Narrow Results

Because the electron transport mechanism in graphene is heavily impacted by the strain and the ferromagnetic metal stripe as well as several other avenues, in this paper we investigate the effects of the strained barrier induced by the strain and the magnetic field generated by the ferromagnetic metal stripe on the valley polarization through numerical calculation. When the strength and the width of the strained barrier as well as the magnitude of the magnetic field are changed, the rapid variation of the valley polarization is observed. This study will be helpful for devising and manufacturing new-style valleytronic devices.

For the massless Dirac equation outside a slow Kerr black hole, we prove asymptotic completeness. We introduce a new Newman–Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are eventually re-interpreted geometrically to provide the solution to a Goursat problem on the Penrose compactified exterior.

The Ghirardi–Rimini–Weber (GRW) theory is a physical theory that, when combined with a suitable ontology, provides an explanation of quantum mechanics. The so-called collapse of the wave function is problematic in conventional quantum theory but not in the GRW theory, in which it is governed by a stochastic law. A possible ontology is the flash ontology, according to which matter consists of random points in space-time, called flashes. The joint distribution of these points, a point process in space-time, is the topic of this work. The mathematical results concern mainly the existence and uniqueness of this distribution for several variants of the theory. Particular attention is paid to the relativistic version of the GRW theory that was developed in 2004.

In this paper, we study the inverse scattering of massive charged Dirac fields in the exterior region of (de Sitter)–Reissner–Nordström black holes. Firstly, we obtain a precise high-energy asymptotic expansion of the diagonal elements of the scattering matrix (i.e. of the transmission coefficients) and we show that the leading terms of this expansion allow to recover uniquely the mass, the charge and the cosmological constant of the black hole. Secondly, in the case of nonzero cosmological constant, we show that the knowledge of the reflection coefficients of the scattering matrix on any interval of energy also permits to recover uniquely these parameters.

The relativistic Lippmann–Schwinger equation was earlier formulated in terms of the limit values of the corresponding resolvent. In the present paper, we write down the limit values of the resolvent in an explicit form, and so the relativistic Lippmann–Schwinger equation is presented as an integral equation. Using this integral equation, we investigate the stationary scattering problems (relativistic case, Dirac equation). We consider the dynamical scattering problems (relativistic case, Dirac equation) as well.

Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function $\psi (x_1,x_2)$ with $x_1,x_2\in {ℝ}^4$ is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half-space, meaning that the Minkowski spacetime is cut off before $t=0$. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at $t=0$ can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in $1+3$ dimensions.

In this paper, we construct a conformal scattering theory for Dirac fields in the interior of a Reissner–Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution.

We consider Dirac equations on even-dimensional Lorentzian manifolds of bounded geometry with a spin structure. For the associated free quantum field theory, we construct pure Hadamard states using global pseudodifferential calculus on a Cauchy surface. We also give two constructions of Hadamard states for Dirac fields for arbitrary spacetimes with a spin structure.

We study Feynman checkers, the most elementary model of electron motion introduced by  Feynman. For the model, we prove that the probability to find an electron vanishes nowhere inside the light cone. We also prove several results on the average electron velocity. In addition, we present a lot of identities related to the model.

In this paper, we study Feynman checkers, one of the most elementary models of electron motion. It is also known as a one-dimensional quantum walk or an Ising model at an imaginary temperature. We add the simplest non-trivial electromagnetic field and find the limits of the resulting model for small lattice step and large time, analogous to the results by Narlikar from 1972 and Grimmet–Jason–Scudo from the 2000s. It turns out that the limits in the model with the added field are obtained from the ones without field by mass renormalization. Also, we find an exact solution of the resulting model.

In this work, we study the Dirac equation with scalar, vector, and tensor interactions. The Dirac Hamiltonian contains quadratic scalar and vector potentials, as well as a tensor potential. The tensor potential is taken as a sum of a linear term and a Coulomb-like term. It is shown that the tensor potential preserves the form of the harmonic oscillator potential and generates spin-orbit terms. The energy eigenvalues and the corresponding eigenfunctions are obtained for different alternatives.

We develop a quantum lattice Boltzmann (QLB) scheme for the Dirac equation with a nonlinear fermion interaction provided by the Nambu–Jona-Lasinio (NJL) model. Numerical simulations in 1 + 1 space-time dimensions, provide evidence of dynamic mass generation, through spontaneous breaking of chiral symmetry.

We describe the numerical methods used to solve the time-dependent Dirac equation on a three-dimensional Cartesian lattice. Efficient algorithms are required for computationally intensive studies of nonperturbative electromagnetic lepton-pair production in relativistic heavy-ion collisions. Discretization is achieved through the lattice basis-spline collocation method, in which quantum-state vectors and coordinate-space operators are expressed in terms of basis-spline functions on a spatial lattice. For relativistic lepton fields on a lattice, the fermion-doubling problem is central in the formulation of the numerical method. All numerical procedures reduce to a series of matrix-vector operations which we perform on the Intel iPSC/860 hypercube, making full use of parallelism. We discuss solutions to the problems of limited node memory and node-to-node communication overhead inherent in using distributed-memory, multiple-instruction, multiple-data stream parallel computers.

We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

We present new procedures in the REDUCE language for algebraic programming of the Dirac equation on curved space-time. The main part of the program is a package of routines defining the Pauli and Dirac matrix algebras. Then the Dirac equation is obtained using the facilities of the EXCALC package. Finally we present some results obtained after running our procedures for the Dirac equation on several curved space-times.

The article presents some new results obtained for the non-relativistic approximation of the Dirac equation in a non-inertial reference frame — rotated and accelerated — and in Schwarzschild gravitational field. These results are obtained with new routines of algebraic programming in REDUCE + EXCALC language for the Dirac equation in a non-inertial reference frame and after three successive Foldy–Wouthuysen transformations.

In this paper we illustrate the existence of genuine bound states for a Dirac particle interacting with a scalar screened Coulomb potential in a 1 + 1 dimension. In contrast to the bare Coulomb potential, the screened Coulomb potential can bind both spin-zero and spin-1/2 particles in the scalar prescription.

We consider solutions to the Dirac equation in the presence of an external pseudoscalar potential V(r) = 1/r. It is found that no normalizable bound state solutions exist.

In the framework of a relativistic constituent quark model we propose some confining potentials behaving like r and r² at large distances which allow for analytical solutions of the Dirac equation. The solutions are given in explicit form.

We seek a quantum-theoretic expression for the probability that a "fermionic" particle which is initially in a well-defined flavor, linear combination of mass-eigenstates, will be found, at later times, in another flavor state. We approach this problem by using the Dirac equation as evolution equation for the mass-eigenstates. The Dirac formalism is useful and essential in keeping clear many of the conceptual aspects of quantum oscillation phenomena that naturally arise in a relativistic spin one-half particle theory. Our study leads to the conclusion that the fermionic nature of the particles and the interference between positive and negative frequency components of mass-eigenstate wave packets modify the standard oscillation probability, obtained by implicitly assuming a "scalar" nature of the mass-eigenstates. Nevertheless, under particular assumptions, i.e. ultra-relativistic particles, strictly peaked momentum distributions and minimal slippage, these modifications introduce correction factors proportional to which are practically un-detectable by any experimental analysis.