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     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Faddeev-Jackiw and the improved methods in quantization of the superconductive system

In this paper, we give the overview on Faddeev-Jackiw method and its improved one, as well as the relative studies recently and realize quantization of the superconductive system by the two methods, we get the same results by the two disposal methods. Furthermore, at convenience of the familiar study in this system, we take it as the application example and compare the two methods by dealing with this system from different aspects, demonstrate the improved Faddeev-Jackiw method is effective and significative, and represent the superiorities of the improved Faddeev-Jackiw method. We show that the improved method may simplify investigations of different complicated constrained systems.     

Self-Dual Conformal Supergravity and the Hamiltonian Formulation

In terms of Dirac matrices the self-dual and anti-self-dual decomposition of conformal supergravity is given and a self-dual conformal supergravity theory is developed as a connection dynamic theory in which the basic dynamic variables include the self-dual spin connection i.e. the Ashtekar connection rather than the triad. The Hamiltonian formulation and the constraints are obtained by using the Dirac-Bergmann algorithm.     

The Quantal Canonical Symmetries for a Singular Lagrangian System with Subsidiary Constraints

The quantization for a system containing subsidiary constraints (in configuration space) with a singular Lagrangian is studied, in certain case which can be brought into the theoretical framework of constrained Hamiltonian system. A modified Dirac-Bergmann algorithm for the calculation of all phase-space constraints in those systems is derived. The path integral quantization is formulated by using the Faddeev-Senjanovic scheme. The classical and quantum canonical symmetries (Noether theorem in canonical formalism) are established for such a system. An example is given to illustrate that the connection between the symmetry and conservation law in classical theory are not always validity in the quantum theory.     

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     

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.     

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.     

Dirac mechanics and Landau two-fluid model in4HE II

This paper is devoted to the development of the Dirac formalism for singular systems when applied to the Landau two-fluid model in superfluid helium. Notably, the Hamiltonian density is weakly zero (in the sense of Dirac). We obtain the physical and gauge variables, and show that all the constraints are of first class, and hence, that the Dirac bracket coincides with the Poisson bracket. We leave the quantization of this system for a later work.     

Explicit Formula for the Natural and Projectively Equivariant Quantization

In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecome conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence using the framework of Thomas–Whitehead connections. In [Lett Math Phys 72(3):183–196, 2005], we gave a new proof of the same theorem thanks to the Cartan connections. After these works, there was no explicit formula for the quantization. In this paper, we give this formula using the formula in terms of Cartan connections given in [Lett Math Phys 72(3):183–196, 2005]. This explicit formula constitutes the generalization to any order of the formulae at second and third orders soon published by Bouarroudj in [Lett Math Phys 51(4):265–274, 2000] and [C R Acad Sci Paris Sér I Math 333(4):343–346, 2001].     

On gauge-invariant variables in QED

Exploiting the path-dependence of the gaugeinvariant variables formalism, we illustrate how the gaugefixing procedure correspond, in this formalism, to choose the path. In particular, we consider two propagators for the gauge field. QED is formulated in terms of gauge-invariant variables and its quantization is carried out using the Dirac’s method for constrained systems.     

A Gauge Invariant Description for the General Conic Constrained Particle from the FJBW Iteration Algorithm

We consider a second-degree algebraic curve describing a general conic constraint imposed on the motion of a massive spinless particle. The problem is trivial at classical level but becomes involved and interesting concerning its quantum counterpart with subtleties in its symplectic structure and symmetries. We start with a second-class version of the general conic constrained particle, which encompasses previous versions of circular and elliptical paths discussed in the literature. By applying the symplectic FJBW iteration program, we proceed on to show how a gauge invariant version for the model can be achieved from the originally second-class system. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelos-Wotzasek iteration program to unravel the essential aspects of the constraint structure. While in the standard Dirac-Bergmann approach there are four second-class constraints, in the FJBW they reduce to two. By using the symplectic potential obtained in the last step of the FJBW iteration process, we construct a gauge invariant model exhibiting explicitly its BRST symmetry. We obtain the quantum BRST charge and write the Green functions generator for the gauge invariant version. Our results reproduce and neatly generalize the known BRST symmetry of the rigid rotor, clearly showing that this last one constitutes a particular case of a broader class of theories.     

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.     

Dirac and Lagrangian reductions in the canonical approach to the first-order form of the Einstein-Hilbert action

It is shown that the Lagrangian reduction, in which solutions of equations of motion that do not involve time derivatives are used to eliminate variables, leads to results quite different from the standard Dirac treatment of the first-order form of the Einstein-Hilbert action when the equations of motion correspond to the first class constraints. A form of the first-order formulation of the Einstein-Hilbert action which is more suitable for the Dirac approach to constrained systems is presented. The Dirac and reduced approaches are compared and contrasted. This general discussion is illustrated by a simple model in which all constraints and the gauge transformations which correspond to first class constraints are completely worked out using both methods to demonstrate explicitly their differences. These results show an inconsistency in the previous treatment of the first-order Einstein-Hilbert action which is likely responsible for problems with its canonical quantization.