Improved Faddeev-Jackiw quantization of the electromagnetic field and Lagrange multiplier fields

We use the improved Faddeev-Jackiw quantization method to quantize the electromagnetic field and its Lagrange multiplier fields.The method's comparison with the usual Faddeev-Jackiw method and the Dirac method is given.We show that this method is equivalent to the Dirac method and also retains all the merits of the usual Faddeev-Jackiw method.Moreover,it is simpler than the usual one if one needs to obtain new secondary constraints.Therefore,the improved Faddeev-Jackiw method is essential.Meanwhile,we find the new meaning of the Lagrange multipliers and explain the Faddeev-Jackiw generalized brackets concerning the Lagrange multipliers.     

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.     

约束系统量子化中Dirac方法与Faddeev-Jackiw方法的等价性

对约束系统量子化中Dirac方法和 Faddeev-Jackiw方法进行了讨论，并对它们的运动方程、正则量子化的等价性进行证明.找出了两种方法中约束的对应关系. 关键词： Faddeev-Jackiw方法 Dirac方法 约束系统 正则量子化     

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.     

相关于弦论的规范不变自对偶场的Faddeev-Jackiw量子化

用Faddeev-Jackiw(FJ)方法对与规范场偶合的规范自对偶场进行了研究, 获得了一个新的辛Lagrangian密度, 导出了此系统的FJ广义括号, 并对其进行了FJ量子化. 进而把FJ方法和Dirac方法进行了比较, 发现在对此系统的量子化中, 两种方法所给出的量子化结果完全是等价的. 通过分析可知FJ方法比Dirac方法要简单, 因FJ方法不需要区分初级约束与次级约束, 而且也不需要区分第一类约束和第二类约束. 故与Dirac方法相比, FJ方法是一种计算上更为经济和有效的量子化方法.     

Abel Chern-Simons项与复标量场耦合系统的正则量子化

应用Faddeev-Jackiw方法对Abel Chern-Simons项与复标量场耦合系统进行正则量子化,它表明用这种方法进行量子化更加直接和优美.     

Faddeev-Jackiw Quantization and Its Application to the Linear σ Model for Disoriented Chiral Condensates

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.     

On Hamiltonian Formulations of the Schrödinger System

We review and compare different variational formulations for the Schrödinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schrödinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated.     

自对偶场与规范场相互作用理论的Faddeev－Jackiw量子化

本文采用Ｆａｄｄｅｅｖ－Ｊａｃｋｉｗ量子化方法，讨论了二维时空中一种自对偶场与规范场的相互作用理论．通过与Ｄｉｒａｃ方法的比较，建立了这两种方法的等价性     

Hamiltonian reduction of theU EM(1) gauged three flavour WZW model

The three-flavour Wess-Zumino model coupled to electromagnetism is treated as a constraint system using the Faddeev-Jackiw method. Expanding into series of powers of the Goldstone boson fields and keeping terms up to second and third order we obtain Coulomb-gauge hamiltonian densities.     

Faddeev-Jackiw quantization method in conformal three-dimensional supergravity

The conformal supergravity in three space-time dimensions is described by a pure Lorentz-Chern-Simons term. This system has constraints on curvatures and so it is a higher-derivative gauge model. The dynamical properties of this model are analyzed by means of the supersymmetric extension of the Faddeev-Jackiw symplectic quantization method. Using this algorithm in the first-order formalism, we study the gauge supersymmetric transformations and we find the constraints of the model.     

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.     

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

The four dimensional SU(2) WZW model coupled to electromagnetism is treated as a constraint system in the context of the Batalin-Fradkin-Vilkovisky formalism. Common features with the Faddeev-Jackiw approach are stressed and the same results are obtained. The Darboux’s transformations which are used to diagonalize the canonical one-form in the Faddeev-Jackiw formalism, are shown to transform the fields of the model into BRST and σ closed. The same analysis is also carried out in the case of spinor electrodynamics.     

Dynamical Model and Path Integral Formalism for Hubbard Operators

In this paper, the possibility to construct apath integral formalism by using the Hubbard operatorsas field dynamical variables is investigated. By meansof arguments coming from the Faddeev-Jackiw symplectic Lagrangian formalism as well as from theHamiltonian Dirac method, it can be shown that it is notpossible to define a classical dynamics consistent withthe full algebra of the Hubbard X-operators. Moreover, from the Faddeev-Jackiw symplectic algorithm,and in order to satisfy the Hubbard X-operatorscommutation rules, it is possible to determine thenumber of constraints that must be included in aclassical dynamical model. Following this approach, it isclear how the constraint conditions that must beintroduced in the classical Lagrangian formulation areweaker than the constraint conditions imposed by the full Hubbard operators algebra. The consequenceof this fact is analyzed in the context of the pathintegral formalism. Finally, in the framework of theperturbative theory, the diagrammatic and the Feynman rules of the model are discussed.     

Hamilton–Jacobi and Symplectic Analysis of a Particle Constrained on a Circle

Hamilton-Jacobi and modified Faddeev-Jackiw methods were applied to investigate the motion of a particle moving on a circle. The results of both methods were found to be equivalent with those of Dirac's formalism. Besides, the importance of the Lagrange multipliers was analyzed and the action of the second-class constrained system was given.     

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.     

Symplectic quantization of open strings in constant background <Emphasis Type="Italic">B</Emphasis>-field

The symplectic quantization (Faddeev-Jackiw) method is reviewed briefly, and then it is applied to the open strings in the D-brane background with a non-vanishing constant B-field. We shall work in the discrete version, and the reduced phase space is obtained directly by solving the mixed boundary conditions. The non-commutativity of coordinates along the D-brane is reproduced. Some ambiguities in the previous papers could be avoided by this method. Received: 3 April 2003, Revised: 18 April 2003, Published online: 18 June 2003     

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.     

An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions

In this paper, we discuss an algebraic multigrid (AMG) method for nearly incompressible elasticity problems in two-dimensions. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and the quartic finite element space. By choosing different smoothers, we obtain two types of two-level methods, namely TL-GS and TL-BGS. The theoretical analysis and numerical results show that the convergence rates of TL-GS and TL-BGS are independent of the mesh size and the Young's modulus, and the convergence of the latter is greatly improved on the order $p$. However, the convergence of both methods still depends on the Poisson's ratio. To fix this, we obtain a coarse level matrix with less rigidity based on selective reduced integration (SRI) method and get some types of two-level methods by combining different smoothers. With the existing AMG method used as a solver on the first coarse level, an AMG method can be finally obtained. Numerical results show that the resulting AMG method has better efficiency for nearly incompressible elasticity problems.