Anjali S, Saurabh Gupta
We analyze the constraints of Christ-Lee model by the 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.
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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.
We quantize the (2+1)-dimensional self-dual and Maxwell-Chern-Simons theories by using the Faddeev-Jackiw formulation and compare the results with those of the Dirac formalism.
Application of the Faddeev-Jackiw formalism to the gauged WZW model
采用Faddeev-Jackiw方法将电磁耦合的Wess-Zumino双味模型作为约束系统处理。我们将π介子场展开为幂级数，并保持项为二阶和三阶，得到了含有非定域项的库仑规范拉格朗日。
Faddeev-Jackiw formalism for a topological-like oscillator in planar dimensions
利用Faddeev和Jackiw提出的约束系统的辛形式，研究了二维空间中谐振子与电磁势和拓扑类（Chern-Simons）质量项的耦合问题.
The two flavour, four dimensional WZW model coupled to electromagnetism, is treated as a constraint system in the context of the Faddeev-Jackiw approach. No approximation is made. Detailed exposition of the calculations is given. Solution of the constraints followed by proper Darboux's transformations leads to an unconstrained Coulomb-gauge Lagrangian density.
BFV analysis of the $U_{EM}(1)$ gauged SU(3) WZW model and the Faddeev-Jackiw approach
在Batalin-Fradkin- Vilkovisky形式下，把耦合到电磁场的四维SU（3）WZW模型作为约束系统处理.结果表明，这种处理是等效的Faddeev-Jackiw（FJ）的方法。它还表明，转换成BRST和$σ$封闭的模型的字段的字段重新定义实际上是达布的FJ形式主义中使用的转换。
The conformal affine sl(2) Toda model coupled to matter field is treated as a constrained system in the context of Faddeev-Jackiw and the (constrained) symplectic schemes. We recover from this theory either, the sine-Gordon or the massive Thirring model, through a process of Hamiltonian reduction, considering the equivalence of the Noether and topological currrents as a constraint and gauge fixing the conformal symmetry.
In this work we have shown that it is possible to construct non-Abelian field theories employing, in a systematic way, the Faddeev-Jackiw symplectic formalism. This approach follows two steps. In the first step, the original Abelian fields are modified in order to introduce the non-Abelian algebra. After that, the Faddeev-Jackiw method is implemented and the gauge symmetry relative to some non-Abelian symmetry group, is introduced through the zero-mode of the symplectic matrix. We construct the SU(2) and SU(2)xU(1) Yang-Mills theories having as starting point the U(1) Maxwell electromagnetic t…
The general conditions for the applicability of the Faddeev-Jackiw approach to gauge theories are studied. When the constraints are effective a new proof in the Lagrangian framework of the equivalence between this method and the Dirac approach is given. We find, however, that the two methods may give different descriptions for the reduced phase space when ineffective constraints are present. In some cases the Faddeev-Jackiw approach may lose some constraints or some equations of motion. We believe that this inequivalence can be related to the failure of the Dirac conjecture (that says that the…
Faddeev-Jackiw Approach to Classical Constrained Systems
本文采用Faddeev-Jackiw方法研究经典约束系统的量化，分析约束结构，揭示规范对称性，并提供辛框架的MATLAB实现方法。
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 ze…
In this paper we analyze two higher-derivative theories, the generalized electrodynamics and the Alekseev-Arbuzov-Baikov's effective Lagrangian from the point of view of Faddeev-Jackiw sympletic approach. It is shown that the full set of constraint is obtained directly from the zero-mode eigenvectors, and that they are in accordance with known results from Dirac's theory, a remnant and recurrent issue in the literature. The method shows to be rather economical in relation to the Dirac's one, obviating thus unnecessary classification and calculations. Afterwards, to conclude we construct the tr…
We obtain a new symplectic Lagrangian density and deduce Faddeev-Jackiw (FJ) generalized brackets of the gauge invariant self-dual fields interacting with gauge fields. We further give FJ quantization of this system. Furthermore, the FJ method is compared with Dirac method, the results show the two methods are equivalent in the quantization of this system. And by the practical research in this letter, it can be found that the FJ method is really simpler than the Dirac method, namely, the FJ method obviates the need to distinguish primary and secondary constraints and first- and second-class co…
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 Faddee…
In this paper the Dirac and Faddeev-Jackiw formulation for Einstein's theory in the $G \rightarrow 0$ limit is performed; the fundamental Dirac's and Faddeev-Jackiw brackets for the theory are obtained. First, the Dirac brackets are constructed by eliminating the second class constraints remaining the first class ones, then we fix the gauge and we convert the first class constraints into second class constraints and the new fundamental Dirac's brackets are computed. Alternatively, we reproduce all relevant Dirac's results by means of the symplectic method. We identify the Faddeev-Jackiw constr…
The equivalence between the Dirac method and Faddeev-Jackiw analysis for gauge theories is proved. In particular we trace out, in a stage by stage procedure, the standard classification of first and second class constraints of Dirac's method in the F-J approach. We also find that the Darboux transformation implied in the F-J reduction process can be viewed as a canonical transformation in Dirac approach. Unlike Dirac's method the F-J analysis is a classical reduction procedure, then the quantization can be achieved only in the framework of reduce and then quantize approach with all the know pr…
We extend the quantization à la Faddeev-Jackiw for non-autonomous singular systems. This leads to a generalization of the Schrödinger equation for those systems. The method is exemplified by the quantization of the damped harmonic oscillator and the relativistic particle in an external electromagnetic field.
Path Integral Quantization of Generalized Stueckelberg Electrodynamics: A Faddeev-Jackiw Approach
本文利用扩展至Grassmann自由度的Faddeev-Jackiw方法，对含四阶导数和规范自由度的广义Stueckelberg电动力学进行路径积分量子化，推导了经典括号、生成泛函及积分测度。
We study the presence of symmetry transformations in the Faddeev-Jackiw approach for constrained systems. Our analysis is based in the case of a particle submitted to a particular potential which depends on an arbitrary function. The method is implemented in a natural way and symmetry generators are identified. These symmetries permit us to obtain the absent elements of the sympletic matrix which complement the set of Dirac brackets of such a theory. The study developed here is applied in two different dual models. First, we discuss the case of a two-dimensional oscillator interacting with an …
FADDEEV-JACKIW APPROACH TO HIDDEN SYMMETRIES
由于缺乏第一类约束作为对称生成元，狄拉克形式主义中隐藏对称的研究没有一个系统的过程。另一方面，在Faddeev-Jackiw方法中，规范和重参数化对称性是由辛矩阵的零本征向量而不是由约束产生的，这表明在整个形式主义中系统地处理隐藏对称性的可能性。本文证明了非不变系统或规范固定系统的隐对称性和显不变性一样，都可以用辛矩阵的零本征向量来描述。因此，Faddeev-Jackiw方法提供了一种系统的算法，以统一的方式处理各种对称性。这种技术在这里说明的SL（2，R）仿射李代数的2-D诱导引力提出的Polyakov，这是一个隐藏的对称性约束系统的正则方法通过狄拉克的方法，后共形和重新参数化不变性已被固定。
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.
A detailed Faddeev-Jackiw quantization of an Abelian topological gravity is performed; we show that this formalism is equivalent and more economical than Dirac's method. In particular, we identify the complete set of constraints of the theory, from which the number of physical degrees of freedom is explicitly computed. We prove that the generalized Faddeev-Jackiw brackets and the Dirac ones coincide to each other. Moreover, we perform the Faddeev-Jackiw analysis of the theory at the chiral point, and the full set of constraints and the generalized Faddeev-Jackiw brackets are constructed. Final…
We consider the gauged model of Siegel type chiral boson with a Lorentz non-covariant mass-like term for the gauge fields which is found to be equivalent to the chiral Schwinger model with Faddeevian anomaly when it is described in terms of Floreanini-Jackiw type chiral boson. We carry out the quantization of gauge non-invariant version this model in both the Lagrangian and Hamiltonian formulation. The quantization of the gauge-invariant version of this model in the extended phase space also has been carried out in the Lagrangian formulation. The gauge-invariant version of this model in the ex…
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 by 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 Podolsky theory.
The method for quantization of constrained theories that was suggested originally by Faddeev and Jackiw along with later modifications is discussed. The particular emphasis of this paper is to show how it is simple to implement their method within the path integral framework using the natural geometric structure that their method utilizes. The procedure is exemplified with the analysis of two models: a quantum mechanical particle constrained to a surface (of which the hypersphere is a special case), and a quantized Schrödinger field interacting with a quantized vector field for both the massiv…
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 intro…
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 $1+1$-dimensional Minkowski in the eikonal case) imply that there might be an interestin…
A detailed Hamiltonian analysis for a five-dimensional St{ü}eckelberg theory with a compact dimension is performed. First, we develop a pure Dirac's analysis of the theory, we show that after performing the compactification, the theory is reduced to four-dimensional St{ü}eckelberg theory plus a tower of Kaluza-Klein modes. We develop a complete analysis of the constraints, we fix the gauge and we show that there are present pseudo-Goldstone bosons. Then we quantize the theory by constructing the Dirac brackets. As complementary work, we perform the Faddeev-Jackiw quantization for the theory un…
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-Schwing…
We analyze the constraints for a system of anti self-dual Yang-Mills (ASDYM) equations by means of the modified Faddeev-Jackiw method in K and J gauges à la Yang. We also establish the Hamiltonian flow for ASDYM system through the hidden BRS invariance in both the gauges. Finally, we remark on the bi-Hamiltonian nature of ASDYM and the compatibility of the symplectic structures therein.
The Legendre transformation on singular Lagrangians, e.g. Lagrangians representing gauge theories, fails due to the presence of constraints. The Faddeev-Jackiw technique, which offers an alternative to that of Dirac, is a symplectic approach to calculating a Hamiltonian paired with a well-defined initial value problem when working with a singular Lagrangian. This phase space coordinate reduction was generalized by Barcelos-Neto and Wotzasek to simplify its application. We present an extension of the Faddeev-Jackiw technique for constraint reduction in gauge field theories and non-gauge field t…
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.
We show how to systematically apply the Faddeev-Jackiw symplectic method to General Relativity (GR) and to GR extensions. This provides a new coherent frame for Hamiltonian analyses of gravitational theories. The emphasis is on the classical dynamics, uncovering the constraints, the gauge transformations and the number of degrees of freedom; but the method results are also relevant for canonical quantization approaches. We illustrate the method with three applications: GR and to two Brans-Dicke cases (the standard case $ω\not= - 3/2$ and the case with one less degree of freedom, $ω= - 3/2$). W…
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.
Faddeev-Jackiw Hamiltonian formulation for general exotic bi-gravity
该研究采用辛几何方法分析一般奇异双引力理论的动力学结构，推导出全部约束与标量关系，并应用于反德西特背景，最终证明该理论具有两个物理自由度。
By applying the Faddeev-Jackiw symplectic approach we systematically show that both the local gauge symmetry and the constraint structure of topologically massive gravity with a cosmological constant $Λ$, elegantly encoded in the zero-modes of the symplectic matrix, can be identified. Thereafter, via an appropriate partial gauge-fixing procedure, the time gauge, we calculate the quantization bracket structure (generalized Faddeev-Jackiw brackets) for the dynamic variables and confirm that the number of physical degrees of freedom is one. This approach provides an alternative to explore the dyn…
We show that the constraint structure in the chain by chain method can be investigated within the symplectic analysis of Faddeev-Jackiw formalism.
The Faddeev-Jackiw Hamiltonian Reduction approach to constrained dynamics is applied to the collective coordinates analysis of non-linear waves, and compared with the alternative procedure known as symplectic formalism.
The path integral quantization method is applied to a relativistically covariant version of the Hopfield model, which represents a very interesting mesoscopic framework for the description of the interaction between quantum light and dielectric quantum matter, with particular reference to the context of analogue gravity. In order to take into account the constraints occurring in the model, we adopt the Faddeev-Jackiw approach to constrained quantization in the path integral formalism. In particular we demonstrate that the propagator obtained with the Faddeev-Jackiw approach is equivalent to th…
We analyse the constraints of an Abelian 2-form gauge theory using Faddeev-Jackiw symplectic formalism. Further, this theory is treated as a constrained system in the context of Batalin-Fradkin-Vilkovisky formalism to retrieve the BRST symmetry. Using the fields decompositions the effective action for Abelian 2-form gauge theory is written in terms of diagonalized uncanonical part and 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.
The symplectic analysis, initiated by Faddeev and Jackiw, is applied to the first order (Palatini) form of the Einstein-Hilbert action in 1 + 1 dimensions. The constraints that arise are shown to result in the same gauge transformations that follow from the first class constraints occurring when the Dirac constraint formalism is applied to this action. Problems associated with gauge fixing are discussed.
The SU(2) collective coordinates quantization of the Born-Infeld Skyrmions Lagrangean is performed. The obtainment of the classical Hamiltonian from this special Lagrangean is made by using an approximate way: it is derived from the expansion of this non-polynomial Lagrangean up to second-order variable in the collective coordinates, using for this some causality arguments. Because this system presents constraints, we use the Dirac Hamiltonian method and the Faddeev-Jackiw Lagrangean approach to quantize this model.
Parity and Front-Form Quantization of Field Theories
最近，我们提出了一种新的前形量子化，它把$x^{+}$和$x^{-}$坐标都作为前形时间。这种量化被发现可以明确地保持奇偶性。在本文中，我们将这种结构推广到费米场的汤川理论的背景下。我们使用最初由Faddeev和Jackiw提出的方法来验证这个理论。我们发现，$P^-$ { \it和} $P^+$成为动态的，该理论是明显不变的宇称。
We deduce the constraint structure and subsequently quantize a free particle system residing on a torus satisfying the geometric constraint $(r-a)=0$, within the framework of modified Faddeev-Jackiw formalism. Further, we reformulate the system as a gauge theory by the means of symplectic gauge invariant formalism. Finally, we establish the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries of the reformulated gauge invariant theory.
Quantization of Gauge Field Theories on the Front-Form without Gauge Constraints I : The Abelian Case
最近，我们提出了一种新的前形量子化，它把$x^{+}$和$x^{-}$坐标都作为前形时间。这种量化被发现可以明确地保持奇偶性。本文将这一构造推广到局部阿贝尔规范场。我们使用最初由Faddeev和Jackiw提出的方法来验证这个理论。我们在这里强调的特点，量子化沿$x^+$和$x^-$，规范理论不需要额外的约束（也称为“规范条件”），以确定唯一的解决方案。
On the Quantization of FLPR Model
本文在修正的Faddeev-Jackiw形式下，通过可容许规范条件对FLPR模型进行量子化，推导出规范对称性并建立离壳幂零且绝对反对易的（反）BRST对称性，同时证明物理态被第一类约束湮灭，与Dirac形式一致。
We quantize a particle confined to move on a torus knot satisfying constraint condition ($p θ+ q ϕ) \approx 0$, 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 formali…
A Darboux-Theorem-Based Derivation of Geometric Structures in Various Non-Abelian Gauge Theories
本文将Faddeev-Jackiw方法应用于多种非阿贝尔规范理论，结合Dirac猜想推导其几何结构，实现了规范结构的统一高效推导。
The derivation of the brackets among coordinates and momenta for classical constrained systems is a necessary step toward their quantization. Here we present a new approach for the determination of the classical brackets which does neither require Dirac's formalism nor the symplectic method of Faddeev and Jackiw. This approach is based on the computation of the brackets between the constants of integration of the exact solutions of the equations of motion. From them all brackets of the dynamical variables of the system can be deduced in a straightforward way.
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 in its quantum counterpart with subtleties in its symplectic structure and symmetries. The model is used here to investigate quantization issues related to the Hamiltonian constraint structure, Dirac brackets, gauge symmetry and BRST transformations. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelo…
The BRST quantization of the Abelian Proca model is performed using the Batalin-Fradkin-Tyutin and the Batalin-Fradkin-Vilkovisky formalism. First, the BFT Hamiltonian method is applied in order to systematically convert a second class constraint system of the model into an effectively first class one by introducing new fields. In finding the involutive Hamiltonian we adopt a new approach which is more simpler than the usual one. We also show that in our model the Dirac brackets of the phase space variables in the original second class constraint system are exactly the same as the Poisson brac…
Symplectic Quantization and General Constraint Structure of a Prototypical Second-Class System
本文研究原型受约束哈密顿系统，运用Dirac-Bergmann算法解析二类约束并获取Dirac括号代数，结合FJBW方法强调子流形几何特征，通过环面几何与洛伦兹破坏模型验证结果，并探讨其在宇宙学中的应用。
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