Faddeev-Jackiw Formalism Research Articles

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

Singular Lagrangians and the Faddeev-Jackiw Formalism in Classical Mechanics

Singular Lagrangians and the Faddeev-Jackiw Formalism in Classical Mechanics

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.

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.

Symplectic gauge-invariant reformulation of a free-particle system on toric geometry

We deduce the constraint structure and subsequently quantize a free-particle system residing on a torus satisfying the geometric constraint , within the framework of the modified Faddeev-Jackiw formalism. Further, we reformulate the system as a gauge theory by means of the symplectic gauge-invariant formalism. Finally, we establish the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries of the reformulated gauge-invariant theory.

CUANTIFICACIÓN CANÓNICA DE FADDEEV-JACKIW EXTENDIDA DE LA ELECTRODINÁMICA NO RELATIVISTA (1+1)-DIMENSIONAL

Some time ago, we proposed an extension of the usual Faddeev-Jackiw formalism for constrained systems with Grassmann dynamic variables in the field theory context. In the present work, we apply this extended formalism to the (1+1)-dimensional nonrelativistic electrodynamics. By comparing the obtained results with those corresponding to the implementation of Dirac formalism on this model, we find the same constraints and generalized brackets. In this way, we can conclude that the extended Faddeev-Jackiw and the Dirac formalisms can be considered equivalent, at least for this model. On the contrary, in this case, we find that there is no equivalence between the usual Faddeev-Jackiw and the Dirac formalisms. On the other hand, we observe that the extended formalism is more economical than the Dirac one regarding the computation of both, constraints and generalized brackets.

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.

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.

A note on the path integral representation for Majorana fermions

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev–Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.

Higher-derivative non-Abelian gauge fields via the Faddeev–Jackiw formalism

In this paper we analyze two higher-derivative theories, the generalized electrodynamics and the Alekseev–Arbuzov–Baikov effective Lagrangian from the point of view of the Faddeev–Jackiw symplectic approach. It is shown that the full set of constraints is obtained directly from the zero-mode eigenvectors, and that they are in accordance with well-known results from Dirac’s theory, a recurrent issue in the literature. The method shows to be rather economical in relation to the Dirac one, obviating thus unnecessary classification and calculations. Afterwards, to conclude we construct the transition amplitude of the non-Abelian theory following a constrained BRST method.

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.

QCD gauge symmetries through Faddeev-Jackiw symplectic method

The interactions between gluons are important in theories such as quantum chromodynamics. Therefore, to rediscover new features of well known methods in order to investigate the SU(3) gauge group can be a new way to deal with Yang-Mills theories. In this work we analyzed YM theories through the well known Faddeev-Jackiw formalism for constrained systems. Besides, we showed precisely that having U(1) Maxwell electromagnetic theory as a starting point we can construct SU(3)-like and SU(3) X SU(2) X U(1) non-Abelian theories.

Hamiltonian Structure of Fractional First Order Lagrangian

In this paper, we show that the fractional constraint Hamiltonian formulation, using Dirac brackets, leads to the same equations as those obtained from fractional Euler-Lagrange equations. Furthermore, the fractional Faddeev-Jackiw formalism was constructed.

Non-equivalence of Faddeev–Jackiw method and Dirac–Bergmann algorithm and the modification of Faddeev–Jackiw method for keeping the equivalence

From the angle of the calculation of constraints, we compare the Faddeev–Jackiw method with Dirac–Bergmann algorithm, study the relations between the Faddeev–Jackiw constraints and Dirac constraints, and demonstrate that Faddeev–Jackiw method is not always equivalent to Dirac method. For some systems, under the assumption of no variables being eliminated in any step in Faddeev–Jackiw formalism, except for the Dirac primary constraints, we are possible to get some Dirac secondary constraints which do not appear in the corresponding Faddeev–Jackiw formalism, which will result in the contradiction between Faddeev–Jackiw quantization and Dirac quantization. At last, accordingly, we propose a modified Faddeev–Jackiw method which keeps the equivalence between Dirac–Bergmann algorithm and Faddeev–Jackiw method. However, one point must be stressed that the Faddeev–Jackiw method and quantization in this paper is these mentioned in [J. Barcelos-Neto, C. Wotzasek, Mod. Phys. Lett. A 7 (1992) 1737], not the initial Faddeev–Jackiw method mentioned in [L. Faddeev, R. Jackiw, Phys. Rev. Lett. 60 (1988) 1692], which is completely on basis of Darboux transformation, and must have the elimination of variables in every step of that, so it is reasonable that the constraints in this Faddeev–Jackiw method is fewer than the Dirac secondary constraints. Thus, we overcome the difficulty of the Non-equivalence of the Faddeev–Jackiw method and Dirac–Bergmann algorithm, and make the equivalence of the Faddeev–Jackiw method and Dirac–Bergmann algorithm restored.

Symplectic algorithm for systems with second-class constraints

The recently modified Faddeev-Jackiw formalism for systems having one chain of four levels of only second-class constraints is applied to the non-triviala=1 bosonized chiral Schwinger model in (1+1) dimensions as well as to one mechanical system. The sets of obtained constraints are in agreement with Dirac’s canonical formulation.

CONSTRAINT STRUCTURE IN MODIFIED FADDEEV–JACKIW METHOD

We show that in modified Faddeev–Jackiw formalism, first- and second-class constraints appear at each level, and the whole constraint structure is in exact correspondence with level by level method of Dirac formalism.

Faddeev-Jackiw analysis of topological mass-generating action

We analyze the gauge symmetry of a topological mass generating action in four dimensions which contains both a vector and a second rank antisymmetric tensor fields. In the Abelian case, this system induces an effective mass for the vector gauge field via a topological coupling $B \wedge F$ in the presence of a kinetic term for the antisymmetric tensor field $B$, while maintaining a gauge symmetry. On the other hand, for the non-Abelian case the $B$ field does not have a gauge symmetry unless an auxiliary vector field is introduced to the system. We analyze this change of symmetry in the Faddeev-Jackiw formalism, and show how the auxiliary vector field enhances the symmetry. At the same time this enhanced gauge symmetry becomes reducible. We also show this phenomenon in this analysis.

Faddeev-Jackiw Quantization and Its Application to the Linear σ Model for Disoriented Chiral Condensates**The project supported in part by the Education Bureau and in part by the Grant LWTZ-1298 of the Chinese Academy of Sciences

The equivalence between the Faddeev–Jackiw formalism and Dirac–Bergmann algorithm is proved. A two-dimensional constrained system and a charged vector field are quantized in the Faddeev–Jackiw formalism. This symplectic method is technically developed, without recourse to Hamiltonian or Lagrangian, to quantize systems whose equations of motion are known. Examples are given to show this role. For constructing quantum approaches to the disoriented chiral condensates, the linear σ model is quantized in the instant form, light-cone form and covariant form.

Nonlinear sigma model in the Faddeev-Jackiw quantization formalism

The key equations of the symplectic Faddeev-Jackiw formalism are written in an alternative way so that the inverse of the symplectic matrix is easily found. The nonlinear sigma model including the Hopf term in the action is treated in the framework of this quantization method. It is shown how the complete dynamics of the system is described by means of the generalized Faddeev-Jackiw quatum brackets.

BRST analysis of the gauged SU(2) WZW model and Darboux's transformations

The four dimensional SU(2) WZW model coupled to elecromagnetism is treated as a constraint system in the context of the BFV approach. We show that the Darboux's transformations which are used to diagonalize the canonical one-form in the Faddeev-Jackiw formalism, transform the fields of the model into BRST invariant ones. The same analysis is also carried out in the case of spinor electrodynamics.