Noncommutativity and Duality through the Symplectic Embedding Formalism

This work is devoted to review the gauge embedding of either commutative and noncommutative (NC) theories using the symplectic formalism framework. To sum up the main features of the method, during the process of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and brings some light on the so-called "arbitrariness problem". This alternative embedding formalism also presents a way to obtain a set of dynamically dual equivalent embedded Lagrangian densities which is obtained after a finite number of steps in the iterative symplectic process, oppositely to the result proposed using the BFFT formalism. On the other hand, we will see precisely that the symplectic embedding formalism can be seen as an alternative and an efficient procedure to the standard introduction of the Moyal product in order to produce in a natural way a NC theory. In order to construct a pedagogical explanation of the method to the nonspecialist we exemplify the formalism showing that the massive NC U(1) theory is embedded in a gauge theory using this alternative systematic path based on the symplectic framework. Further, as other applications of the method, we describe exactly how to obtain a Lagrangian description for the NC version of some systems reproducing well known theories. Naming some of them, we use the procedure in the Proca model, the irrotational fluid model and the noncommutative self-dual model in order to obtain dual equivalent actions for these theories. To illustrate the process of noncommutativity introduction we use the chiral oscillator and the nondegenerate mechanics

Tsallis and Kaniadakis statistics from a point of view of the holographic equipartition law

In this work, we have illustrated the difference between both Tsallis and Kaniadakis entropies through cosmological models obtained from the formalism proposed by Padmanabhan, which is called holographic equipartition law. Similarly to the formalism proposed by Komatsu, we have obtained an extra driving constant term in the Friedmann equation if we deform the Tsallis entropy by Kaniadakis' formalism. We have considered initially Tsallis entropy as the Black Hole (BH) area entropy. This constant term may lead the universe to be in an accelerated mode. On the other hand, if we start with the Kaniadakis entropy as the BH area entropy and then by modifying the Kappa expression by Tsallis' formalism, the same constant, which shows that the universe have an acceleration is obtained. In an opposite limit, no driving inflation term of the early universe was derived from both deformations.Comment: 8 pages, preprint format. Final version to appear in Europhysics Letter

The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space

This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity ($\theta^{\mu\nu}$) is a variable of the NC system and has a canonical conjugate momentum. The DFRA formalism is constructed in an extended space-time with independent degrees of freedom associated with the object of noncommutativity $\theta^{\mu\nu}$. A consistent algebra involving the enlarged set of canonical operators is described, which permits one to construct theories that are dynamically invariant under the action of the rotation group. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC quantum theory with interesting results. The Dirac formalism for constrained Hamiltonian systems is considered and the object of noncommutativity $\theta^{ij}$ plays a fundamental role as an independent quantity. It is also explained about the generalized Dirac equation issue, that the fermionic field depends not only on the ordinary coordinates but on $\theta^{\mu\nu}$ as well. The dynamical symmetry content of such fermionic theory is discussed, and we show that its action is invariant under ${\cal P}'$. In the last part of this work we analyze the complex scalar fields using this new framework. As said above, in a first quantized formalism, $\theta^{\mu\nu}$ and its canonical momentum $\pi_{\mu\nu}$ are seen as operators living in some Hilbert space. In a second quantized formalism perspective, we show an explicit form for the extended Poincar\'e generators and the same algebra is generated via generalized Heisenberg relations. We also consider a source term and construct the general solution for the complex scalar fields using the Green function technique