Faddeev Jackiw Symplectic Formalism Research Articles

Hořava–Lifshitz cosmological models with noncommutative phase space variables

In this work, we will analyze a noncommutative (NC) version of the Friedmann-Robert-Walker cosmological models within the gravitational Ho\v{r}ava-Lifshitz theory. The matter content of the models is described by a perfect fluid and the constant curvature of the spatial sections may be positive, negative or zero. In order to obtain this theory, we will use the Faddeev-Jackiw symplectic formalism to introduce, naturally, space-time noncommutativity inside the equations that provide the dynamics of the theory. We will investigate, in details, the classical field equations of a particular version of the NC models. The equations will be modified, with respect to the commutative ones, by the introduction of a NC parameter. We will demonstrate that various NC models, with different types of matter and spatial constant curvatures, show several interesting and new results relative to the corresponding commutative ones. We will pay special attention to some cases, where the NC model predicts a scale factor accelerated expansion, which may describe the current state of our Universe.

Constrained dynamics of two interacting relativistic particles in the Faddeev-Jackiw symplectic framework

The Faddeev-Jackiw symplectic formalism for constrained systems is applied to analyze the dynamical content of a model describing two massive relativistic particles with interaction, which can also be interpreted as a bigravity model in one dimension. We systematically investigate the nature of the physical constraints, for which we also determine the zero-modes structure of the corresponding symplectic matrix. After identifying the whole set of constraints, we find out the transformation laws for all the set of dynamical variables corresponding to gauge symmetries, encoded in the remaining zero-modes. In addition, we use an appropriate gauge-fixing procedure, the conformal gauge, to compute the quantization brackets (Faddeev-Jackiw brackets) and also obtain the number of physical degree of freedom. Finally, we argue that this symplectic approach can be helpful for assessing physical constraints and understanding the gauge structure of theories of interacting spin-2 fields.

BRST symmetry and Darboux transformations in Abelian 2-form gauge theory

We analyse the constraints of an Abelian 2-form gauge theory using the Faddeev-Jackiw symplectic formalism. Further, this theory is treated as a constrained system in the context of the Batalin-Fradkin-Vilkovisky formalism to retrieve the BRST symmetry. Using the fields decompositions the effective action for the Abelian 2-form gauge theory is written in terms of a diagonalized uncanonical part and the BRST exact one. The nilpotent BRST and contracting homotopy σ closed transformations with field redefinitions are shown as the Darboux transformations used in the Faddeev-Jackiw formalism.

LAGRANGIAN FORMULATION FOR NONCOMMUTATIVE NONLINEAR SYSTEMS

In this work we study Lagrangian formulations for the noncommutative versions of the SU(2) Skyrme model and O(3) nonlinear sigma model. These formulations will be obtained using the Faddeev–Jackiw symplectic formalism. Other noncommutative Lagrangian formulations can be proposed and different noncommutative versions for these nonlinear systems can be obtained.

Path integrals for dimerized quantum spin systems

Dimerized quantum spin systems may appear under several circumstances, e.g. by a modulation of the antiferromagnetic exchange coupling in space, or in frustrated quantum antiferromagnets. In general, such systems display a quantum phase transition to a Néel state as a function of a suitable coupling constant. We present here two path-integral formulations appropriate for spin S = 1 / 2 dimerized systems. The first one deals with a description of the dimers degrees of freedom in an SO ( 4 ) manifold, while the second one provides a path-integral for the bond-operators introduced by Sachdev and Bhatt. The path-integral quantization is performed using the Faddeev–Jackiw symplectic formalism for constrained systems, such that the measures and constraints that result from the algebra of the operators is provided in both cases. As an example we consider a spin-Peierls chain, and show how to arrive at the corresponding field-theory, starting with both an SO ( 4 ) formulation and bond-operators.

Lagrangian formulation of noncommutative fluid models

In this work we study Lagrangian formulations for the noncommutative versions of the irrotational and rotational fluid models. These formulations will be obtained by using the Faddeev–Jackiw symplectic formalism. In this context, interesting results will be revealed, for example, a chiral behavior into both fluid models comes up. In fact, distinct and non-dynamically equivalent Lagrangian descriptions of the noncommutative versions of the fluid models can be proposed.

Obtaining non-Abelian field theories via the Faddeev–Jackiw symplectic formalism

In this Letter we construct non-Abelian field theories employing the Faddeev–Jackiw symplectic formalism. The original Abelian fields were modified in order to introduce the non-Abelian algebra. We construct the SU ( 2 ) and SU ( 2 ) ⊗ U ( 1 ) Yang–Mills theories having as starting point the U ( 1 ) Maxwell electromagnetic theory.

PERTURBATIVE EXPANSION FOR THE t-J MODEL CONSTRUCTED FROM THE GENERATORS OF THE SUPERSYMMETRIC HUBBARD ALGEBRA

By using the Hubbard [Formula: see text]-operators as field variables along with the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In order to satisfy the Hubbard [Formula: see text]-operator commutation rules satisfying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for this model are presented. The model is also analyzed via path-integral formalism, where the correlation-generating functional and the effective Lagrangian are constructed. In this context, the introduction of a proper ghost field is needed to render the model renormalizable. The perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator arising in the present formalism is discussed. As an example, the thermal softening of the magnon frequency is computed.

First-Order Lagrangians and Path-Integral Quantization in the t–J Model

By using the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t–J model is found. In this approach the Hubbard X-operators are used as field variables. In this framework, we first study the spinless fermion model which satisfies the graded algebra spl(1, 1). Later on, in order to satisfy the Hubbard X-operator commutation rules satisfying the graded algebra spl(2, 1), we find the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for the t–J model. This model is also analyzed in the context of the path-integral formalism, and so the correlation generating functional and the effective Lagrangian are constructed.

Dissipation and Topologically Massive Gauge Theories in the Pseudo-Euclidean Plane

In pseudo-Euclidean metrics the Chern–Simons gauge theory in the infrared region is found to be associated with dissipative dynamics. In the infrared limit the Lagrangian of (2+1)-dimensional pseudo-euclidean topologically massive electrodynamics has indeed the same form as the Lagrangian of the damped harmonic oscillator. On the hyperbolic plane a set of two damped harmonic oscillators, each time-reversed from the other, is shown to be equivalent to a single undamped harmonic oscillator. The equations for the damped oscillators are proven to be the same as the ones for the Lorentz force acting on two particles carrying opposite charge in a constant magnetic field and in the electric harmonic potential. This provides an immediate link with Chern–Simons-like dynamics of Bloch electrons in solids propagating along the lattice plane with a hyperbolic energy surface. The symplectic structure of the reduced theory is finally discussed in the Dirac constrained canonical formalism and in the Faddeev–Jackiw symplectic formalism.