Dirac Quantization in Kantowski—Sachs Spacetime

By forming the square root of the Wheeler-DeWitt equation and applying itto a minisuperspace composed of a Kantowski-Sachs universe, we derive acosmological wave function with conserved current and positive-definiteprobability density.     

A Search on Dirac Equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.     

Bell’s inequality with Dirac particles

We study Bell’s inequality using the Bell states constructed from four component Dirac spinors. Spin operator is related to the Pauli-Lubanski pseudo vector which is relativistic invariant operator. By using Lorentz transformation, in both Bell states and spin operator, we obtain an observer independent Bell’s inequality, so that it is maximally violated as long as it is violated maximally in the rest frame. The article is published in the original.     

Dirac Equation in Lemaįtre—Tolman—Bondi Models

The Dirac equation is studied for a sufficiently large class of Lematre—Tolman—Bondi cosmological models. While the angular equation (whose solution is known) separates directly, the spatial and temporal dependence de-couples only after a suitable separation procedure. The separated time equation is integrated by series. The separated spatial equation still depends on an arbitrary function relative to the integration of the cosmological model.     

Quantum Inequality for Negative Energy Density States of Massive Dirac Field in Four-Dimensional Spacetime

Negative energy density and the quantum inequality are examined for the Dirac field. A proof is given of the quantum inequality for negative energy densities in the massive Dirac field produced by the superposition of two single particle electron states.     

On the Accelerated Observer’s Proper Coordinates and the Rigid Motion Problem in Minkowski Spacetime

Physicists have been interested in accelerated observers for quite some time. Since the advent of special relativity, many authors have tried to understand these observers in the framework of Minkowski spacetime. One of the most important issues related to these observers is the problematic definition of rigid motion. In this paper, I write the metric in terms of the Frenet–Serret curvatures and the proper coordinate system of a general accelerated observer. Then, I use this approach to create a systematic way to construct a rigid motion in Minkowski spacetime. Finally, I exemplify the benefits of this procedure by applying it to two well-known observers, namely, the Rindler and the rotating ones, and also by creating a set of observers that, perhaps, may be interpreted as a rigid cylinder which rotates while accelerating along the axis of rotation.     

Maxwell’s Equations as the One-Photon Quantum Equation

Maxwell's equations (the Faraday and Ampère-Maxwell laws) can be presented as a three-component equation in a way similar to the two-component neutrino equation. However, in this case, the electric and magnetic Gauss laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.     

Renormalization and Asymptotic Expansion of Dirac’s Polarized Vacuum

We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density ρ _ph including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no ‘real’ electron. We show that ρ _ph admits an asymptotic expansion to any order in powers of the physical coupling constant α _ph, provided that the ultraviolet cut-off behaves as L ~ e^{3p(1-Z₃)/2a_ph} >> 1{\Lambda\sim e^{3\pi(1-Z_3)/2\alpha_{\rm ph}} \gg 1}. The renormalization parameter 0 < Z ₃ < 1 is defined by Z ₃ = α _ph/α, where α is the bare coupling constant. The coefficients of the expansion of ρ _ph are independent of Z ₃, as expected. The first order term gives rise to the well-known Uehling potential, whereas the higher order terms satisfy an explicit recursion relation.     

Schrödinger’s Equation with Gauge Coupling Derived from a Continuity Equation

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ? S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable χ. Using these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out.     

Existence of Self-Similar Solutions to Smoluchowski’s Coagulation Equation

The existence of self-similar solutions to Smoluchowskis coagulation equation has been conjectured for several years by physicists, and numerical simulations have confirmed the validity of this conjecture. Still, there was no existence result up to now, except for the constant and additive kernels for which explicit formulae are available. In this paper, the existence of self-similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels.     

Green’s Function of the Schrödinger Equation in the Potential Quantization Method

Journal of Experimental and Theoretical Physics - The problem for determining Green’s function G(r, r') for the time-independent Schrödinger equation is considered using the...     

Effective Mass and Pseudoscalar Interaction in the Dirac Equation with Woods–Saxon Potential

We consider the one-dimensional Dirac equation with the Woods–Saxon potential in the Framework of position dependent mass and pseudoscalar interaction. By imposing appropriate constraints on the mass function and the pseudoscalar term new exact solvable models are obtained. A detailed study of the scattering and bound-states problems for these models is presented. Meanwhile, we work out the exact expressions for the transmission and reflection probabilities of scattered states and obtain the exact equation for the energy eigenvalues associated to bound states. In particular, transmission resonance at zero-momentum is observed for supercritical states.     

Charge Conservation,Klein’s Paradox and the Concept of Paulions in the Dirac Electron Theory

An algebraic block-diagonalization of the Dirac Hamiltonian in a time-independent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects—paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the Dirac equation physics, including new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law.     

Noncommutative Integrability of the Klein–Gordon and Dirac Equations in (2+1)-Dimensional Spacetime

Russian Physics Journal - Noncommutative integration of the Klein–Gordon and Dirac relativistic wave equations in (2+1)-dimensional Minkowski space is considered. It is shown that for all...     

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces

The aim of this paper is to accurately describe the blowup of Jang’s equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence with an identifiable rate near a strictly stable MOTS.