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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Separability of the Dirac equation in the STU black hole spacetime: Pairwise-equal charge case study

We study the separability of the Dirac equation in the background of four dimensional charged rotating asymptotically flat black hole solution of supergravity and the low energy string theory known as STU black hole. In particular, we analyze in detail the separability conditions in the pairwise equal charge STU black hole space-time. While in the latter case the minimally coupled Dirac equation is not separable, the introduction of a specific torsion term ensures the separability. The source of the torsion is the Kalb-Ramond field, which is an integral part of string theory, but further aspects of its properties and coupling to fermionic fields remain to be studied. To derive the torsion, two different approaches are used in conformally related frames, showing that the torsion is not unique. The correspondingly modified Dirac equations in the Einstein and string frames are shown to be separable. Furthermore, the massless radial and angular wave equations are examined; they show close similarity with corresponding equations for the standard Kerr background. A generalization of the Teukolsky equation for the pairwise equal case is conjectured. We also briefly analyze a technically sophisticated radial equation in the massive case.

Dirac equation↗
DOE OSTI2014

Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form. - Highlights: • Maxwell–Dirac equations derived in manifestly gauge invariant tensor form. • Invariant scalar and four vector fields for four Poincaré subgroups derived, including two unusual cases. • Symmetry reduction imposed on Maxwell–Dirac equations under example subgroups. • Magnetic monopole arises for spherically symmetric case, consistent with charge quantization condition.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Spin eigen-states of Dirac equation for quasi-two-dimensional electrons

Dirac equation for electrons in a potential created by quantum well is solved and the three sets of the eigen-functions are obtained. In each set the wavefunction is at the same time the eigen-function of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigen-functions of Dirac equation describe three independent spin eigen-states. The energy spectrum of electrons confined by the rectangular quantum well is calculated for each of these spin states at the values of energies relevant for solid state physics. It is shown that the standard Rashba spin splitting takes place in one of such states only. In another one, 2D electron subbands remain spin degenerate, and for the third one the spin splitting is anisotropic for different directions of 2D wave vector.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Bound state solution of Dirac equation for Hulthen plus trigonometric Rosen Morse non-central potential using Romanovski polynomial

The bound state solutions of Dirac equation for Hulthen and trigonometric Rosen Morse non-central potential are obtained using finite Romanovski polynomials. The approximate relativistic energy spectrum and the radial wave functions which are given in terms of Romanovski polynomials are obtained from solution of radial Dirac equation. The angular wave functions and the orbital quantum number are found from angular Dirac equation solution. In non-relativistic limit, the relativistic energy spectrum reduces into non-relativistic energy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Dirac equation in low dimensions: The factorization method

We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials. - Highlights: • The low-dimensional Dirac equation in the presence of static potentials is solved. • The factorization method is generalized for energy-dependent Hamiltonians. • The shape invariance is generalized for energy-dependent Hamiltonians. • The stability of the Dirac sea is related to the existence of supersymmetric partner Hamiltonians.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

The total angular momentum algebra related to the S{sub 3} Dunkl Dirac equation

We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S{sub 3} Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A{sub 2}, with corresponding Weyl group S{sub 3}, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra so(3), incorporating elements of S{sub 3}. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Solitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equation constrained to planar and space curves

Here, we study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature. We obtain for various curves in two and three dimensions the transformation of the Dirac equation to the 1 + 1 dimensional flat space Dirac equation written in terms of the arc length. Depending on the curvature, the soliton profile either narrows or expands. Our results may be applicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Bound state solution of Dirac equation for 3D harmonics oscillator plus trigonometric scarf noncentral potential using SUSY QM approach

Dirac equation of 3D harmonics oscillator plus trigonometric Scarf non-central potential for spin symmetric case is solved using supersymmetric quantum mechanics approach. The Dirac equation for exact spin symmetry reduces to Schrodinger like equation. The relativistic energy and wave function for spin symmetric case are simply obtained using SUSY quantum mechanics method and idea of shape invariance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Relativistic Lagrangians for the Lorentz–Dirac equation

We present two types of relativistic Lagrangians for the Lorentz–Dirac equation written in terms of an arbitrary world-line parameter. One of the Lagrangians contains an exponential damping function of the proper time and explicitly depends on the world-line parameter. Another Lagrangian includes additional cross-terms consisting of auxiliary dynamical variables and does not depend explicitly on the world-line parameter. We demonstrate that both the Lagrangians actually yield the Lorentz–Dirac equation with a source-like term.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

The Dirac equation as a model of topological insulators

The Dirac equation with a spatially dependent mass can be used as a simple, exactly soluble, continuum model of a three-dimensional Topological Insulator. For a bulk system, the sign of the mass determines the parity at the only time-reversal point ( k _ = 0 ) and, thus, leads to the designation of the bulk as being topologically trivial or non-trivial. Since the mass changes sign at the interface between a topologically trivial and non-trivial materials, topological surface states appear on that boundary. We present that electron scattering experiments may provide an alternate probe of the topological character of the surface states. For infinitely thick slabs, the states on the opposite sides of the slab decouple. The spatial decoupling results in the surface states become gapless, non-degenerate and, due to the Rashba spin–orbit coupling generated by the loss of inversion symmetry, exhibit spin-momentum locking. We review several characteristic properties of the topological surface states which are dependent on the topological quantum numbers and show that, using this model, they can be calculated exactly using simple approaches.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2014

The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials

We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)].

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Solution of D dimensional Dirac equation for hyperbolic tangent potential using NU method and its application in material properties

The analytical solution of D-dimensional Dirac equation for hyperbolic tangent potential is investigated using Nikiforov-Uvarov method. In the case of spin symmetry the D dimensional Dirac equation reduces to the D dimensional Schrodinger equation. The D dimensional relativistic energy spectra are obtained from D dimensional relativistic energy eigen value equation by using Mat Lab software. The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi polynomials. The thermodynamically properties of materials are generated from the non-relativistic energy eigen-values in the classical limit. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy. The thermal quantities of the system, partition function and specific heat, are expressed in terms of error function and imaginary error function which are numerically calculated using Mat Lab software.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Applications of Two Body Dirac Equations to Hadron and Positronium Spectroscopy

We review recent applications of the Two Body Dirac equations of constraint dynamics to meson spectroscopy and describe new extensions to three-body problems in their use in the study of baryon spectroscopy. We outline unique aspects of these equations for QED bound states that distinguish them among the various other approaches to the relativistic two body problem. Finally we discuss recent theoretical solutions of new peculiar bound states for positronium arising from the Two Body Dirac equations of constraint dynamics, assuming point particles for the electron and the positron.

Crater, H. W.↗
DOE OSTI2017

Approximate analytic solutions to coupled nonlinear Dirac equations

Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2 /2($\bar{ψ}$ψ) 2 + g 2 2 /2($\bar{Φ}$Φ) 2 + g 2 3 ($\bar{ψ}$ψ)($\bar{Φ}$Φ) as well as vector–vector interactions g 1 2 /2($\bar{ψ}$γμψ)($\bar{ψ}$γμψ) + g 2 2 /2($\bar{Φ}$γμΦ)($\bar{Φ}$γμΦ) + g 2 3 ($\bar{ψ}$γμψ)($\bar{Φ}$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1t R 1 cosθ,R 1 sinθΦ=e –iω2t R 2 cosη,R 2 sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2 /g 2 2 and g 3 2 /g 1 2 . In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter 0 < ε &NestedLessLess; 1 inversely proportional to the speed of light. In the massless and nonrelativistic regime, the solution exhibits rapid motion in space and is highly oscillatory in time. Specifically, the wavelength of the propagating waves in time is at O(ε), while in space, it is at O(1) with the wave speed at O(ε -1 ). We adopt one leap-frog, two semi-implicit, and one conservative Crank-Nicolson finite difference methods to numerically discretize the Dirac equation in one dimension and establish rigorously the error estimates which depend explicitly on the time step τ, mesh size h, and the small parameter ε. The error bounds indicate that, to obtain the “correct” numerical solution in the massless and nonrelativistic regime, i.e., 0 < ε &NestedLessLess; 1, all these finite difference methods share the same ε-scalability as time step τ = O(ε 3/2 ) and mesh size h = O(ε 1/2 ). A large number of numerical results are reported to verify the error estimates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Quantum structure emerging from the transformation design of the Dirac equation

It is shown that a quantum structure can be created by a set of chosen constraint conditions that emerge from the transformation design of the Dirac equation in general relativity. As an explanation, the constraints that cause novel bound states with the quantization rule of a 2D Coulomb system are presented. The discussion in this paper provides a systematic way to look for constraints that generate a required quantization rule. -- Highlights: •We perform the transformation design of space and time for spin-1/2 matter waves. •A quantum rule could naturally emerge as constraints imposed on the Dirac equation itself. •New fermion states share the quantum spectrum of a 2D Coulomb system. •Transformation design uncovers a new exact solvable model. •A quantum spectrum can be created by a geometric structure.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗