arXiv subjects

Jayme Vaz Jr

Publications and source records attributed to Jayme Vaz Jr.

13 recordsLinked to original sources

arXiv2020

On the mistake in defining fractional derivative using a non-singular kernel

Definitions of fractional derivative of order $α$ ($0 < α\leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in modelling problems with a initial value condition (like, for example, the fractional diffusion equation) because the solutions obtained for these equations do not satisfy the initial condition (except for the integer case $α= 1$). In order to satisfy an arbitrary initial condition the definitions of fractional derivative must necessarily involve a singular kernel.

math.CA↗

arXiv2017

Life in the Rindler Reference Frame: Does an Uniformly Accelerated Charge Radiates? Is there a Bell `Paradox'? Is Unruh Effect Real?

The determination of the electromagnetic field generated by a charge in hyperbolic motion is a classical problem for which the majority view is that the Liénard-Wiechert solution which implies that the charge radiates) is the correct one. However we analyze in this paper a less known solution due to Turakulov that differs from the Liénard-Wiechert one and which according to him does not radiate. We prove his conclusion to be wrong. We analyze the implications of both solutions concerning the validity of the Equivalence Principle. We analyze also two other issues related to hyperbolic motion, the so-called Bell's "paradox" which is as yet source of misunderstandings and the Unruh effect, which according to its standard derivation in the majority of the texts, is a correct prediction of quantum field theory. We recall that the standard derivation of the Unruh effect does not resist any tentative of any rigorous mathematical investigation, in particular the one based in the algebraic approach to field theory which we also recall. These results make us to align with some researchers that also conclude that the Unruh effect does not exist.

physics.gen-ph↗

arXiv2014

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

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 monotonicty 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 response functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters.

cond-mat.stat-mech↗

arXiv2012

Non-Associativity in the Clifford Bundle on the Parallelizable Torsion 7-Sphere

In this paper we discuss generalized properties of non-associativity in Clifford bundles on the 7-sphere S7. Novel and prominent properties inherited from the non-associative structure of the Clifford bundle on S7 are demonstrated. They naturally lead to general transformations of the spinor fields on S7 and have dramatic consequences for the associated Kac-Moody current algebras. All additional properties concerning the non-associative structure in the Clifford bundle on S7 are considered. We further discuss and explore their applications.

math-ph↗

arXiv2011

Tunneling in Fractional Quantum Mechanics

We study the tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schrödinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behaviour. 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 $\mathcal{T}_0 = \cos^2{π/α}$, where $α$ is the order of the derivative ($1 < α\leq 2$).

math-ph↗

arXiv2010

k-deformed Poincare algebras and quantum Clifford-Hopf algebras

The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.

math-ph↗

arXiv2007

Isotopic liftings of Clifford algebras and applications in elementary particle mass matrices

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact, when the generators of the isotopic Lie algebra su(6) are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses.

math-ph↗

arXiv2007

Conformal structures and twistors in the paravector model of spacetime

Some properties of the Clifford algebras Cl(3,0), Cl(1,3), Cl(1,3)(C), Cl(4,1) and Cl(2,4) are presented, and three isomorphisms between the Dirac-Clifford algebra C x Cl(1,3) and Cl(4,1) are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group Spin+(2,4) is also investigated, in the light of a suitable isomorphism between C x Cl(1,3) and Cl(4,1). After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of Spin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian R(4,1) spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac-Clifford algebra C x Cl(1,3) using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose the Clifford algebra over R(4,1) is also used to describe conformal maps, instead of R(2,4). Our formalism sheds some new light on the use of the paravector model and generalizations.

math-ph↗

arXiv2006

On Clifford Subalgebras, Spacetime Splittings and Applications

Z2-gradings of Clifford algebras are reviewed and we shall be concerned with an alpha-grading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced alpha-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.

math-ph↗

arXiv2004

Revisiting Clifford algebras and spinors II: Weyl spinors in Cl(3,0) and Cl(0,3) and the Dirac equation

This paper is the second one of a series of three and it is the continuation of math-ph/0412074. We review some properties of the algebraic spinors in Cl(3,0) and Cl(0,3) and how Weyl, Pauli and Dirac spinors are constructed in Cl(3,0) (and Cl(0,3) in the case of Weyl spinors. A plane wave solution for the Dirac equation is obtained, and the Dirac equation is written in terms of Weyl spinors, and alternatively, in terms of Pauli spinors. Finally the covariant and contravariant undotted spinors in Cl(0,3), isomorphic to two copies of the quaternion algebra (H), are constructed. We prove that there exists an application that maps the even subalgebra Cl+(0,3) of Cl(0,3), viewed as a right H-module, onto Cl+(0,3), but now viewed as a left H-module.

math-ph↗

arXiv2003

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.

quant-ph↗

arXiv2003

Z_2-gradings of Clifford algebras and multivector structures

Let Cl(V,g) be the real Clifford algebra associated to the real vector space V, endowed with a nondegenerate metric g. In this paper, we study the class of Z_2-gradings of Cl(V,g) which are somehow compatible with the multivector structure of the Grassmann algebra over V. A complete characterization for such Z_2-gradings is obtained by classifying all the even subalgebras coming from them. An expression relating such subalgebras to the usual even part of Cl(V,g) is also obtained. Finally, we employ this framework to define spinor spaces, and to parametrize all the possible signature changes on Cl(V,g) by Z_2-gradings of this algebra.

math-ph↗

arXiv2000

Signature Change and Clifford Algebras

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the minkowskian Dirac equation over the euclidean spacetime and to define a new duality operation in terms of which one can find self-dual and anti-self-dual solutions of gauge fields over Minkowski spacetime analogous to the ones over Euclidean spacetime and without needing to complexify the original real algebra.

math-ph↗