Dirac Brackets Research Articles

FIRST-ORDER ACTIONS AND DUALITY

We consider some aspects of classical S-duality transformations in first-order actions taking into account the general covariance of the Dirac algorithm and the transformation properties of the Dirac bracket. By classical S-duality transformations we mean a field redefinition that interchanges the equations of motion and its associated Bianchi identities. By working from a first-order variational principle and performing the corresponding Dirac analysis we find that the standard electromagnetic duality can be reformulated as a canonical local transformation. The reduction from this phase space to the original phase space variables coincides with the well-known result about duality as a canonical nonlocal transformation. We have also applied our ideas to the bosonic string. These dualities are not canonical transformations for the Dirac bracket and relate actions with different kinetic terms in the reduced space.

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 Backlund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.

Quantum mechanics of the doubled torus

We investigate the quantum mechanics of the doubled torus system, introduced by Hull [1] to describe T-folds in a more geometric way. Classically, this system consists of a world-sheet Lagrangian together with some constraints, which reduce the number of degrees of freedom to the correct physical number. We consider this system from the point of view of constrained Hamiltonian dynamics. In this case the constraints are second class, and we can quantize on the constrained surface using Dirac brackets. We perform the quantization for a simple T-fold background and compare to results for the conventional non-doubled torus system. Finally, we formulate a consistent supersymmetric version of the doubled torus system, including supersymmetric constraints.

Star Products and Branes in Poisson-Sigma Models

We prove that non-coisotropic branes in the Poisson-Sigma model are allowed at the quantum level. When the brane is defined by second-class constraints, the perturbative quantization of the model yields the Kontsevich's star product associated to the Dirac bracket on the brane. We also discuss the quantization when both first and second-class constraints are present.

Polynomial Hamiltonian form of general relativity

Phase space of General Relativity is extended to a Poisson manifold by inclusion of the determinant of the metric and conjugate momentum as additional independent variables. As a result, the action and the constraints take a polynomial form. New expression for the generating functional for the Green functions is proposed. We show that the Dirac bracket defines degenerate Poisson structure on a manifold, and a second class constraints are the Casimir functions with respect to this structure. As an application of the new variables, we consider the Friedmann universe.

The reduced phase space of an open string in the background B-field

The problem of an open string in background B-field is discussed. Using the discretized model in details we show that the system is influenced by an infinite number of second class constraints. We interpret the allowed Fourier modes as the coordinates of the reduced phase space. This enables us to compute the Dirac brackets more easily. We prove that the coordinates of the string are non-commutative at the boundaries. We argue that in order to find the Dirac bracket or commutator algebra of the physical variables, one should not expand the fields in terms of the solutions of the equations of motion. Instead, one should impose a set of constraints in suitable coordinates.

Reality conditions for Ashtekar gravity from Lorentz-covariant formulation

We study the limit of the Lorentz-covariant canonical formulation where the Immirzi parameter approaches β = i. We show that, formulated in terms of a shifted spacetime connection, which also plays a crucial role in the covariant quantization, the limit is smooth and reproduces the canonical structure of the self-dual Ashtekar gravity. The reality conditions of Ashtekar gravity can be incorporated by means of the Dirac brackets derived from the covariant formulation and defined on an extended phase space which involves, besides the self-dual variables, also their anti-self-dual counterparts.

A complete solution of a constrained system: SUSY monopole quantum mechanics

We solve the quantum mechanical problem of a charged particle on S^2 in the background of a magnetic monopole for both bosonic and supersymmetric cases by constructing Hilbert space and realizing the fundamental operators obeying complicated Dirac bracket relations in terms of differential operators. We find the complete energy eigenfunctions. Using the lowest energy eigenstates we count the number of degeneracies and examine the supersymmetric structure of the ground states in detail.

Publisher’s Note: Quantum deformation of the Dirac bracket [Phys. Rev. D73, 025008 (2006)

Received 18 January 2006DOI:https://doi.org/10.1103/PhysRevD.73.029905©2006 American Physical Society

Quantum deformation of the Dirac bracket

The quantum deformation of the Poisson bracket is the Moyal bracket. We construct quantum deformation of the Dirac bracket for systems which admit global symplectic basis for constraint functions. Equivalently, it can be considered as an extension of the Moyal bracket to second-class constraints systems and to gauge-invariant systems which become second class when gauge-fixing conditions are imposed.

Statistical mechanics of quantum-classical systems with holonomic constraints

The statistical mechanics of quantum-classical systems with holonomic constraints is formulated rigorously by unifying the classical Dirac bracket and the quantum-classical bracket in matrix form. The resulting Dirac quantum-classical theory, which conserves the holonomic constraints exactly, is then used to formulate time evolution and statistical mechanics. The correct momentum-jump approximation for constrained systems arises naturally from this formalism. Finally, in analogy with what was found in the classical case, it is shown that the rigorous linear-response function of constrained quantum-classical systems contains nontrivial additional terms which are absent in the response of unconstrained systems.

OPERATOR ORDERING AND CLASSICAL SOLITON PATH IN TWO-DIMENSIONAL N = 2 SUPERSYMMETRY WITH KÄHLER POTENTIAL

We investigate a two-dimensional N = 2 supersymmetric model which consists of n chiral superfields with Kähler potential. When we define quantum observables, we are always plagued by operator ordering problem. Among various ways to fix the operator order, we rely upon the supersymmetry. We demonstrate that the correct operator order is given by requiring the super-Poincaré algebra by carrying out the canonical Dirac bracket quantization. This is shown to be also true when the supersymmetry algebra has a central extension by the presence of topological soliton. It is also shown that the path of soliton is a straight line in the complex plane of superpotential W and triangular mass inequality holds. One half of supersymmetry is broken by the presence of soliton.

New realizations of observables in dynamical systems with second class constraints

In the Dirac bracket approach to dynamical systems with second class constraints observables are represented by elements of a quotient Dirac bracket algebra. We describe families of new realizations of this algebra through quotients of the original Poisson bracket algebra. Explicit expressions for generators and brackets of the algebras under consideration are found.

Improved BFT embedding having chain-structure

We newly revisit the gauge non-invariant chiral Schwinger model with a=1 in view of the chain structure. As a result, we show that the Dirac brackets can be easily read off from the exact symplectic algebra of second-class constraints. Furthermore, by using an improved BFT embedding preserving the chain structure, we obtain the desired gauge invariant action including a new type of Wess–Zumino term.

Non-Abelian conversion and quantization of nonscalar second-class constraints

We propose a general method for deformation quantization of any second-class constrained system on a symplectic manifold. The constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a non-trivial vector bundle over the phase-space manifold. The covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the ``constraint bundle'', which becomes a key ingredient of the conversion procedure for the non-scalar constraints. Unlike in the case of scalar second-class constraints, no Abelian conversion is possible in general. Within the BRST framework, a systematic procedure is worked out for converting non-scalar second-class constraints into non-Abelian first-class ones. The BRST-extended system is quantized, yielding an explicitly covariant quantization of the original system. An important feature of second-class systems with non-scalar constraints is that the appropriately generalized Dirac bracket satisfies the Jacobi identity only on the constraint surface. At the quantum level, this results in a weakly associative star-product on the phase space.

Hamiltonian and action principle formulations of plasma physics

Hamiltonian and action principle formulations of the basic equations of plasma physics are reviewed. Various types of Lagrangian and Poisson bracket formulations for kinetic and fluid theories are discussed, and it is described how such formulations can be used to derive and approximate physical models. Additional uses are also described. Two applications are treated in greater detail: an algorithm based on Dirac brackets for the calculation of V-states of contour dynamics and the calculation of fluctuation spectra of Vlasov theory and shear flow dynamics by methods of statistical mechanics.

Chern–Simons soliton and a critical study of the energy momentum tensor in noncommutative field theory

We have shown unambiguously the existence of solitons in the non-commutative (NC) extension of Chern–Simons–Higgs model. The analysis is done at the classical level (since solitons are essentially classical objects) and in the first non-trivial order in θ, the only spatial noncommutativity parameter. At the same time, we have exposed an inadequacy in the conventional definitions of the energy momentum tensor (EMT) in the present context but this pathology appears to be generic to NC field theories. This is reflected in the fact that the BPS soliton equations (obtained from the EMT) are not compatible with the full variational equations of motion, requiring further imposition a restriction on the form of the Higgs field, contrary to the commutative spacetime case. Both in the Lagrangian and Hamiltonian formulations of the problem, we concentrate on the canonical and symmetric forms of the energy–momentum tensor. In the Hamiltonian scheme, constraint analysis and the induced Dirac brackets are derived. In fact the EMT behaves properly as the spacetime translation generators and their actions on the fields are discussed in detail. The effects of noncommutativity on the soliton solutions have been analyzed carefully and we have come up with some interesting results. Comparing the relative strengths of the noncommutative effects, we have shown that there is a universal character in the noncommutative correction to the magnetic field—it depends only on θ. On the other hand, in the cases of all other observables of physical interest, such as the potential profile, soliton mass or the electric field, the parameters θ as well as τ (the latter comprising solely of commutative Chern–Simons–Higgs model parameters) appear with similar weightage.

Two-dimensional [formula omitted] supersymmetric mechanics in superspace

We construct a two-dimensional N=8 supersymmetric quantum mechanics which inherits the most interesting properties of N=2, d=4 supersymmetric Yang–Mills theory. Analyzing the superfield constraints, we show that only complex scalar fields from the N=2,d=4 vector multiplet become physical bosons in d=1. The rest of the bosonic components are reduced to auxiliary fields, thus giving rise to the (2,8,6) supermultiplet in d=1. We construct the most general superfield action for this supermultiplet and demonstrate that it possesses duality symmetry extended to the fermionic sector of theory. We also explicitly present the Dirac brackets for the canonical variables and construct the supercharges and Hamiltonian which form a N=8 super-Poincaré algebra with central charges. Finally, we discuss the duality transformations which relate the (2,8,6) supermultiplet with the (4,8,4) one.

Third post-Newtonian constrained canonical dynamics for binary point masses in harmonic coordinates

The conservative dynamics of two point masses given in harmonic coordinates up to the third post-Newtonian order is treated within the framework of constrained canonical dynamics. A representation of the approximate Poincar\'e algebra is constructed with the aid of Dirac brackets. Uniqueness of the generators of the Poincar\'e group or the integrals of motion is achieved by imposing their action on the point mass coordinates to be identical with that of the usual infinitesimal Poincar\'e transformations. The second post-Coulombian approximation to the dynamics of two point charges as predicted by Feynman-Wheeler electrodynamics in Lorentz gauge is treated similarly.

Canonical quantization of the Dirac field in a finite volume

The 1+1 dimensional Dirac field in a finite volume is quantized canonically by using two different methods in this paper. First, we take the boundary conditions (BCs) as primary Dirac constraints, and prove that those BCs as well as the intrinsic constraints are entangled and they form the second class constraints. The quantization is performed canonically by using Dirac's procedure. And then, we study this model in the reduced phase space. It shows that the Poisson brackets in the reduced phase space are equivalent to the Dirac brackets in the original phase space.