid some ambiguities arising in the study of the gauge structure via the Dirac formalism. In particular, the complete set of constraints and the gauge symmetry generators of the theory are obtained straightforwardly via the zero-modes of the symplectic matrix. In order to obtain the generalized Faddeev-Jackiw brackets and calculate the local physical degrees of freedom of this model, an appropriate gauge-fixing procedure is introduced. Finally, the similarities and advantages between Faddeev-Jackiw method and Dirac's formalism are briefly discussed.

Faddeev–Jackiw quantization of four dimensional BF theory

The symplectic analysis of a four dimensional BF theory in the context of the Faddeev–Jackiw symplectic approach is performed. It is shown that this method is more economical than Dirac’s formalism. In particular, the complete set of Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported. In addition, we show that the generalized Faddeev–Jackiw brackets and the Dirac ones coincide to each other. Finally, the similarities and advantages between Faddeev–Jackiw method and Dirac’s formalism are briefly discussed.

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

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.

Dirac and Faddeev–Jackiw quantization of a five-dimensional Stüeckelberg theory with a compact dimension

A detailed Hamiltonian analysis for a five-dimensional Stüeckelberg theory with a compact dimension is performed. First, we develop a pure Dirac’s analysis of the theory; we show that after performing the compactification, the theory is reduced to four-dimensional Stüeckelberg theory plus a tower of Kaluza–Klein modes. We develop a complete analysis of the constraints, we fix the gauge and we show that there are present pseudo-Goldstone bosons. Then we quantize the theory by constructing the Dirac brackets. As complementary work, we perform the Faddeev–Jackiw quantization for the theory under study, and we calculate the generalized Faddeev–Jackiw brackets, we show that both the Faddeev–Jackiw and Dirac’s brackets are the same. Finally we discuss some remarks and prospects.

The CP1nonlinear sigma model with Chern–Simons term in the Faddeev–Jachiw quantization formalism

Using the Faddeev–Jackiw (FJ) quantization method, this paper treats the CP1nonlinear sigma model with Chern–Simons term. The generalized FJ brackets are obtained in the framework of this quantization method, which agree with the results obtained by using the Dirac's method.