Faddeev Jackiw Research Articles

Faddeev–Jackiw Hamiltonian formulation for general exotic bi-gravity

General exotic bi-gravity, obtained in Ozkan et al. (Phys Rev Lett 123(3):031303, 2019), is a unitary parity-preserving model which describes two interacting spin-two fields in three-dimensional spacetime. Adopting a symplectic viewpoint, we investigate the dynamical structure of general exotic bi-gravity theory. In particular, by exploiting the properties of the corresponding pre-symplectic matrix and its associated zero-modes, we explicitly derive all constraints of the theory, including the integrability conditions and scalar relationships between all the parameters and fields defining the model. Then, as an application, these scalar relationships are used for studying the anti-de Sitter background. After that, we derive the gauge transformations for the dynamical variables from the structure of the remaining zero-modes, meaning that such zero-modes are indeed the generators of the gauge symmetry of the theory. Finally, by switching off one of the four coupling constants βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{n}$$\\end{document} and assuming the invertibility of some of the dreibeins, we find that the general exotic bi-gravity theory has two physical degrees of freedom.

Hamiltonian analysis and Faddeev–Jackiw formalism for two dimensional quadratic gravity expressed as BF theory

Hamiltonian analysis and Faddeev–Jackiw formalism for two dimensional quadratic gravity expressed as BF theory

On the quantization of FLPR model

In this paper, we quantize the Friedberg–Lee–Pang–Ren (FLPR) model, using an admissible gauge condition, within the framework of modified Faddeev–Jackiw formalism. Further, we deduce the gauge symmetries and establish off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries. We also show that the physical states of the theory are annihilated by the first class constraints which is consistent à la Dirac formalism.

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.

Extended Faddeev–Jackiw canonical quantization for the Podolsky electrodynamics

We perform the Faddeev–Jackiw (FJ) canonical quantization for the Podolsky electrodynamics. To this end, we use an extension of the usual FJ formalism for constrained systems with Grassmann dynamical field variables, proposed by us some time ago. Besides, we compare the obtained results with those corresponding to the implementation of the Dirac formalism to this issue. In this way, we see that the extended FJ and the Dirac formalisms provide the same constraints and generalized brackets, thus suggesting the equivalence between these formalisms, at least for the present case. Furthermore, we find that the extended FJ formalism is more economical than the Dirac one as regards the calculation of both the constraints and the generalized brackets. On the other hand, we also compare the mentioned obtained results with the ones corresponding to the analysis of the issue in study by means of the usual FJ formalism, showing that between the extended and the usual FJ formalisms there are significant differences.

Alternative symplectic structures in second–class constrained systems

The classical description of second-class constrained systems using alternative pairs, comprising a symplectic structure and a primary Hamiltonian other than the canonical ones, is investigated. It is exhibited that the dynamical equations do not fix a unique pair and instead there may exist infinitely many of them whose difference is not exhausted by canonical transformations. The consequences on the symplectic structure of the reduced phase space are addressed, namely the effect of different pairs on the construction of Dirac bracket is explored. Since constrained systems can also be studied via the Faddeev–Jackiw symplectic analysis, the implementation of alternative pairs on this formalism is considered as well. It is shown that Dirac’s algorithm outcomes are independent of the type of alternative pairs considered here irrespectively to whether there are also first-class constraints in the system. Finally, some comments on the implications of our results in the path integral quantisation scheme are pointed out.

Particle on a torus knot: symplectic analysis

We quantize a particle confined to move on a torus knot satisfying constraint condition ( $$p\theta +q\phi ) \approx 0$$ , within the context of a geometrically motivated approach—the Faddeev–Jackiw formalism. We also deduce the constraint spectrum and discern the basic brackets of the theory. We further reformulate the original gauge non-invariant theory into a physically equivalent gauge theory, which is free from any additional Wess–Zumino variables, by employing symplectic gauge invariant formalism. In addition, we analyze the reformulated gauge invariant theory within the framework of BRST formalism to establish the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries. Finally, we construct the conserved (anti-)BRST charges which satisfy the physicality criteria and turn out to be the generators of corresponding symmetries.

Conserved laws and dynamical structure of axions coupled to photons

In this work, we carry out a study of the conserved quantities and dynamical structure of the four-dimensional modified axion electrodynamics theory described by the axion-photon coupling. In the first part of the analysis, we employ the covariant phase space method to construct the conserved currents and to derive the Noether charges associated with the gauge symmetry of the theory. We further derive the improved energy–momentum tensor using the Belinfante–Rosenfeld procedure, which leads us to the expressions for the energy, momentum, and energy flux densities. Thereafter, with the help of Faddeev–Jackiw’s Hamiltonian reduction formalism, we obtain the relevant fundamental brackets structure for the dynamic variables and the functional measure for determining the quantum transition amplitude. We also confirm that modified axion electrodynamics has three physical degrees of freedom per space point. Moreover, using this symplectic framework, we yield the gauge transformations and the structure of the constraints directly from the zero-modes of the corresponding pre-symplectic matrix.

Faddeev–Jackiw quantization of the higher derivative Chern–Simons gauge theory in the first-order formalism

We analyse the physical constraints of the higher derivative Chern–Simons gauge model by means of Faddeev–Jackiw symplectic approach in the first-order formalism. Within such framework, we systematically determine the zero-mode structure of the corresponding symplectic matrix. In addition, we calculate the Faddeev–Jackiw quantum brackets by choosing appropriate gauge-fixing conditions and evaluate the determinant of the non-singular symplectic matrix as well as the transition-amplitude. Finally, we present a detailed Hamiltonian analysis using Dirac–Bergmann algorithm method and show that the Dirac brackets coincide with the FJ brackets when all the second-class constraints are treated as zero equations.

Equivalent Hamiltonian approach to quantum cosmology of integrable models

We propose an approach to quantum cosmology of integrable models. To analyze the models with two dynamical variables, we introduce equivalent Hamiltonians in reduced phase spaces, which are obtained with the aid of the Faddeev–Jackiw method. Quantum dynamics of the models can be studied by using the equivalent Hamiltonians with various techniques.

Faddeev–Jackiw quantization of Christ–Lee model

We analyze the constraints of Christ–Lee model by means of modified Faddeev–Jackiw formalism in Cartesian as well as polar coordinates. Further, we accomplish quantization à la Faddeev–Jackiw by choosing appropriate gauge conditions in both the coordinate systems. Finally, we establish gauge symmetries of Christ–Lee model with the help of zero-modes of the symplectic matrix.

Faddeev–Jackiw approach of noncommutative space–time electromagnetic theories

The interest in higher derivatives field theories has its origin mainly in their influence concerning the renormalization properties of physical models and to remove ultraviolet divergences. The noncommutative Podolsky theory is a constrained system that cannot be directly quantized by the canonical way. In this work, we have used the Faddeev–Jackiw method in order to obtain the Dirac brackets of the NC space–time Maxwell, Proca and Podolsky theories.

Note on soft photons and Faddeev–Jackiw symplectic reduction of quantum electrodynamics in the eikonal limit

In this note, we go over the recent soft photon model and Faddeev–Jackiw quantization of the massless quantum electrodynamics in the eikonal limit to some extent. Throughout our readdressing, we observe that the gauge potentials in both approaches become pure gauges and the associated eikonal Faddeev–Jackiw quantum bracket matches with the soft quantum bracket. These observations and the fact that the gauge fields in two cases localize in two-dimensional plane (even if it is spatial in soft photon case and (1[Formula: see text]+[Formula: see text]1)-dimensional Minkowski in the eikonal case) imply that there might be an interesting relation between these two distinct perspectives.

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.

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.

On the equivalence and non-equivalence of Dirac and Faddeev–Jackiw formalisms for constrained systems

We present a comparative analysis between the Dirac method and the original Faddeev–Jackiw formalism for constrained systems, such as Dirac free field, Proca model, electromagnetism coupled to matter, and source-free Maxwell field. We establish the possible differences between the approaches and show that they are not completely equivalent.

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 Hamiltonian reduction for free and gauged Rarita–Schwinger theories

We study the Faddeev-Jackiw symplectic Hamiltonian reduction for 3+1-dimensional free and Abelian gauged Rarita-Schwinger theories that comprise Grassmannian fermionic fields. We obtain the relevant fundamental brackets and find that they are in convenient forms for quantization. The brackets are independent of whether the theories contain mass or gauge fields, and the structure of constraints and symplectic potentials largely determine characteristic behaviors of the theories. We also note that, in contrast to the free massive theory, the Dirac field equations for free massless Rarita-Schwinger theory cannot be obtained in a covariant way.

Faddeev-Jackiw analysis for the charged compressible fluid in a higher-derivative electromagnetic field background

In the present paper we will discuss the Faddeev-Jackiw (FJ) symplectic approach in the analysis of a charged compressible fluid immersed in a higher-derivative electromagnetic field theory. In other words, we have analyzed, from the FJ point of view, the minimal coupling between the strength tensor of the fluid and the higher derivative of the electromagnetic field. We have obtained the full set of constraints directly from the zero-mode eigenvectors. Besides, we have computed the Dirac brackets for the dynamic variables of the compressible fluid. Finally, as a result of the coupling between the charged compressible fluid and the electromagnetic field we have calculated two Dirac brackets between the fluid and electromagnetic fields, which are both zero when there is no coupling between them.