E. M. C. Abreu

Obtaining non-Abelian field theories via the Faddeev―Jackiw symplectic formalism

Abstract 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.

Noncommutative relativistic particles

We present a relativistic formulation of noncommutative mechanics where the object of noncommutativity {theta}{sup {mu}{nu}}is considered as an independent quantity. Its canonical conjugate momentum is also introduced so that it permits one to obtain an explicit form for the generators of the extended Poincare group in the noncommutative case. The theory, which is invariant under reparametrization, generalizes recent nonrelativistic results. Free noncommutative bosonic particles satisfy an extended Klein-Gordon equation depending on two parameters.

Quantum complex scalar fields and noncommutativity

In this work we analyze complex scalar fields using a new framework where the object of noncommutativity {theta}{sup {mu}}{sup {nu}} represents independent degrees of freedom. In a first quantized formalism, {theta}{sup {mu}}{sup {nu}} and its canonical momentum {pi}{sub {mu}}{sub {nu}} are seen as operators living in some Hilbert space. This structure is compatible with the minimal canonical extension of the Doplicher-Fredenhagen-Roberts algebra and is invariant under an extended Poincare group of symmetry. In a second quantized formalism perspective, we present an explicit form for the extended Poincare generators and the same algebra is generated via generalized Heisenberg relations. We also introduce a source term and construct the general solution for the complex scalar fields using the Greens function technique.

DUALITY THROUGH THE SYMPLECTIC EMBEDDING FORMALISM

In this work we show that we can obtain dual equivalent actions following the symplectic formalism with the introduction of extra variables which enlarge the phase space. We show that the results are equal as the one obtained with the recently developed gauging iterative Noether dualization method. We believe that, with the arbitrariness property of the zero mode, the symplectic embedding method is more profound since it can reveal a whole family of dual equivalent actions. We illustrate the method demonstrating that the gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the introduction of an explicit mass term and a topological term can be restored to obtain the dual equivalent and gauge-invariant version of the theory.

Fractional Dirac bracket and quantization for constrained systems

So far, it is not well known how to deal with dissipative systems. There are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem. On the other hand, it is well known that the fractional formalism is a powerful alternative when treating dissipative problems. In this paper, we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the Dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way. We believe that, by using the extended Dirac bracket definition, it will be possible to analyze more deeply gauge theories starting with second-class systems.

A SUPERSYMMETRIC MODEL FOR GRAPHENE

In this work we use the fermionic structure of the low-energy excitations of graphene (a monolayer of carbon atoms) to construct a new supersymmetric graphene field theory. In the current literature, concerning the inhomogeneities between neighbor carbon atoms, Jackiw et al. constructed a useful chiral Abelian gauge theory. Other supersymmetries in a theory for graphene were analyzed at the level of supersymmetric quantum mechanics. In our formulation, all these gauge theories constructed before are shown precisely to be sectors of a wider and richer planar gauge theory given here. A detailed analysis of the spectrum of this graphene field theory is also carried out.

THE DUAL EMBEDDING METHOD IN D = 3

Improving the beginning steps of a previous work, we settle the dual embedding method (DEM) as an alternative and efficient method for obtaining dual equivalent actions also in D = 3. We show that we can obtain dual equivalent actions which were previously obtained in the literature using the gauging iterative Noether dualization method (NDM). We believe that, with the arbitrariness property of the zero mode, the DEM is more profound since it can reveal a whole family of dual equivalent actions. The result confirms the one obtained previously which is important since it has the same structure that appears in the Abelian Higgs model with an anomalous magnetic interaction.

Noncommutative Particles in Curved Spaces

We present a formulation in a curved background of noncommutative mechanics, where the object of noncommutativity θμν is considered as an independent quantity having a canonical conjugate momentum. We introduce a noncommutative first-order action in D = 10 curved spacetime and the covariant equations of motions are computed. This model, invariant under diffeomorphism, generalizes recent relativistic results.

Open string with a background B field as the first order mechanics, noncommutativity, and soldering formalism

To study noncommutativity properties of the open string with constant B field, we construct a mechanical action that reproduces classical dynamics of the string sector under consideration. It allows one to apply the Dirac quantization procedure for constrained systems in a direct and unambiguous way. The mechanical action turns out to be the first order system without taking the strong field limit B{yields}{infinity}. In particular, it is true for the zero mode of the string coordinate, which means that the noncommutativity is an intrinsic property of this mechanical system. We describe the arbitrariness in the relation existing between the mechanical and the string variables and show that noncommutativity of the string variables on the boundary can be removed. This is in correspondence with the result of Seiberg and Witten on the relation among noncommutative and ordinary Yang-Mills theories. The recently developed soldering formalism helps us to establish a connection between the original open string action and the Polyakov action.

New considerations about the Maxwell-Podolsky-like theory through the symplectic embedding formalism

The Podolsky theory can be modified by changing the sign of the Maxwell term. In this case we have a Podolsky-like action with a different spectrum. This new theory was already analyzed by the literature at the propagator level. However, we believe that the results are inconclusive since a precise and entire comprehension, at the Lagrangian level, of this theory is still lacking. In this paper we show in an exact way, using the symplectic embedding formalism, that this Podolsky-like action is in fact dual equivalent to the Proca theory showing the same physical features.