Jayme Vaz

Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

Abstract.We revisit the Mittag-Leffler functions of a real variable t, with one, two and three order-parameters {α,β,γ}, as far as their Laplace transform pairs and complete monotonicity properties are concerned. These functions, subjected to the requirement to be completely monotone for t > 0, are shown to be suitable models for non–Debye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole–Cole, Davidson–Cole and Havriliak–Negami. We show 3D plots of the relaxations functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters.

The fractional Schrödinger equation for delta potentials

The fractional Schrodinger equation is solved for the delta potential and the double delta potential for all energies. The solutions are given in terms of Foxs H-function.

About Zitterbewegung and Electron Structure

Abstract We start from the spinning electron model by Barut and Zanghi, which has been recently translated into the Clifford algebra language. We “complete” such a translation, first of all, by expressing in the Clifford formalism a particular Barut-Zanghi (BZ) solution, which refers (at the classical limit) to an “internal” helical motion with a time-like speed (and is here shown to originate from the superposition of positive and negative frequency solutions of the Dirac equation). Then, we show how to construct solutions of the Dirac equation describing helical motions with light-like speed, which meet very well the standard interpretation of the velocity operator in the Dirac equation theory (and agree with the solution proposed by Hestenes, on the basis — however — of ad-hoc assumptions that are unnecessary in the present approach). The above results appear to support the conjecture that the zitterbewegung motion (a helical motion, at the classical limit) is responsible for the electron spin.

Tunneling in fractional quantum mechanics

We study tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schrodinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behavior. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmission coefficient is given by , where α is the order of the derivative (1 < α ≤ 2).

On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator

The fractional generalized Langevin equation (FGLE) is proposed to discuss the anomalous diffusive behavior of a harmonic oscillator driven by a two-parameter Mittag–Leffler noise. The solution of this FGLE is discussed by means of the Laplace transform methodology and the kernels are presented in terms of the three-parameter Mittag–Leffler functions. Recent results associated with a generalized Langevin equation are recovered.

HIDDEN CONSEQUENCE OF ACTIVE LOCAL LORENTZ INVARIANCE

In this paper we investigate a hidden consequence of the hypothesis that Lagrangians and field equations must be invariant under active local Lorentz transformations. We show that this hypothesis implies in an equivalence between spacetime structures with several curvature and torsion possibilities.

Spinor Fields and Superfields as Equivalence Classes of Exterior Algebra Fields

In this paper we show that spinor fields with the right transformation properties and also supernelds can be represented as equivalence classes of exterior algebra fields. We construct explicitly these spinors for Lorentzian manifolds and present superfields through the hyperbolic Clifford algebra bundle (some interesting results concerning to this structure are presented) and its associated real spinor bundle. We compare our spinor fields with Crumeyrolle’s amorphic spinors (ideal sections of a Clifford algebra bundle) and also Dirac-Kahler spinors (sections of a Clifford algebra bundle) which cannot be used in physical theories of spinning matter.

Zitterbewegung and the electromagnetic field of the electron

Abstract By using the spacetime algebra, we explain the helical motion of the electron (zitterbewegung) and its Coulomb field by introducing a mechanism that breaks locally the electromagnetic gauge invariance. We show that this gauge invariance is broken in all points of spacetime, except for those that correspond to that cylindrical helix which is the electrons world line, and that it gives rise to an oscillating Coulomb-like field with frequency equal to the zitterbewegung one. This field is found to satisfy the so-called Maxwell-London equations. This oscillating field, when averaged over a zitterbewegung period, is the usual Coulomb field of the electron.

Clifford algebra-parametrized octonions and generalizations

Abstract Introducing products between multivectors of C l 0 , 7 (the Clifford algebra over the metric vector space R 0 , 7 ) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere S 7 , and the XY-product. This generalization is accomplished in the u- and ( u , v ) -products, where u , v ∈ C l 0 , 7 are fixed, but arbitrary. Moreover, we extend these original products in order to encompass the most general—non-associative—products ( R ⊕ R 0 , 7 ) × C l 0 , 7 → R ⊕ R 0 , 7 , C l 0 , 7 × ( R ⊕ R 0 , 7 ) → R ⊕ R 0 , 7 and C l 0 , 7 × C l 0 , 7 → R ⊕ R 0 , 7 . We also present the formalism necessary to construct Clifford algebra-parametrized octonions, which provides the structure to present the O 1 , u algebra. Finally we introduce a method to construct O -algebras endowed with the ( u , v ) -product from O -algebras endowed with the u-product. These algebras are called O -like algebras and their octonionic units are parametrized by arbitrary Clifford multivectors. When u is restricted to the underlying paravector space R ⊕ R 0 , 7 ↪ C l 0 , 7 of the octonion algebra O , these algebras are shown to be isomorphic. The products between Clifford multivectors and octonions, leading to an octonion, are shown to share graded-associative, supersymmetric properties. We also investigate the generalization of Moufang identities, for each one of the products introduced.

Quantum tomography for Dirac spinors

We present a tomographic scheme, based on spacetime symmetries, for the reconstruction of the internal degrees of freedom of a Dirac spinor. We discuss the circumstances under which the tomographic group can be taken as SU(2), and how this crucially depends on the choice of the gamma matrix representation. A tomographic reconstruction process based on discrete rotations is considered, as well as a continuous alternative.