In [1], some of the statements concerning odd paragrassmann algebras are misleading, due to the presence of negative values of the deformations of integers [n] q . We hereby redefine the odd algebras not only to ensure that all deformations [n] q are positive, but also to provide a unified treatment of both even and odd algebras, as can be done in the representation theory of the q-oscillator, where q is a primitive root of unity [2].Changes in section 3 'The essential paragrassmann algebra'. The leading paragraphs must be changed to the following.'In this paper, we consider as an observation set X the paragrassmann algebra k for k even [3][4][5][6]. We recall that a Grassmann (or exterior) algebra of a given vector space V over a field is the algebra generated by the exterior (or wedge) product for which all elements are nilpotent, θ 2 = 0. Paragrassmann algebras are generalizations for which, given an even number k > 2, all elements obey θ k = 0, where k = k/2 for even k. For a given k, we define k as the linear span of {1, . . . , θ n , . . . , θ k −1 } and of their respective conjugates θ n : here θ is a paragrassmann variable satisfying θ k = 0. Variables θ and θ do not commute:

We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches. * baldiott@fma.if.usp.br †

In the present work we develop a strictly Hamiltonian approach to Thermodynamics. A thermodynamic description based on symplectic geometry is introduced, where all thermodynamic processes can be described within the framework of Analytic Mechanics. Our proposal is constructed on top of a usual symplectic manifold, where phase space is even dimensional and one has well-defined Poisson brackets. The main idea is the introduction of an extended phase space where thermodynamic equations of state are realized as constraints. We are then able to apply the canonical transformation toolkit to thermodynamic problems. Throughout this development, Dirac's theory of constrained systems is extensively used. To illustrate the formalism, we consider paradigmatic examples, namely, the ideal, van der Waals and Clausius gases.

In this work we study the Thermodynamics of D-dimensional Schwarzschild-anti de Sitter (SAdS) black holes. The minimal Thermodynamics of the SAdS spacetime is briefly discussed, highlighting some of its strong points and shortcomings. The minimal SAdS Thermodynamics is extended within a Hamiltonian approach, by means of the introduction of an additional degree of freedom. We demonstrate that the cosmological constant can be introduced in the thermodynamic description of the SAdS black hole with a canonical transformation of the Schwarzschild problem, closely related to the introduction of an anti-de Sitter thermodynamic volume. The treatment presented is consistent, in the sense that it is compatible with the introduction of new thermodynamic potentials, and respects the laws of black hole Thermodynamics. By demanding homogeneity of the thermodynamic variables, we are able to construct a new equation of state that completely characterizes the Thermodynamics of SAdS black holes. The treatment naturally generates phenomenological constants that can be associated with different boundary conditions in underlying microscopic theories. A whole new set of phenomena can be expected from the proposed generalization of SAdS Thermodynamics.Comment: 13 pages, published versio

We find dispersion laws for the photon propagating in the presence of mutually orthogonal constant external electric and magnetic fields in the context of the $\theta $-expanded noncommutative QED. We show that there is no birefringence to the first order in the noncommutativity parameter $% \theta .$ By analyzing the group velocities of the photon eigenmodes we show that there occurs superluminal propagation for any direction. This phenomenon depends on the mutual orientation of the external electromagnetic fields and the noncommutativity vector. We argue that the propagation of signals with superluminal group velocity violates causality in spite of the fact that the noncommutative theory is not Lorentz-invariant and speculate about possible workarounds