Hamiltonian BRST quantization of the interacting antisymmetric tensor field

We study the hamiltonian BRST quantization of the non-abelian antisymmetric tensor field. We find the constrained system which arises from the standard action by Dirac's procedure, and eliminate the second-class constraints by introducing Dirac brackets. Having isolated the underlying first-class constrained system, we quantize it using the hamiltonian BRST techniques of Batalin and Fradkin. We study the Lorentz covariant gauge fixing of this system, and discuss the relationship between our results and other recent studies of the interacting antisymmetric tensor field.     

Faddeev–Jackiw quantization of the higher derivative Chern–Simons gauge theory in the first-order formalism

We analyse the physical constraints of the higher derivative Chern–Simons gauge model by means of Faddeev–Jackiw symplectic approach in the first-order formalism. Within such framework, we systematically determine the zero-mode structure of the corresponding symplectic matrix. In addition, we calculate the Faddeev–Jackiw quantum brackets by choosing appropriate gauge-fixing conditions and evaluate the determinant of the non-singular symplectic matrix as well as the transition-amplitude. Finally, we present a detailed Hamiltonian analysis using Dirac–Bergmann algorithm method and show that the Dirac brackets coincide with the FJ brackets when all the second-class constraints are treated as zero equations.     

On BRST quantization of second class constraint algebras

A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechaism. As an application the possibility of string theories in subcritical dimensions is considered.     

Hidden Symmetries in the Two-Dimensional Isotropic Antiferromagnet

We discuss the two-dimensional isotropic antiferromagnet in the framework of gauge invariance. Gauge invariance is one of the most subtle useful concepts in theoretical physics, since it allows one to describe the time evolution of complex physical system in arbitrary sequences of reference frames. All theories of the fundamental interactions rely on gauge invariance. In Dirac’s approach, the two-dimensional isotropic antiferromagnet is subject to second-class constraints, which are independent of the Hamiltonian symmetries and can be used to eliminate certain canonical variables from the theory. We have used the symplectic embedding formalism developed by a few of us to make the system under study gauge invariant. After carrying out the embedding and Dirac analysis, we systematically show how second-class constraints can generate hidden symmetries. We obtain the invariant second-order Lagrangian and the gauge-invariant model Hamiltonian. Finally, for a particular choice of factor ordering, we derive the functional Schröodinger equations for the original Hamiltonian and for the first-class Hamiltonian and show them to be identical, which justifies our choice of factor ordering.     

Dirac Reduction Revisited

Abstract

The procedure of Dirac reduction of Poisson operators on submanifolds is discussed within a particularly useful special realization of the general Marsden-Ratiu reduction procedure. The Dirac classification of constraints on ‘first-class’ constraints and ‘second-class’ constraints is reexamined.     

Front-Form Hamiltonian, Path Integral, and BRST Formulations of the Nonlinear Sigma Model

The nonlinear sigma model in one-space one-time dimension is considered on the light-front. The front-form theory is seen to possess a set of three first-class constraints and consequently it possesses a local vector gauge symmetry. This is in contrast to the usual instant-form theory, which is well known to be a gauge noninvariant theory possessing a set of four second-class constraints. The front-form Hamiltonian, path integral, and BRST formulations of this theory are investigated under some specific gauge choices.     

On the Hamiltonian formalism of the tetrad-gravity with fermions

We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero (Lagraa et al. in Class Quantum Gravity 34:115010, 2017). Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form leading to the same algebra of the first-class constraints.     

有限空间中Dirac场的正则量子化

研究当存在边界的情形下 Dirac场的正则量子化问题. 采用文献[1]的观点, 将边界条件当作Dirac初级约束.与已有研究不同的是, 本文从离散的角度研究此问题. 将Dirac场的拉氏量和内在约束进行离散化, 并且将离散的边界条件当作初级Dirac约束. 因此, 从约束的起源来看, 这个模型中存在两种不同的约束: 一种是由于模型的奇异性而带来的约束, 即内在约束; 另一种是边界条件. 在对此模型进行正则量子化过程中提出一种能够平等地处理内在约束和边界条件的方法. 为了证明该方法能够平等地对待这两种起源不同的约束, 在计算Dirac 括号时分别选取了两个不同的子集合来构造"中间Dirac括号", 最后得到了相同的结果. 关键词： 边界条件 Dirac约束 Dirac括号     

Dirac Canonical Quantization of Composite Fermions QED

Composite Fermions QED is quantized by using the Dirac’s canonical formalism for constrained systems. As a strategy, we first work out the constraints (including primary and secondary constraints), combine two first-class constraints, introduce Coulomb gauge and its stationary as gauge conditions, and then quantize, replacing the Dirac brackets with quantum commutators.     

有限空间中经典场的正则量子化

从离散的角度研究带边界的1+1维经典标量场和Dirac场的正则量子化问题. 与以往不同的是, 这里将时间和空间两个变量同时进行变步长的离散, 应用变步长离散的变分原理, 得到离散形式的运动方程、边界条件和能量守恒的表达式. 然后, 根据Dirac理论, 将边界条件当作初级约束, 将边界条件和内在约束统一处理. 研究表明, 采用此方法, 不仅在每个离散的时空格点上能够建立起Dirac括号, 从而可以完成该模型的正则量子化;而且, 该方法还保持了离散情况下的能量守恒.     

The Limit of Noether Conserved Charges is the Number of Primary First-Class Constraints in a Constrained System

We discuss the stationarity of generator G for gauge symmetries in two directions.One is to the motion equations defined by total Hamiltonian H_T,and gives that the number of the independent coefficients in the generator G is not greater than the number of the primary first-class constraints,and the number of Noether conserved charges is not greater than that of the primary first-class constraints,too.The other is to the variances of canonical variables deduced from the generator G,and gives the variances of Lagrangian multipliers contained in extended Hamiltonian H_E.And a second-class constraint generated by a first-class constraint may imply a new first-class constraint which can be combined by introducing other second-class constraints.Finally,we supply two examples.One with three first-class constraints (two is primary and one is secondary) has two Noether conserved charges,and the secondary first-class constraint is combined by three second-class constraints which are a secondary and two primary second-class constraints.The other with two first-class constraints (one is primary and one is secondary) has one Noehter conserved charge.     

Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang’s J- and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac’s theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J- and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac’s theory. Finally, we present the Bäcklund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.     

S Q E D 4 and Q E D 4 on the Null-Plane

We study the scalar electrodynamics (S Q E D ₄) and the spinor electrodynamics (Q E D ₄) in the null-plane formalism. We follow Dirac’s technique for constrained systems to analyze the constraint structure in both theories in detail. We impose the appropriate boundary conditions on the fields to fix the hidden subset first class constraints that generate improper gauge transformations and obtain a unique inverse of the second-class constraint matrix. Finally, choosing the null-plane gauge condition, we determine the generalized Dirac brackets of the independent dynamical variables, which via the correspondence principle give the (anti)-commutators for posterior quantization.     

Complex Canonical Gravity and Reality Constraints

Ashtekar canonical variables for generalrelativity yielding low degree polynomial constraintsare complex and describe complex canonical gravity. Topick the real sector, a la Dirac, one must introduce reality constraints; they turn out to besecond-class. It is shown here that this holds not onlyfor pure gravity but also for a scalar fieldnon-minimally coupled to gravity. The originalsimplicity produced by the complex variables is spoiled if one getsrid of the second-class constraints via Dirac brackets,however. To circumvent such an undesirable feature,alternative possibilities are pointed out.     

Superfield formulation of the phase space path integral

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.     

Covariant Quantization of Spinor Fields in a Given Gravitational Field

In coupling gravity with the quantum field theory, unitary transformations, depending on space-time-points, were considered and derivatives were introduced, which imply a nonintegrable parallel transport of the state vectors of Hilbert space [1]. The Dirac equation, built with these generalized derivatives, is quantized in a prescribed classical gravitational field. The quantization can be performed in complete analogy to the usual procedure in Minkowski space, but the quantum state vector becomes path dependent. In carrying out the quantization, two two-component classical spinor fields necessarily occur, which obey Weyl's equation. The considered quantized Dirac equations are also picture-covariant, that is they have the same from in each physical picture, especially in the Heisenberg picture and the Schrödinger picture.     

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.     

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.     

Proper Dirac quantization of a free particle on a D-dimensional sphere

We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in D dimensions is possible only by converting the constraints to Abelian gauge constraints, which are of first class in Dirac's classification scheme. The energy spectrum is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. A quantization of Dirac's modified Poisson brackets for second-class constraints is also possible and unique, but must be rejected since the resulting energy spectrum is physically incorrect.