Dirac and Faddeev–Jackiw quantization of a five-dimensional Stüeckelberg theory with a compact dimension

A detailed Hamiltonian analysis for a five-dimensional Stüeckelberg 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 Stüeckelberg 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 under study, and we calculate the generalized Faddeev–Jackiw brackets, we show that both the Faddeev–Jackiw and Dirac’s brackets are the same. Finally we discuss some remarks and prospects.     

Faddeev–Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions

A detailed Faddeev–Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions is performed. We obtain for the theories under study the constraints, the gauge transformations, the generalized Faddeev–Jackiw brackets and we perform the counting of physical degrees of freedom. In addition, we compare our results with those found in the literature where the canonical analysis is developed, in particular, we show that both the generalized Faddeev–Jackiw brackets and Dirac’s brackets coincide to each other. Finally we discuss some remarks and prospects.     

Incomplete Dirac reduction of constrained Hamiltonian systems

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.     

Hamiltonian Dynamics for Einstein’s Action in <Emphasis Type="Italic">G</Emphasis>→0 Limit

The Hamiltonian analysis for the Einstein’s action in G→0 limit is performed. Considering the original configuration space without involve the usual ADM variables we show that the version G→0 for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.     

Simplectic Geometry and the Canonical Variables for Dirac–Nambu–Goto and Gauss–Bonnet System in String Theory

Using a strongly covariant formalism given by Carter for the deformations dynamics of p-branes in a curved background and a covariant and gauge invariant geometric structure constructed on the corresponding Witten's phase space, we identify the canonical variables for Dirac–Nambu–Goto (DNG) and Gauss–Bonnet (GB) system in string theory. Future extensions of the present results are outlined.     

Hamilton-Jacobi theory for Hamiltonian systems with non-canonical symplectic structures

A proposal for the Hamilton-Jacobi theory in the context of the covariant formulation of Hamiltonian systems is done. The current approach consists in applying Dirac’s method to the corresponding action which implies the inclusion of second-class constraints in the formalism which are handled using the procedure of Rothe and Scholtz recently reported. The current method is applied to the non-relativistic two-dimensional isotropic harmonic oscillator employing the various symplectic structures for this dynamical system recently reported.     

Deformation Dynamics and the Gauss–Bonnet Topological Term in String Theory

We show that there exists a nontrivial contribution on the Witten covariant phase space when the Gauss–Bonnet topological term is added to the Dirac–Nambu–Goto action describing strings, because the geometry of deformations is modified, and on such space we construct a symplectic structure. Future extensions of the present results are outlined.     

Faddeev–Jackiw quantization of four dimensional BF theory

The symplectic analysis of a four dimensional BF$BF$ theory in the context of the Faddeev–Jackiw symplectic approach is performed. It is shown that this method is more economical than Dirac’s formalism. In particular, the complete set of Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported. In addition, we show that the generalized Faddeev–Jackiw brackets and the Dirac ones coincide to each other. Finally, the similarities and advantages between Faddeev–Jackiw method and Dirac’s formalism are briefly discussed.     

On the Dirac eigenvalues as observables of the on-shell <Emphasis Type="Italic">N</Emphasis> = 2<Emphasis Type="Italic">D</Emphasis> = 4 Euclidean supergravity

We generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.      

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.     

Particle on a torus knot: Constrained dynamics and semi-classical quantization in a magnetic field

Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac’s Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied.     

Symmetries in constrained Hamiltonian systems

We derive an algorithm for the construction of all the gauge generators of a constrained hamiltonian theory. Dirac's conjecture that all secondary first-class constraints generate symmetries is revisited and replaced by a theorem. The algorithm is applied to Yang-Mills theories and metric gravity, and we find generators which operate on the complete set of canonical variables, thus producing the correct transformation laws also for the unphysical coordinates. Finally we discuss the general structure of the Hamiltonian for constrained theories. We show how in most cases one can read off the first-class constraints directly from the Hamiltonian.     

Lorentz-Covariant Hamiltonian Formalism

The dynamics of a classical system can be expressed by means of Poissonbrackets. In this paper we generalize the relation between the usual noncovariantHamiltonian and the Poisson brackets to a covariant Hamiltonian and new bracketsin the frame of Minkowski space. These brackets can be related to those usedby Feynman in his derivation Maxwell's equations. The case of curved space isalso considered with the introduction of Christoffel symbols, covariant derivatives,and curvature tensors.     

Electric and Magnetic Monopoles from a Lorentz-Covariant Hamiltonian

In previous work we generalized the relation between the usual noncovariantHamiltonian and the Poisson brackets to a covariant Hamiltonian and new bracketsin the frame of Minkowski space. In the present paper we study the consequencesof this new algebraic structure on the Lorentz Lie algebra defined in terms ofthese brackets. We show how a monopole with a dual electric—magnetic chargeappears as a consequence of the conservation of the form of the standard Lorentzalgebra symmetry. The breakdown of this symmetry is also envisaged.     

A note on the Holst action, the time gauge, and the Barbero–Immirzi parameter

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero–Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero–Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.     

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.