Dirac reduced radial equations and the Problem of Additional Solutions
Khelashvili, Anzor
2016-01-01
For spinless particles there appear additional solutions in the framework of Schrodinger and Klein-Gordon equations. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions. Because the question is to determine the asymptotic behaviour of wave function at the origin, using the radial equations, is natural. The s...
Trzetrzelewski, Maciej
2016-11-01
Starting with a Nambu-Goto action, a Dirac-like equation can be constructed by taking the square-root of the momentum constraint. The eigenvalues of the resulting Hamiltonian are real and correspond to masses of the excited string. In particular there are no tachyons. A special case of radial oscillations of a closed string in Minkowski space-time admits exact solutions in terms of wave functions of the harmonic oscillator.
Bruning, J.; Dobrokhotov, S.Y.; Katsnelson, M.I.; Minenkov, D.S.
2016-01-01
We consider the two-dimensional stationary Schrodinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of st
Amplitude-phase calculations of Regge poles obtained from coupled radial Dirac equations
Thylwe, K-E [KTH-Mechanics, Royal lnstitute of Technology, S-100 44 Stockholm (Sweden); McCabe, P, E-mail: ket@mech.kth.se [CCDC, 12 Union Road, CB2 1EZ, Cambridge (United Kingdom)
2011-07-08
A recently developed amplitude-phase method for spinor-wave solutions is applied to the calculations of Regge pole positions and residues of Dirac particles. At a given energy the Dirac spin causes two sets of Regge poles that tend to coalesce in the non-relativistic limit. For the particular case of equal Lorentz-type vector and scalar potentials there is only one pole string, located very close to the non-relativistic pole string.
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Decoherence in the Dirac equation
Meyer, D A
1998-01-01
A Dirac particle is represented by a unitarily evolving state vector in a Hilbert space which factors as $H_{spin} \\otimes H_{position}$. Motivated by the similarity to simple models of decoherence consisting of a two state system coupled to an environment, we investigate the occurence of decoherence in the Dirac equation upon tracing over position. We conclude that the physics of this mathematically exact model for decoherence is closely related to Zitterbewegung.
Structure of Dirac matrices and invariants for nonlinear Dirac equations
2004-01-01
We present invariants for nonlinear Dirac equations in space-time ${\\mathbb R}^{n+1}$, by which we prove that a special choice of the Cauchy data yields free solutions. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. Related problems on the structure of Dirac matrices are studied.
M. Ko(c)ak; B. G(o)nül
2007-01-01
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 Schr(o)dinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.
Dirac equations in n + 1 dimensions
Jiang Yu [Departamento de FIsica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, 09340 Mexico DF (Mexico)
2005-02-04
The Dirac equation in n + 1 dimensions is derived by a simple algebraic approach. The similarity in the structure of the arbitrary n-dimensional Dirac equations in a central field and their solutions is discussed.
Dirac and Maxwell equations in Split Octonions
Beradze, Revaz
2016-01-01
The split octonionic form of Dirac and Maxwell equations are found. In contrast with the previous attempts these equations are derived from the octonionic analyticity condition and also we use different basis of the 8-dimensional space of split octonions.
Contemplations on Dirac's equation in quaternionic coordinates
Schuricht, Dirk; Greiter, Martin
2004-11-01
A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several peculiarities, which we investigate and attempt to interpret.
On the Dirac equation for a quark
Pestov, I B
2003-01-01
It is argued from geometrical, group-theoretical and physical points of view that in the framework of QCD it is not only necessary but also possible to modify the Dirac equation so that correspondence principle holds valid. The Dirac wave equation for a quark is proposed and some consequences are considered. In particular, it is shown that interquark potential expresses the Coulomb law for the quarks and, in fact, coincides with the known Cornell potential.
Suparmi, A., E-mail: suparmiuns@gmail.com; Cari, C., E-mail: suparmiuns@gmail.com [Physics Department, Post Graduate Study, Sebelas Maret University (Indonesia); Angraini, L. M. [Physics Department, Mataram University (Indonesia)
2014-09-30
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.
Krylov subspace methods for the Dirac equation
Beerwerth, Randolf; Bauke, Heiko
2015-03-01
The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard.
Krylov subspace methods for the Dirac equation
Beerwerth, Randolf
2014-01-01
The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The Dirac Hamiltonian's property of not being bounded does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard.
Dirac equation on a curved surface
Brandt, F. T.; Sánchez-Monroy, J. A.
2016-09-01
The dynamics of Dirac particles confined to a curved surface is examined employing the thin-layer method. We perform a perturbative expansion to first-order and split the Dirac field into normal and tangential components to the surface. In contrast to the known behavior of second order equations like Schrödinger, Maxwell and Klein-Gordon, we find that there is no geometric potential for the Dirac equation on a surface. This implies that the non-relativistic limit does not commute with the thin-layer method. Although this problem can be overcome when second-order terms are retained in the perturbative expansion, this would preclude the decoupling of the normal and tangential degrees of freedom. Therefore, we propose to introduce a first-order term which rescues the non-relativistic limit and also clarifies the effect of the intrinsic and extrinsic curvatures on the dynamics of the Dirac particles.
Gravitationally Coupled Dirac Equation for Antimatter
Jentschura, U D
2013-01-01
The coupling of antimatter to gravity is of general interest because of conceivable cosmological consequences ("surprises") related to dark energy and the cosmological constant. Here, we revisit the derivation of the gravitationally coupled Dirac equation and find that the prefactor of a result given previously in [D.R. Brill and J.A. Wheeler, Rev. Mod. Phys., vol. 29, p. 465 (1957)] for the affine connection matrix is in need of a correction. We also discuss the conversion the curved-space Dirac equation from East-Coast to West-Coast conventions, in order to bring the gravitationally coupled Dirac equation to a form where it can easily be unified with the electromagnetic coupling as it is commonly used in modern particle physics calculations. The Dirac equation describes anti-particles as negative-energy states. We find a symmetry of the gravitationally coupled Dirac equation, which connects particle and antiparticle solutions for a general space-time metric of the Schwarzschild type and implies that particl...
Prastyaningrum, I.; Cari, C.; Suparmi, A.
2016-11-01
The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part.
Quantum simulation of the Dirac equation
Gerritsma, Rene; Kirchmair, Gerhard; Zaehringer, Florian; Blatt, Rainer; Roos, Christian [Institut fuer Quantenoptik und Quanteninformation, 6020 Innsbruck (Austria); Solano, Enrique [Departamento de Quimica Fisica, Universidad del Pais Vasco - Euskal Herriko Unibertsitatea, Bilbao (Spain)
2010-07-01
The Dirac equation is a cornerstone in the history of physics, merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and predicting the existence of anti-matter. However, the Dirac equation also predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an unexpected quivering motion of a free relativistic quantum particle first examined by Schroedinger. In this talk, we report on a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion, which is set to behave as a free relativistic quantum particle. We measure as a function of time the particle position and study Zitterbewegung for different initial superpositions of positive and negative energy spinor states, as well as the cross-over from relativistic to nonrelativistic dynamics.
Qualitative Properties of the Dirac Equation in a Central Potential
Esposito, G; Esposito, Giampiero; Santorelli, Pietro
1999-01-01
The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a perturbation resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the squared Dirac operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The underlying motivation for this qualitative analysis is given by the possib...
Quantum simulation of the Dirac equation.
Gerritsma, R; Kirchmair, G; Zähringer, F; Solano, E; Blatt, R; Roos, C F
2010-01-07
The Dirac equation successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter and is able to reproduce accurately the spectrum of the hydrogen atom. The realm of the Dirac equation-relativistic quantum mechanics-is considered to be the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects, such as Klein's paradox and 'Zitterbewegung', an unexpected quivering motion of a free relativistic quantum particle. These and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. In recent years, there has been increased interest in simulations of relativistic quantum effects using different physical set-ups, in which parameter tunability allows access to different physical regimes. Here we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion set to behave as a free relativistic quantum particle. We measure the particle position as a function of time and study Zitterbewegung for different initial superpositions of positive- and negative-energy spinor states, as well as the crossover from relativistic to non-relativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate textbook examples of relativistic quantum physics.
Building Atomic Nuclei with the Dirac Equation
Serot, Brian D.
2003-01-01
The relevance of the Dirac equation for computations of nuclear structure is motivated and discussed. Quantitatively successful results for medium- and heavy-mass nuclei are described, and modern ideas of effective field theory and density functional theory are used to justify them.
Solution of the Dirac Equation with Special Hulthen Potentials
郭建友; 孟杰; 徐辅新
2003-01-01
The Dirac equation for the special case of a spinor in the relativistic potential with the even and odd components related by a constraint is solved exactly when the even component is chosen to be the Hulthen potential.The corresponding radial wavefunctions for two-component spinor are obtained in terms of the hypergeometric function, and the energy spectrum of the bound states is obtained as a solution to a given equation by boundary constraints, in which the nonrelativistic limit reproduces the usual Hulthen potential.
Röken, Christian
2015-01-01
The separability of the massive Dirac equation in a rotating Kerr black hole background in advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr spacetime is described in the framework of the Newman-Penrose formalism by a local Carter tetrad, and the Dirac wave functions are given on a spin bundle in a chiral Newman-Penrose dyad representation. Applying mode analysis techniques, the Dirac equation is separated into coupled systems of radial and angular ordinary differential equations. Asymptotic radial solutions at infinity and the event and Cauchy horizons are explicitly derived and, by means of error estimates, the decay properties are analyzed. Solutions of the angular ordinary differential equations matching the Chandrasekhar-Page equation are discussed. These solutions are used in order to study the scattering of Dirac waves by the gravitational field of a Kerr black hole for an observer described by a frame without coordinate singularities at the inner horizon boundaries such t...
Qualitative properties of the Dirac equation in a central potential
Esposito, Giampiero; Santorelli, Pietro
1999-07-01
The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wavefunction are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a `perturbation' resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the second-order operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The same methods are applied to the analysis of coupled eigenvalue equations when the anomalous magnetic moment of the electron is not neglected.
Quantum simulation of the Dirac equation
Gerritsma, R; Zähringer, F; Solano, E; Blatt, R; Roos, C F
2009-01-01
The Dirac equation is a cornerstone in the history of physics, merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and predicting the existence of anti-matter. Furthermore, it is able to reproduce accurately the spectrum of the hydrogen atom and its realm, relativistic quantum mechanics, is considered as the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an unexpected quivering motion of a free relativistic quantum particle first examined by Schr\\"odinger. These and other predictions would be difficult to observe in real particles, while constituting key fundamental examples to understand relativistic quantum effects. Recent years have seen an increased interest in simulations of relativistic quantum effects in different physical setups, where parameter tunability allows accessibility to different physical regimes. Here, we perform a proof-of...
Dirac equation and the Melvin metric
Santos, L.C.N.; Barros, C.C. [Universidade Federal de Santa Catarina, Depto de Fisica-CFM, CP. 476, Florianopolis, SC (Brazil)
2016-10-15
A relativistic wave equation for spin 1/2 particles in the Melvin space-time, a space-time where the metric is determined by a magnetic field, is obtained. The energy levels for these particles are obtained as functions of the magnetic field and compared with the ones calculated with the Dirac equation in the flat Minkowski space-time. The numeric values for some magnetic fields of interest are shown. With these results, the effects of very intense magnetic fields on the energy levels, as intense as the ones expected to be produced in magnetars or in ultra-relativistic heavy-ion collisions, are investigated. (orig.)
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
I.Yu. Krivsky
2010-01-01
Full Text Available We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration is proved. In order to provide the corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.
Steen-Ermakov-Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
Prykarpatskyy, Yarema
2017-01-01
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide.
Lorentz-Dirac equation and circularly moving charges
Comay, E.
1987-09-01
The Lorentz-Dirac equation of radiation reaction is tested in a system of circularly moving changes. It is shown that this equation together with the Lienard-Wiechert retarded fields is consistent with energy conservation. Therefore, in this particular experiment, any alternative expression of radiation reaction must agree with the Lorentz-Dirac equation.
New exactly solvable periodic potentials for the Dirac equation
Samsonov, B F; Pozdeeva, E O; Glasser, M L
2003-01-01
A new exactly solvable relativistic periodic potential is obtained by the periodic extension of a well-known transparent scalar potential. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Dirac Kronig-Penney model. The solutions of the Dirac equation are expressed in terms of elementary functions.
Two-body Dirac equation approach to the deuteron
Galeao, A.P.; Castilho A, J.A.; Ferreira, P. Leal
1996-06-01
The two-body Dirac (Breit) equation with potentials associated to one-boson-exchanges with cutoff masses is solved for the deuteron and its observables calculated. The 16-component wave-function for the J{sup {pi}} = 1{sup +} state contains four independent radial functions which satisfy a system of four coupled differential equations of firs order. This system is numerically integrated, from infinity towards the origin, by fixing the value of the deuteron binding energy and imposing appropriate boundary conditions at infinity. For the exchange potential of the pion, a mixture of direct plus derivative couplings to the nucleon is considered. We varied the pion-nucleon coupling constant, and the best results of our calculations agree with the lower values recently determined for this constant. The present treatment differs from the more conventional ones in that non-relativistic reductions up to the order c{sup -2} are not used. (author). 20 refs., 1 fig., 2 tabs.
Time Delay for the Dirac Equation
Naumkin, Ivan; Weder, Ricardo
2016-10-01
We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator {intlimits0 ^{∞}e^{iH0t}ζ(\\vert x\\vert /R) e^{-iH0t}dt}, as {R → ∞}, is presented. Here, H 0 is the free Dirac operator and {ζ(t)} is such that {ζ(t) = 1} for {0 ≤ t ≤ 1} and {ζ(t) = 0} for {t > 1}. This approach allows us to obtain the time delay operator {δ {T}(f)} for initial states f in {{H} 2^{3/2+ɛ}({R}3;{C}4)}, {ɛ > 0}, the Sobolev space of order {3/2+ɛ} and weight 2. The relation between the time delay operator {δ{T}(f)} and the Eisenbud-Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.
Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation
A. A. Deriglazov
2011-01-01
Full Text Available We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. Classical equations for the spin tensor are the same as those of the Barut-Zanghi model of spinning particle.
Topological Insulators Dirac Equation in Condensed Matters
Shen, Shun-Qing
2012-01-01
Topological insulators are insulating in the bulk, but process metallic states around its boundary owing to the topological origin of the band structure. The metallic edge or surface states are immune to weak disorder or impurities, and robust against the deformation of the system geometry. This book, Topological insulators, presents a unified description of topological insulators from one to three dimensions based on the modified Dirac equation. A series of solutions of the bound states near the boundary are derived, and the existing conditions of these solutions are described. Topological invariants and their applications to a variety of systems from one-dimensional polyacetalene, to two-dimensional quantum spin Hall effect and p-wave superconductors, and three-dimensional topological insulators and superconductors or superfluids are introduced, helping readers to better understand this fascinating new field. This book is intended for researchers and graduate students working in the field of topological in...
Topological insulators Dirac equation in condensed matter
Shen, Shun-Qing
2017-01-01
This new edition presents a unified description of these insulators from one to three dimensions based on the modified Dirac equation. It derives a series of solutions of the bound states near the boundary, and describes the current status of these solutions. Readers are introduced to topological invariants and their applications to a variety of systems from one-dimensional polyacetylene, to two-dimensional quantum spin Hall effect and p-wave superconductors, three-dimensional topological insulators and superconductors or superfluids, and topological Weyl semimetals, helping them to better understand this fascinating field. To reflect research advances in topological insulators, several parts of the book have been updated for the second edition, including: Spin-Triplet Superconductors, Superconductivity in Doped Topological Insulators, Detection of Majorana Fermions and so on. In particular, the book features a new chapter on Weyl semimetals, a topic that has attracted considerable attention and has already b...
Solutions of relativistic radial quasipotential equations
Minh, V.X.; Kadyshevskii, V.G.; Zhidkov, E.P.
1985-11-01
A systematic approach to the investigation of relativistic radial quasipotential equations is developed. The quasipotential equations can be interpreted either as linear equations in finite differences of fourth and second orders, respectively, or as differential equations of infinite order.
Geometric Structures and Field Equations of Dirac-Lu Space
REN Xin-An; ZHANG Li-You
2008-01-01
In this paper, a -invariant Lorentz metric on the Dirac-Lu space is given, and then the geodesic equation is investigated. Finally, we discuss the field equations and find their solutions by the method of separating variables.
Transparent boundary conditions for the wave equation in one dimension and for a Dirac-like equation
M. Pilar Velasco
2015-11-01
Full Text Available We present a method to achieve transparent boundary conditions for the one-dimensional wave equation, and show its numerical implementation using a finite-difference method. We also present an alternative method for building the same transparent boundary conditions using a Dirac-like equation and a Spinor-like formalism. Finally, we extend our method to the three-dimensional wave equation with radial symmetry.
Discrete Dirac equation on a finite half-integer lattice
Smalley, L. L.
1986-01-01
The formulation of the Dirac equation on a discrete lattice with half-integer spacing and periodic boundary conditions is investigated analytically. The importance of lattice formulations for problems in field theory and quantum mechanics is explained; the concept of half-integer Fourier representation is introduced; the discrete Dirac equation for the two-dimensional case is derived; dispersion relations for the four-dimensional case are developed; and the spinor formulation for the Dirac fields on the half-integer lattice and the discrete time variable for the four-dimensional time-dependent Dirac equation are obtained. It is argued that the half-integer lattice, because it takes the Dirac Lagrangian into account, is more than a mere relabeling of the integer lattice and may have fundamental physical meaning (e.g., for the statistics of fermions). It is noted that the present formulation does not lead to species doubling, except in the continuum limit.
Letter: On the Solutions of the Lorentz-Dirac Equation
Vogt, D.; Letelier, P. S.
2003-12-01
We discuss the unstable character of the solutions of the Lorentz-Dirac equation and stress the need of methods like order reduction to derive a physically acceptable equation of motion. The discussion is illustrated with the paradigmatic example of the non-relativistic harmonic oscillator with radiation reaction. We also illustrate the removal of the noncausal pre-acceleration with the introduction of a small correction in the Lorentz-Dirac equation.
Intertwining technique for the one-dimensional stationary Dirac equation
Nieto, L M; Samsonov, B F; Samsonov, Boris F.
2003-01-01
The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples.
Canonical conjugated Dirac equation in a curved space
Dzhunushaliev, Vladimir
2012-01-01
It is shown that the calculation of Dirac operator for the spherical coordinate system with spherical Dirac matrices and using the spin connection formalism is in the contradiction with the definition of standard Dirac operator in the spherical Minkowski coordinate system. It is shown that such contradiction one can avoid by introducing a canonical conjugated covariant derivative for the spinor field. The Dirac equation solution on the Reissner - Nordstr\\"om background is obtained. The solution describes a bound state of a charged particle.
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Lorz, Alexander
2011-01-17
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
Dirac mass dynamics in a multidimensional nonlocal parabolic equation
Lorz, Alexander; Perthame, Benoit
2010-01-01
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a structure of gradient flow. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models darwinian evolution.
Large-j Expansion Method for Two-Body Dirac Equation
Askold Duviryak
2006-02-01
Full Text Available By using symmetry properties, the two-body Dirac equation in coordinate representation is reduced to the coupled pair of radial second-order differential equations. Then the large-j expansion technique is used to solve a bound state problem. Linear-plus-Coulomb potentials of different spin structure are examined in order to describe the asymptotic degeneracy and fine splitting of light meson spectra.
ON GROUND STATE SOLUTIONS FOR SUPERLINEAR DIRAC EQUATION
张建; 唐先华; 张文
2014-01-01
This article is concerned with the nonlinear Dirac equations Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth.
The Lorentz-Dirac equation and the structure of spacetime
De Souza, M M
1995-01-01
A new interpretation of the causality implementation in the Lienard-Wiechert solution raises new doubts against the validity of the Lorentz-Dirac equation and the limits of validity of the Minkowski structure of spacetime.
The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials
Contreras-Astorga, Alonso, E-mail: aloncont@iun.edu; Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, E-mail: xbataxel@gmail.com [Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States)
2014-10-15
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)].
E.Maghsoodi; H.Hassanabadi; S.Zarrinkamar
2013-01-01
Exact analytical solutions of the Dirac equation are reported for the P(o)schl-Teller double-ring-shaped Coulomb potential.The radial,polar,and azimuthal parts of the Dirac equation are solved using the Nikiforov-Uvarov method,and the exact bound-state energy eigenvalues and corresponding two-component spinor wavefunctions are reported.
Dirac equation in low dimensions: The factorization method
Sánchez-Monroy, J. A.; Quimbay, C. J.
2014-11-01
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.
Dirac-like equations for barotropic FRW cosmologies
Rosu, H C; Reyes, M; Jimenez, D
2002-01-01
Simple Schroedinger-like equations have been written down for FRW cosmologies with barotropic fluids by Faraoni. His results have been extended by Rosu, who employed techniques belonging to nonrelativistic supersymmetry. Further extensions are presented herein using the known connection between Schroedinger-like equations and Dirac-like equations in the same supersymmetric context
The Dirac equation as one fourth-order equation for one function -- a general form
Akhmeteli, Andrey
2015-01-01
Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma-matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma-matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also much more symmetric than that of the previous work and should be useful in applications of the Dirac equation.
Generalized de Broglie Relations for Dirac Equations in Curved Spacetimes
Arminjon, Mayeul
2011-01-01
One may ask whether the special relativistic relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We do this by applying Whitham's Lagrangian method to derive covariant equations describing wave packet motion which preserve the symmetries of the Dirac Lagrangian, and in particular, conserve the probability current. We show that generalized de Broglie relations emerge from the Whitham equations after transforming each Dirac equation into a canonical form via a local similarity transformation of the type first introduced by Pauli. This gives the de Broglie relations a universal character for spin-half particles in a curved spacetime. We show that COW and Sagnac type terms also appear in the Whitham equations. We further discuss the classical-quantum correspondence in a curved spa...
Conjugated Molecules Described by a One-Dimensional Dirac Equation.
Ernzerhof, Matthias; Goyer, Francois
2010-06-08
Starting from the Hückel Hamiltonian of conjugated hydrocarbon chains (ethylene, allyl radical, butadiene, pentadienyl radical, hexatriene, etc.), we perform a simple unitary transformation and obtain a Dirac matrix Hamiltonian. Thus already small molecules are described exactly in terms of a discrete Dirac equation, the continuum limit of which yields a one-dimensional Dirac Hamiltonian. Augmenting this Hamiltonian with specially adapted boundary conditions, we find that all the orbitals of the unsaturated hydrocarbon chains are reproduced by the continuous Dirac equation. However, only orbital energies close to the highest occupied molecular orbital/lowest unoccupied molecular orbital energy are accurately predicted by the Dirac equation. Since it is known that a continuous Dirac equation describes the electronic structure of graphene around the Fermi energy, our findings answer the question to what extent this peculiar electronic structure is already developed in small molecules containing a delocalized π-electron system. We illustrate how the electronic structure of small polyenes carries over to a certain class of rectangular graphene sheets and eventually to graphene itself. Thus the peculiar electronic structure of graphene extends to a large degree to the smallest unsaturated molecule (ethylene).
Dirac equation in low dimensions: The factorization method
Sánchez-Monroy, J.A., E-mail: antosan@if.usp.br [Instituto de Física, Universidade de São Paulo, 05508-090, São Paulo, SP (Brazil); Quimbay, C.J., E-mail: cjquimbayh@unal.edu.co [Departamento de Física, Universidad Nacional de Colombia, Bogotá, D. C. (Colombia); CIF, Bogotá (Colombia)
2014-11-15
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.
Digital quantum simulation of Dirac equation with a trapped ion
Shen, Yangchao; Zhang, Xiang; Zhang, Junhua; Casanova, Jorge; Lamata, Lucas; Solano, Enrique; Yung, Man-Hong; Zhang, Jingning; Kim, Kihwan; Department Of Physical Chemistry Collaboration
2014-05-01
Recently there has been growing interest in simulating relativistic effects in controllable physical system. We digitally simulate the Dirac equation in 3 +1 dimensions with a single trapped ion. We map four internal levels of 171Yb+ ion to the Dirac bispinor. The time evolution of the Dirac equation is implemented by trotter expansion. In the 3 +1 dimension, we can observe a helicoidal motion of a free Dirac particle which reduces to Zitterbewegung in 1 +1 dimension. This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540. KK acknowledge the support from the recruitment program of global youth experts.
Geometric Correlation between Dirac Equation and Yang-mills Equation/ Maxwell Equation
Yu, Xuegang
2011-01-01
The problem about geometric correspondence of Dirac particle and contain quality item of Yang-Mills equation has always not been solved.This paper introduced the hyperbolic imaginary unit in Minkowski space, established a classes of Dirac wave equations with t'Hooft matrices.In lightlike region of Minkowski space,we can discuss the hermitian conjugate transformation of Dirac positive particle and antiparticle, find the space-time corresponding points of Dirac particle,and draw Feynman clip-art though the geometrical relation between timelike region and lightlike region.The coupling of motion equation of Dirac positive particle and antiparticle can get Klein-Gordon equation, when it reach classical approximate we can get Schrodinger equation,and this illustrated that p meson or m meson may be composite particle. Using the relation of timelike region and lightlike region in Minkowski momentum space to renormalize the rest mass of particles,we can describe the geometric relation between rest mass and electromagn...
Moving potential for Dirac and Klein–Gordon equations
Hamil B; Chetouani L
2016-04-01
Using the Lorentz transformation, the Klein–Gordon and Dirac equations with moving potentials are reduced to one standard where the potential is time-independent. As application, the reflection and transmission coefficients are determined by considering the moving step with a constant velocity $v$. It has been found that $R \\pm T = 1$ only at $x = vt$. The problem of massless (2+1) Dirac particle is also considerered.
Klein-Gordon and Dirac Equations with Thermodynamic Quantities
Arda, Altuğ; Tezcan, Cevdet; Sever, Ramazan
2016-03-01
We study the thermodynamic quantities such as the Helmholtz free energy, the mean energy and the specific heat for both the Klein-Gordon, and Dirac equations. Our analyze includes two main subsections: (1) statistical functions for the Klein-Gordon equation with a linear potential having Lorentz vector, and Lorentz scalar parts (2) thermodynamic functions for the Dirac equation with a Lorentz scalar, inverse-linear potential by assuming that the scalar potential field is strong ( A ≫ 1). We restrict ourselves to the case where only the positive part of the spectrum gives a contribution to the sum in partition function. We give the analytical results for high temperatures.
Generalization of the Lorentz-Dirac equation to include spin
Barut, A. O.; Unal, Nuri
1989-11-01
For the classical point electron with Zitterbewegung (hence spin) we derive, after regularization, the radiation reaction force and covariant equations for the dynamical variables (xμ, πμ, vμ, and Sμν), which reduce to the Lorentz-Dirac equation in the spinless limit.
New applications of pseudoanalytic function theory to the Dirac equation
Castaneda, Antonio; Kravchenko, Vladislav V [Seccion de Posgrado e Investigacion, Escuela Superior de IngenierIa Mecanica y Electrica, Instituto Politecnico Nacional, CP07738 Mexico DF (Mexico)
2005-10-21
In the present work, we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potential in a two-dimensional case and a pair of decoupled Vekua equations. In general, these Vekua equations are bicomplex. However, we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain nonrestrictive condition. As an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions. One of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result. Consider the Dirac equation with a scalar potential depending on one variable with fixed energy and mass. In general, this equation cannot be solved explicitly even if one looks for wavefunctions of one variable. Nevertheless, for such Dirac equation, we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions (depending on two variables). These solutions generalize the system of powers 1, z, z{sup 2}, ... in complex analysis and are called formal powers. With their aid any regular solution of the Dirac equation can be represented by its Taylor series in formal powers.
Dirac equation on coordinate dependent noncommutative space–time
Kupriyanov, V. G.
2014-01-01
We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properti...
Axel Schulze-Halberg
2006-01-01
New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form of a Riccati equation. The present note generalizes a series of former articles.
Cari, C., E-mail: cari@staff.uns.ac.id; Suparmi, A., E-mail: soeparmi@staff.uns.ac.id; Yunianto, M., E-mail: muhtaryunianto@staff.uns.ac.id; Husein, A. S. [Physics Department, Faculty of Mathematics and Science, SebelasMaret University, Jl. Ir. Sutami 36A Kentingan Surakarta 57126 (Indonesia)
2016-02-08
The analytical solution of Ddimensional Dirac equation for Coulombic potential is investigated using Nikiforov-Uvarov method. The D dimensional relativistic energy spectra are obtained from relativistic energy eigenvalue equation by using Mat Lab software.The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi and Laguerre Polynomials. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy which will be applied to determine some thermodynamical properties of the system. The thermodynamical properties of the system are expressed in terms of error function and imaginary error function.
On radiation reaction and the Abraham-Lorentz-Dirac equation
de Oca, Alejandro Cabo Montes
2013-01-01
It is underlined that the Lienard-Wiechert solutions indicate that after the external force is instantly removed from a small charged particle, the field in its close neighborhood becomes a Lorentz boosted Coulomb field. It suggests that the force of the self-field on the particle should instantaneously vanish after a sudden removal of the external force. A minimal modification of Abraham-Lorentz-Dirac equation is searched seeking to implement this property. A term assuring this behavior is added to the equation by maintaining Lorentz covariance and vanishing scalar product with the four-velocity. The simple Dirac constant force example does not show runaway acceleration.
Relativistic Lagrangians for the Lorentz–Dirac equation
Deguchi, Shinichi, E-mail: deguchi@phys.cst.nihon-u.ac.jp [Institute of Quantum Science, College of Science and Technology, Nihon University, Chiyoda-ku, Tokyo 101-8308 (Japan); Nakano, Kunihiko [Institute of Quantum Science, College of Science and Technology, Nihon University, Chiyoda-ku, Tokyo 101-8308 (Japan); Suzuki, Takafumi [Junior College Funabashi Campus, Nihon University, Narashinodai, Funabashi, Chiba 274-8501 (Japan)
2015-09-15
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.
Stability problem for singular Dirac equation system on finite interval
Ercan, Ahu; Panakhov, Etibar
2017-01-01
In this study, we show the stability problem for the singular Dirac equation system respect to two spectra on finite interval. The meaning of the stability problem of differential operators is to estimate difference of the spectral functions which considered problems when a finite number of eigenvalues of these problems coincide. The method is based on work by Ryabushko in [12]. The author in [12] studied to what extent only finitely many eigenvalues in one or both spectra determine the potential. We obtain a bound on variation of difference of the spectral functions for singular Dirac equation system.
General spin and pseudospin symmetries of the Dirac equation
Alberto, P; Frederico, T; de Castro, A
2015-01-01
In the 70's Smith and Tassie, and Bell and Ruegg independently found SU(2) symmetries of the Dirac equation with scalar and vector potentials. These symmetries, known as pseudospin and spin symmetries, have been extensively researched and applied to several physical systems. Twenty years after, in 1997, the pseudospin symmetry has been revealed by Ginocchio as a relativistic symmetry of the atomic nuclei when it is described by relativistic mean field hadronic models. The main feature of these symmetries is the suppression of the spin-orbit coupling either in the upper or lower components of the Dirac spinor, thereby turning the respective second-order equations into Schr\\"odinger-like equations, i.e, without a matrix structure. In this paper we propose a generalization of these SU(2) symmetries for potentials in the Dirac equation with several Lorentz structures, which also allow for the suppression of the matrix structure of second-order equation equation of either the upper or lower components of the Dirac...
Analytic Representation of Relativistic Wave Equations I The Dirac Case
Tepper, L; Zachary, W W
2003-01-01
In this paper we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wave functions, as is done by the Foldy-Wouthuysen method, and reveals the nonlocal time behavior of the particle-antiparticle relationship. It is well known that the Foldy-Wouthuysen transformation leads to a diagonalization that is nonlocal in space. We interpret the zitterbewegung, and the result that a velocity measurement (of a Dirac particle) at any instant in time is +(-)c, as reflections of the fact that the Dirac equation makes a spatially extended particle appear as a point in the present by forcing it to oscillate between the past and future at speed c. This suggests that although the Dirac Hamiltonian and the square-root Hamiltonian, are mathematically, they are not physically, equivalent. Furthermore, we see that alt! ho! ugh the form of the Dirac equation serve...
Non-Relativistic Limit of the Dirac Equation
Ajaib, Muhammad Adeel
2016-01-01
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.
Dirac equation on coordinate dependent noncommutative space-time
Kupriyanov, V G
2014-01-01
We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properties of the star product we derive the corresponding probability current density and prove its conservation. The energy-momentum tensor for the free noncommutative spinor field is calculated. We solve the free noncommutative Dirac equation and show that the standard energy-momentum dispersion relation remains valid in the noncommutative case.
Higher dimensional supersymmetric quantum mechanics and Dirac equation
L P Singh; B Ram
2002-04-01
We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering `mass' as a function of coordinates. Its usefulness in solving potential problems is discussed with speciﬁc examples. We also discuss the `physical' signiﬁcance of the supersymmetric states in this formalism.
An extended Dirac equation in noncommutative space-time
Mendes, R Vilela
2015-01-01
Stabilizing, by deformation, the algebra of relativistic quantum mechanics a non-commutative space-time geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed, as well as the effects of coupling the two solutions.
Dirac equation on coordinate dependent noncommutative space–time
Kupriyanov, V.G., E-mail: vladislav.kupriyanov@gmail.com
2014-05-01
In this paper we discuss classical aspects of spinor field theory on the coordinate dependent noncommutative space–time. The noncommutative Dirac equation describing spinning particle in an external vector field and the corresponding action principle are proposed. The specific choice of a star product allows us to derive a conserved noncommutative probability current and to obtain the energy–momentum tensor for free noncommutative spinor field. Finally, we consider a free noncommutative Dirac fermion and show that if the Poisson structure is Lorentz-covariant, the standard energy–momentum dispersion relation remains valid.
Dirac equation on coordinate dependent noncommutative space-time
Kupriyanov, V. G.
2014-05-01
In this paper we discuss classical aspects of spinor field theory on the coordinate dependent noncommutative space-time. The noncommutative Dirac equation describing spinning particle in an external vector field and the corresponding action principle are proposed. The specific choice of a star product allows us to derive a conserved noncommutative probability current and to obtain the energy-momentum tensor for free noncommutative spinor field. Finally, we consider a free noncommutative Dirac fermion and show that if the Poisson structure is Lorentz-covariant, the standard energy-momentum dispersion relation remains valid.
Solution of Dirac equation in Reissner-Nordström de Sitter space
Lyu, Yan; Cui, Song
2009-02-01
The radial parts of the Dirac equation between the outer black hole horizon and the cosmological horizon are solved in Reissner-Nordström de Sitter (RNdS) space numerically. An accurate approximation, the polynomial approximation, is used to approximate the modified tortoise coordinate \\hat r_* , which leads to the inverse function r = r(\\hat r_* ) and the potential V(\\hat r_* ). The potential V(\\hat r_* ) is replaced by a collection of step functions in sequence. Then the solution of the wave equation as well as the reflection and transmission coefficients is computed by a quantum mechanical method.
Dirac Equation in Four Time and Four Space Dimensions
Nieto, J A
2016-01-01
The Dirac equation in four time and four space dimensions (or (4+4)-dimensions) is considered. Step by step we show that such an equation admits Majorana and Weyl solutions. In order to obtain the Majorana or Weyl spinors we used a method based on the construction of Clifford algebra in terms of 2x2-matrices. We argue that our approach can be useful in supergravity, superstrings and qubit theory.
Adjunctation and Scalar Product in the Dirac Equation - II
Dima, M.
2017-02-01
Part-I Dima (Int. J. Theor. Phys. 55, 949, 2016) of this paper showed in a representation independent way that γ 0 is the Bergmann-Pauli adjunctator of the Dirac { γ μ } set. The distiction was made between similarity (MATH) transformations and PHYS transformations - related to the (covariant) transformations of physical quantities. Covariance is due solely to the gauging of scalar products between systems of reference and not to the particular action of γ 0 on Lorentz boosts - a matter that in the past led inadvertently to the definition of a second scalar product (the Dirac-bar product). Part-II shows how two scalar products lead to contradictions and eliminates this un-natural duality in favour of the canonical scalar product and its gauge between systems of reference. What constitutes a proper observable is analysed and for instance spin is revealed not to embody one (except as projection on the boost direction - helicity). A thorough investigation into finding a proper-observable current for the theory shows that the Dirac equation does not possess one in operator form. A number of problems with the Dirac current operator are revealed - its Klein-Gordon counterpart being significantly more physical. The alternative suggested is finding a current for the Dirac theory in scalar form j^{μ } = < ρ rangle _{_{ψ }}v^{μ }_{ψ }.
On integrable rational potentials of the Dirac equation
Stachowiak, Tomasz, E-mail: stachowiak@cft.edu.pl [Center for Theoretical Physics PAS, Al. Lotnikow 32/46, 02-668 Warszawa (Poland); Przybylska, Maria, E-mail: M.Przybylska@proton.if.uz.zgora.pl [Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra (Poland)
2013-05-03
The one-dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.
Non-Grassmann mechanical model of the Dirac equation
Deriglazov, A. A.; Zamudio, G. P.; Castro, P. S. [Department de Matematica, ICE, Universidade Federal de Juiz de Fora, MG (Brazil); Rizzuti, B. F. [ISB, Universidade Federal do Amazonas, Coari-AM (Brazil)
2012-12-15
We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both {Gamma}{sup {mu}} and {Gamma}{sup {mu}{nu}}-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing Zitterbewegung of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wavelength. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.
On integrable rational potentials of the Dirac equation
Stachowiak, Tomasz
2012-01-01
The Dirac equation, when reducible to an ordinary second order linear equation, exhibits a form of quasi-integrability, i.e. exact solutions exist only for a particular subset of energies. The differential Galois theory can be used to identify the integrable cases, recover integrable rational potentials, explicit solutions and strictly rule out the remaining cases as non-integrable. The effectiveness of this approach is demonstrated by providing a new class of potentials for which the equation in question can be transformed to the Whittaker form.
Essence of Special Relativity, Reduced Dirac Equation and Antigravity
Ni, Guang-jiong; Lou, Senyue; Xu, Jianjun
2010-01-01
The essence of special relativity is hiding in the equal existence of particle and antiparticle, which can be expressed by two discrete symmetries within one inertial frame --- the invariance under the (newly defined) space-time inversion (${\\bf x}\\to -{\\bf x},t\\to -t$), or equivalently, the invariance under a mass inversion ($m\\to -m$). The problems discussed are: the evolution of the $CPT$ invariance into a basic postulate, an unique solution to the original puzzle in Einstein-Podolsky-Rosen paradox, the reduced Dirac equation for hydrogenlike atoms, and the negative mass paradox leading to the prediction of antigravity between matter and antimatter. {\\bf Keywords}: Special relativity, Reduced Dirac Equation, Antiparticle, Antigravity
Dirac Equation in Noncommutative Space for Hydrogen Atom
Adorno, T C; Chaichian, M; Gitman, D M; Tureanu, A
2009-01-01
We consider the energy levels of a hydrogen-like atom in the framework of $\\theta $-modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels $2S_{1/2}, 2P_{1/2}$ and $ 2P_{3/2}$ is lifted completely, such that new transition channels are allowed.
Dirac equation in noncommutative space for hydrogen atom
Adorno, T.C., E-mail: tadorno@nonada.if.usp.b [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, CEP 05508-090 Sao Paulo, SP (Brazil); Baldiotti, M.C., E-mail: baldiott@fma.if.usp.b [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, CEP 05508-090 Sao Paulo, SP (Brazil); Chaichian, M., E-mail: Masud.Chaichian@helsinki.f [Department of Physics, University of Helsinki and Helsinki Institute of Physics, PO Box 64, FIN-00014 Helsinki (Finland); Gitman, D.M., E-mail: gitman@dfn.if.usp.b [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, CEP 05508-090 Sao Paulo, SP (Brazil); Tureanu, A., E-mail: Anca.Tureanu@helsinki.f [Department of Physics, University of Helsinki and Helsinki Institute of Physics, PO Box 64, FIN-00014 Helsinki (Finland)
2009-11-30
We consider the energy levels of a hydrogen-like atom in the framework of theta-modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels 2S{sub 1/2}, 2P{sub 1/2} and 2P{sub 3/2} is lifted completely, such that new transition channels are allowed.
Solving Dirac equation using the tridiagonal matrix representation approach
Alhaidari, A.D. [Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438 (Saudi Arabia); Bahlouli, H., E-mail: bahlouli@kfupm.edu.sa [Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438 (Saudi Arabia); Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261 (Saudi Arabia); Assi, I.A. [Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261 (Saudi Arabia)
2016-04-22
The aim of this work is to find exact solutions of the one dimensional Dirac equation using the tridiagonal matrix representation. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. We are honored to dedicate this work to Prof. Hashim A. Yamani on the occasion of his 70th birthday. - Highlights: • We choose L2 basis such that the Dirac wave operator is tridiagonal matrix. • We use the tridiagonal-matrix-representation approach. • The wave equation becomes a symmetric three-term recursion relation. • We solve the associated three-term recursion relation exactly. • The energy spectrum formula is obtained.
Noncommutativity into Dirac Equation with mass dependent on the position
Bastos, Samuel Batista; Almeida, Carlos Alberto Santos [Universidade Federal do Ceara - UFC, Fortaleza, CE (Brazil); Nunes, Luciana Angelica da Silva [Universidade Federal Rural do Semi-arido - UFERSA, Mossoro, RN (Brazil)
2013-07-01
Full text: In recent years, there is growing interest in the study of theories in non-commutative spaces. Non-commutative fields theories are related with compactifications of M theory, string theory and the quantum Hall effect. Moreover, the role of the non-commutativity of theories of a particle finds large applications when analyzed in scenarios of quantum mechanics and relativistic quantum mechanics. In these contexts investigations on the Schrodinger and Dirac equations with mass depending on the position (MDP) has attracted much attention in the literature. Systems endowed with MDP models are useful for the study of many physical problems. In particular, they are used to study the energy density in problems of many bodies, determining the electronic properties of semiconductor heterostructures and also to describe the properties of heterojunctions and quantum dots. In particular, the investigation of relativistic effects it is important for systems containing heavy atoms or doping by heavy ions. For these types of materials, the study of the properties of the Dirac equation, in the case where the mass becomes variable is of great interest. In this paper, we seek for the non-relativistic limit of the Dirac Hamiltonian in the context of a theory of effective mass, through a Foldy-Wouthuysen transformation. We analyse the Dirac equation with mass dependent on the position, in a smooth step shape mass distribution, in non-commutative space (NC). This potential type kink was recently discussed by several authors in the commutative context and now we present our results in the non-commutative context. (author)
Abstract Wave Equations and Associated Dirac-Type Operators
Gesztesy, Fritz; Holden, Helge; Teschl, Gerald
2010-01-01
We discuss the unitary equivalence of generators $G_{A,R}$ associated with abstract damped wave equations of the type $\\ddot{u} + R \\dot{u} + A^*A u = 0$ in some Hilbert space $\\mathcal{H}_1$ and certain non-self-adjoint Dirac-type operators $Q_{A,R}$ (away from the nullspace of the latter) in $\\mathcal{H}_1 \\oplus \\mathcal{H}_2$. The operator $Q_{A,R}$ represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of $A^*A$. In addition to the unitary equivalence results concerning $G_{A,R}$ and $Q_{A,R}$, we provide a detailed study of the domain of the generator $G_{A,R}$, consider spectral properties of the underlying quadratic operator pencil $M(z) = |A|^2 - iz R - z^2 I_{\\mathcal{H}_1}$, $z\\in\\mathbb{C}$, derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The...
Quantum mechanics for three versions of the Dirac equation in a curved spacetime
Arminjon, Mayeul
2008-01-01
We present a recent work on the Dirac equation in a curved spacetime. In addition to the standard equation, two alternative versions are considered, derived from wave mechanics, and based on the tensor representation of the Dirac field. The latter considers the Dirac wave function as a spacetime vector and the set of the Dirac matrices as a third-order tensor. Having the probability current conserved for any solution of the Dirac equation gives an equation to be satisfied by the coefficient fields. A positive definite scalar product is defined and a hermiticity condition for the Dirac Hamiltonian is derived for a general coordinate system in a general curved spacetime. For the standard equation, the hermiticity of the Dirac Hamiltonian is not preserved under all admissible changes of the coefficient fields.
BETA NUR PRATIWI; A SUPARMI; C CARI; ANDRI SOFYAN HUSEIN
2017-02-01
Analytical solution of the Dirac equation for the modified Pöschl–Teller potential and trigonometric Scarf II non-central potential for spin symmetry is studied using asymptotic iteration method. One-dimensional Dirac equation consisting of the radial and angular parts can be obtained by the separation of variables. By usingasymptotic iteration method, the relativistic energy equation and orbital quantum number (l) equation can be obtained, where both are interrelated. Relativistic energy equation is calculated numerically by the Matlab software. The increase in the radial quantum number $n_r$ causes a decrease in the energy value, and the wave functions of the radial and the angular parts are expressed in terms of hypergeometric functions. Some thermodynamical properties of the system can be determined by reducing the relativistic energy equation to the non-relativisticenergy equation. Thermodynamical properties such as vibrational partition function, vibrational specific heat function and vibrational mean energy function are expressed in terms of error function.
Pratiwi, Beta Nur; Suparmi, A.; Cari, C.; Husein, Andri Sofyan
2017-02-01
Analytical solution of the Dirac equation for the modified Pöschl-Teller potential and trigonometric Scarf II non-central potential for spin symmetry is studied using asymptotic iteration method. One-dimensional Dirac equation consisting of the radial and angular parts can be obtained by the separation of variables. By using asymptotic iteration method, the relativistic energy equation and orbital quantum number ( l) equation can be obtained, where both are interrelated. Relativistic energy equation is calculated numerically by the Matlab software. The increase in the radial quantum number n r causes a decrease in the energy value, and the wave functions of the radial and the angular parts are expressed in terms of hypergeometric functions. Some thermodynamical properties of the system can be determined by reducing the relativistic energy equation to the non-relativistic energy equation. Thermodynamical properties such as vibrational partition function, vibrational specific heat function and vibrational mean energy function are expressed in terms of error function.
The square root of the Dirac operator on the superspace and the Maxwell equations
Bzdak, A; Bzdak, Adam; Hadasz, Leszek
2003-01-01
We re-consider the procedure of ``taking a square root of the Dirac equation'' on the superspace and show that it leads to the well known superfield W_\\alpha and to the proper equations of motion for the components, i.e. the Maxwell equations and the massless Dirac equation.
The square root of the Dirac operator on superspace and the Maxwell equations
Bzdak, Adam; Hadasz, Leszek
2004-02-26
We re-consider the procedure of 'taking a square root of the Dirac equation' on superspace and show that it leads to the well-known superfield W{sub {alpha}} and to the proper equations of motion for the components, i.e., the Maxwell equations and the massless Dirac equation.
Suparmi, A., E-mail: soeparmi@staff.uns.ac.id; Cari, C., E-mail: cari@staff.uns.ac.id; Pratiwi, B. N., E-mail: namakubetanurpratiwi@gmail.com [Physics Department, Faculty of Mathematics and Science, Sebelas Maret University, Jl. Ir. Sutami 36A Kentingan Surakarta 57126 (Indonesia); Deta, U. A. [Physics Department, Faculty of Science and Mathematics Education and Teacher Training, Surabaya State University, Surabaya (Indonesia)
2016-02-08
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.
Solution of One-dimensional Dirac Equation via Poincare Map
Bahlouli, Hocine; Jellal, Ahmed
2011-01-01
We solve the general one-dimensional Dirac equation using a "Poincare Map" approach which avoids any approximation to the spacial derivatives and reduces the problem to a simple recursive relation which is very practical from the numerical implementation point of view. To test the efficiency and rapid convergence of this approach we apply it to a vector coupling Woods--Saxon potential, which is exactly solvable. Comparison with available analytical results is impressive and hence validates the accuracy and efficiency of this method.
The supersymmetric Dirac equation the application to hydrogenic atoms
Hirshfeld, Allen
2012-01-01
The solution of the Dirac equation for an electron in a Coulomb field is systematically treated here by utilizing new insights provided by supersymmetry. It is shown that each of the concepts has its analogue in the non-relativistic case. Indeed, the non-relativistic case is developed first, in order to introduce the new concepts in a familiar context. The symmetry of the non-relativistic model is already present in the classical limit, so the classical Kepler problem is first discussed in order to bring out the role played by the Laplace vector, one of the central concepts of the whole book.
General Electromagnetic Nonminimal Couplings in the Dirac Equation
Araujo, J B; Ferreira, M M
2016-01-01
We examine a new class of CPT-even and dimension-five nonminimal interactions between fermions and photons, deprived of higher-order derivatives, yielding electric dipole moment and magnetic dipole moment in the context of the Dirac equation. These couplings are Lorentz-violating nonminimal structures, composed of a rank-2 tensor, the electromagnetic tensor, and gamma matrices, being addressed in its axial and non-axial hermitian versions. We use the electron's anomalous magnetic and electric dipole moment measurements to reach upper bounds of $1$ part in $10^{11}$ and $10^{16}$ GeV$^{-1}$.
Nonlocal separable potential in the one-dimensional Dirac equation
Calkin, M.G.; Kiang, D.; Nogami, Y.
1988-08-01
The one-dimensional Dirac equation is solved for a separable potential of the form of Lorentz scalar plus vector, (..beta..g+h)v(x)v(x'). Exact analytic solutions are obtained for bound and scattering states for arbitrary v(x). For a particular combination of the values of g and h, degeneracy of the bound state occurs, and total reflection also takes place for a certain incident energy. The limiting case, in which v(x) becomes a delta function, is discussed in detail.
Abraham-Lorentz-Dirac Equation in 5D Stuekelberg Electrodynamics
Land, Martin
2016-01-01
We derive the Abraham-Lorentz-Dirac (ALD) equation in the framework of the electrodynamic theory associated with Stueckelberg manifestly covariant canonical mechanics. In this framework, a particle worldline is traced out through the evolution of an event $x^\\mu(\\tau)$. By admitting unconstrained commutation relations between the positions and velocities, the associated electromagnetic gauge fields are in general dependent on the parameter $\\tau$, which plays the role of time in Newtonian mechanics. Standard Maxwell theory emerges from this system as a $\\tau$-independent equilibrium limit. In this paper, we calculate the $\\tau$-dependent field induced by an arbitrarily evolving event, and study the long-range radiation part, which is seen to be an on-shell plane wave of the Maxwell type. Following Dirac's method, we obtain an expression for the finite part of the self-interaction, which leads to the ALD equation that generalizes the Lorentz force. This third-order differential equation is then converted to an...
Nonrelativistic limit of solution of radial quasipotential equations
Minh, Vu.X.; Kadyshevskii, V.G.; Zhidkov, E.P.
1986-10-01
For the S-wave case, solutions of relativistic radial quasipotential equations that degenerate in the limit c ..-->.. infinity into the Jost solutions of the corresponding nonrelativistic radial Schrodinger equations are found.
Suparmi
2014-12-01
Full Text Available The bound state solution of the Dirac equation for generalized PöschlTeller and trigonometric Pöschl-Teller non-central potentials was obtained using SUSY quantum mechanics and the idea of shape invariance potential. The approximate relativistic energy spectrum was expressed in the closed form. The radial and polar wave functions were obtained using raising and lowering of radial and polar operators. The orbital quantum numbers were found from the polar Dirac equation, which was solved using SUSY quantum mechanics and the idea of shape invariance.
A novel quantum-mechanical interpretation of the Dirac equation
K-H Kiessling, M.; Tahvildar-Zadeh, A. S.
2016-04-01
A novel interpretation is given of Dirac’s ‘wave equation for the relativistic electron’ as a quantum-mechanical one-particle equation. In this interpretation the electron and the positron are merely the two different ‘topological spin’ states of a single more fundamental particle, not distinct particles in their own right. The new interpretation is backed up by the existence of such ‘bi-particle’ structures in general relativity, in particular the ring singularity present in any spacelike section of the spacetime singularity of the maximal-analytically extended, topologically non-trivial, electromagnetic Kerr-Newman (KN)spacetime in the zero-gravity limit (here, ‘zero-gravity’ means the limit G\\to 0, where G is Newton’s constant of universal gravitation). This novel interpretation resolves the dilemma that Dirac’s wave equation seems to be capable of describing both the electron and the positron in ‘external’ fields in many relevant situations, while the bi-spinorial wave function has only a single position variable in its argument, not two—as it should if it were a quantum-mechanical two-particle wave equation. A Dirac equation is formulated for such a ring-like bi-particle which interacts with a static point charge located elsewhere in the topologically non-trivial physical space associated with the moving ring particle, the motion being governed by a de Broglie-Bohm type law extracted from the Dirac equation. As an application, the pertinent general-relativistic zero-gravity hydrogen problem is studied in the usual Born-Oppenheimer approximation. Its spectral results suggest that the zero-G KN magnetic moment be identified with the so-called ‘anomalous magnetic moment of the physical electron,’ not with the Bohr magneton, so that the ring radius is only a tiny fraction of the electron’s reduced Compton wavelength.
Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation
Bogoslovsky, G Yu
2004-01-01
The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space-time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented.
Considerations concering the generalization of the Dirac equations to unstable fermions
Kniehl, Bernd A.; Sirlin, Alberto [Max-Planck-Institut fuer Physik (Werner-Heisenberg-Institut), Muenchen (Germany)
2014-08-15
We discuss the generalization of the Dirac equations and spinors in momentum space to free unstable spin-1/2 fermions taking into account the fundamental requirement of Lorentz covariance. We derive the generalized adjoint Dirac equations and spinors, and explain the very simple relation that exists, in our formulation, between the unstable and stable cases. As an application of the generalized spinors, we evaluate the probability density. We also discuss the behavior of the generalized Dirac equations under time reversal.
Relativistic Dirac equation for particles with arbitrary half-integral spin
Guseinov, I I
2008-01-01
The sets of 2(2s+1)-component matrices through the four-component Dirac matrices are suggested, where s=3/2, 5/2,.... Using these matrices sets the Dirac relativistic equation for a description of arbitrary half-integral spin particles is constructed. The new Dirac equation of motion leads to an equation of continuity with a positive-definite probability density.
Yang-Mills gauge fields conserving symmetry algebra of Dirac equation in homogeneous space
Breev, A I
2014-01-01
We consider the Dirac equation with external Yang-Mills gauge field in a homogeneous space with invariant metric. The Yang-Mills fields for which the motion group of the space serves as symmetry group of the Dirac equation are found by comparison of the Dirac equation with a invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The basis of eigenfunctions and corresponding spectrum are obtained for the Dirac equation in the space $\\mathbb{R}^2 \\times \\mathbb{S}^2$ in the framework of the noncommutative integration method.
Partially Ordered Sets of Quantum Measurements and the Dirac Equation
Knuth, Kevin H.
2012-02-01
Events can be ordered according to whether one event influences another. This results in a partially ordered set (poset) of events often referred to as a causal set. In this framework, an observer can be represented by a chain of events. Quantification of events and pairs of events, referred to as intervals, can be performed by projecting them onto an observer chain, or even a pair of observer chains, which in specific situations leads to a Minkowski metric replete with Lorentz transformations (Bahreyni & Knuth, 2011. APS B21.00007). In this work, we unify this picture with the Process Calculus, which coincides with the Feynman rules of quantum mechanics (Goyal, Knuth, Skilling, 2010, arXiv:0907.0909; Goyal & Knuth, Symmetry 2011, 3(2), 171), by considering quantum measurements to be events. This is performed by quantifying pairs of events, which represent transitions, with a pair of numbers, or a quantum amplitude. In the 1+1D case this results in the Feynman checkerboard model of the Dirac equation (Feynman & Hibbs, 1965). We further demonstrate that in the case of 3+1 dimensions, we recover Bialnycki-Birula's (1994, Phys. Rev. D, 49(12), 6920) body-centered cubic cellular automata model of the Dirac equation studied more recently by Earle (2011, arXiv:1102.1200v1).
Dirac and Higher-Spin Equations of Negative Energies
Dvoeglazov, Valeriy V
2011-01-01
It is easy to check that both algebraic equation Det (\\hat p - m) =0 and Det (\\hat p + m) =0 for 4-spinors u- and v- have solutions with p_0= \\pm E_p =\\pm \\sqrt{{\\bf p}^2 +m^2}. The same is true for higher-spin equations. Meanwhile, every book considers the p_0=E_p only for both u- and v- spinors of the (1/2,0)\\oplus (0,1/2)) representation, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of negative-energy solutions. Recent works of Ziino (and, independently, of several others) show that the Fock space can be doubled. We re-consider this possibility on the quantum field level for both s=1/2 and higher spins particles.
A B-spline Galerkin method for the Dirac equation
Froese Fischer, Charlotte; Zatsarinny, Oleg
2009-06-01
The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y=-λy, that can also be written as a pair of first-order equations y=λz, z=-λy. Expanding both y(r) and z(r) in the B basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the B basis and z(r) in the dB/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method ( B,B) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.
The Solution of Dirac Equation in Quasi-Extreme REISSNER-NORDSTRÖM de Sitter Space
Lyu, Yan; Cui, Song; Liu, Ling
The radial parts of Dirac equation between the outer black hole horizon and the cosmological horizon in quasi-extreme Reissner-Nordström de Sitter (RNdS) geometry is solved numerically. We use an accurate polynomial approximation to mimic the modified tortoise coordinate hat r*(r), for obtaining the inverse function r=r(hat r*) and V=V(hat r*). We then use a quantum mechanical method to solve the wave equation and give the reflection and transmission coefficients. We concentrate on two limiting cases. The first case is when the two horizons are close to each other, and the second case is when the horizons are far apart.
Analytical solutions for Dirac and Klein-Gordon equations using Backlund transformations
Zabadal, Jorge R.; Borges, Volnei, E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Dept. de Engenharia Mecanica; Ribeiro, Vinicius G., E-mail: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio, E-mail: marciophd@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Centro de Estudos Interdisciplinares
2015-07-01
This work presents a new analytical method for solving Klein-Gordon type equations via Backlund transformations. The method consists in mapping the Klein-Gordon model into a first order system of partial differential equations, which contains a generalized velocity field instead of the Dirac matrices. This system is a tensor model for quantum field theory whose space solution is wider than the Dirac model in the original form. Thus, after finding analytical expressions for the wave functions, the Maxwell field can be readily obtained from the Dirac equations, furnishing a self-consistent field solution for the Maxwell-Dirac system. Analytical and numerical results are reported. (author)
Duarte, Celso de Araujo
2015-01-01
Traditionally, the electromagnetic theory dictates the well-known second order differential equation for the components of the scalar and the vector potentials, or in other words, for the four-vector electromagnetic potential $\\phi^{\\mu}$. But the second order is not obligatory at least with respect to the electromagnetic radiation fields: actually, a heuristic first order differential equation can be constructed to describe the electromagnetic radiation, supported on the phenomenology of its electric and magnetic fields. Due to a formal similarity, such an equation suggests a direct comparative analysis with Dirac's equation for half spin fermions, conducting to the finding that the Dirac's spinor field $\\Psi$ for massive or massless fermions is equivalent to a set of two potential-like four vector fields $\\psi^{\\mu}$ and $\\chi^{\\mu}$. Under this point of view, striking similarities with the electromagnetic theory emerge with a category of "pseudo electric'' and "pseudo magnetic'' vector fermionic fields.
Searching for an equation: Dirac, Majorana and the others
Esposito, S
2011-01-01
We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein-Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz-Pauli, Kemmer, Rarita-Schwinger and many others. A particular interest comes out for the general problem of the description of particles with \\textit{arbitrary} spin, introduced (and solved) by Majorana as early as 1932, and later reconsidered, within a different approach, by Dirac in 1936 and by Fierz-Pauli in 1939. The final settlement of the problem in 1945 by Bhabha, who came back to the general ideas introduced by Majorana in 1932, is discussed as well, and, by making recourse also to unpublished documents by Majorana, we are able to reconstruct the line of reasoning behind the Majorana and the Bhabha equations, ...
Smoothing Effects for the Classical Solutions to the Landau-Fermi-Dirac Equation
Shuangqian LIU
2012-01-01
The smoothness of the solutions to the full Landau equation for Fermi-Dirac particles is investigated.It is shown that the classical solutions near equilibrium to the Landau-Fermi-Dirac equation have a regularizing effects in all variables (time,space and velocity),that is,they become immediately smooth with respect to all variables.
Pratiwi, B. N.; Suparmi, A.; Cari, C.; Husein, A. S.; Yunianto, M.
2016-08-01
We apllied asymptotic iteration method (AIM) to obtain the analytical solution of the Dirac equation in case exact pseudospin symmetry in the presence of modified Pcischl- Teller potential and trigonometric Scarf II non-central potential. The Dirac equation was solved by variables separation into one dimensional Dirac equation, the radial part and angular part equation. The radial and angular part equation can be reduced into hypergeometric type equation by variable substitution and wavefunction substitution and then transform it into AIM type equation to obtain relativistic energy eigenvalue and wavefunctions. Relativistic energy was calculated numerically by Matlab software. And then relativistic energy spectrum and wavefunctions were visualized by Matlab software. The results show that the increase in the radial quantum number nr causes decrease in the relativistic energy spectrum. The negative value of energy is taken due to the pseudospin symmetry limit. Several quantum wavefunctions were presented in terms of the hypergeometric functions.
Dirac equation in very special relativity for hydrogen atom
Maluf, R V; Cruz, W T; Almeida, C A S
2014-01-01
In this work, we study the modified Dirac equation in the framework of very special relativity (VSR). The low-energy regime is accessed and the nonrelativistic Hamiltonian is obtained. It turns out that this Hamiltonian is similar to that achieved from the Standard Model Extension (SME) via coupling of the spinor field to a Lorentz-violating term, but new features arise inherited from the non-local character of the VSR. In addition, the implications of the VSR-modified Lorentz symmetry on the spectrum of a hydrogen atom is determined by calculating the first-order energy corrections in the context of standard quantum mechanics. Among the results, we highlight that the modified Hamiltonian provides non-vanishing corrections which lift the degeneracy of the energy levels and allow us to find an upper bound upon the VSR-parameter.
Lorentz invariant CPT breaking in the Dirac equation
Fujikawa, Kazuo
2016-01-01
If one modifies the Dirac equation in momentum space to $[\\gamma^{\\mu}p_{\\mu}-m-\\Delta m(\\theta(p_{0})-\\theta(-p_{0})) \\theta(p_{\\mu}^{2})]\\psi(p)=0$, the symmetry of positive and negative energy eigenvalues is lifted by $m\\pm \\Delta m$ for a small $\\Delta m$. The mass degeneracy of the particle and antiparticle is thus lifted in a Lorentz invariant manner since the combinations $\\theta(\\pm p_{0})\\theta(p_{\\mu}^{2})$ with step functions are manifestly Lorentz invariant. We explain an explicit construction of this CPT breaking term in coordinate space, which is Lorentz invariant but non-local at a distance scale of the Planck length. The application of this Lorentz invariant CPT breaking mechanism to the possible mass splitting of the neutrino and antineutrino in the Standard Model is briefly discussed.
Dirac equation in very special relativity for hydrogen atom
R.V. Maluf
2014-11-01
Full Text Available In this work, we study the modified Dirac equation in the framework of very special relativity (VSR. The low-energy regime is accessed and the nonrelativistic Hamiltonian is obtained. It turns out that this Hamiltonian is similar to that achieved from the Standard Model Extension (SME via coupling of the spinor field to a Lorentz-violating term, but new features arise inherited from the non-local character of the VSR. In addition, the implications of the VSR-modified Lorentz symmetry on the spectrum of a hydrogen atom are determined by calculating the first-order energy corrections in the context of standard quantum mechanics. Among the results, we highlight that the modified Hamiltonian provides non-vanishing corrections which lift the degeneracy of the energy levels and allow us to find an upper bound upon the VSR-parameter.
J J THOMSON'S ELECTRON: The Dirac equation. Cosmic implications of a tidy electron
Miller, David
1997-07-01
Dirac devised the quantum theory of the electron itself, which required him to generalize SchrÃ¶dinger's famous equation to cover relativistic motion. He interpreted the resulting equation as showing that an antiparticle to the electron must exist.
Pathology-Free Modification of the Lorentz-Dirac Equation
Blinder, S. M.
2001-04-01
The Lorentz-Dirac equation for the force on an accelerating electron is conventionally written in covariant form F_ext^λ=ma^λ-2 e^2\\over 3 c^3(dot a^λ+1\\over c^2 a^2 v^λ) However, this equation has fallen into disfavor in recent years because it admits pathological solutions representing runaway behavior or preacceleration violating classical causality. For example, force-free motion can exhibit unphysical runaway solutions of the form a(t)= a(0)exp(t/ τ_0), where τ_0≡ 2e^2/3mc^3≈ 6.26× 10-24 sec. Note that the first two terms of the L-D equation could originate from expansion of ma^λ(τ-τ_0) in powers of τ_0. We propose the following differential-difference equation as a compact non-pathologial alternative to the L-D equation: F^λ_ext(τ)=m [a^λ β^μ -a^μ β^λ]_τ-τ0 β_μ(τ) where β^λ=v^λ/c. Expansion of the bracketed quantity reacquires the conventional equation, apart from higher-order terms in τ_0. It can be demonstrated that F=0 unambiguously implies a=0. Moreover the occurrence of the retarded time variable τ-τ0 precludes any solutions with preacceleration. A more detailed derivation is given in a forthcoming paper [S. M. Blinder, ``Classical electrodynamics with vacuum polarization: electron self-energy and radiation reaction," Repts. Math. Phys., in press].
Calculation of the Dirac equation in curved spacetimes with possible torsion using MAPLE and REDUCE
Vulcanov, Dumitru N.
2003-08-01
The article presents computer algebra procedures and routines applied to the study of the Dirac field on curved spacetimes. The main part of the procedures is devoted to the construction of Pauli and Dirac matrices algebra on an anholonomic orthonormal reference frame. Then these procedures are used to compute the Dirac equation on curved spacetimes in a sequence of special dedicated routines. A comparative review of such procedures obtained for two computer algebra platforms (REDUCE+EXCALC and MAPLE+GRTensorII) is carried out. Applications for the calculus of Dirac equation on specific examples of spacetimes with or without torsion are pointed out.
Relativistic (Dirac equation) effects in microscopic elastic scattering calculations
Hynes, M. V.; Picklesimer, A.; Tandy, P. C.; Thaler, R. M.
1985-04-01
A simple relativistic extension of the first-order multiple scattering mechanism for the optical potential is employed within the context of a Dirac equation description of elastic nucleon-nucleus scattering. A formulation of this problem in terms of a momentum-space integral equation displaying an identifiable nonrelativistic sector is described and applied. Extensive calculations are presented for proton scattering from 40Ca and 16O at energies between 100 and 500 MeV. Effects arising from the relativistic description of the propagation of the projectile are isolated and are shown to be responsible for most of the departures from typical nonrelativistic (Schrödinger) results. Off-shell and nonlocal effects are included and these, together with uncertainties in the nuclear densities, are shown not to compromise the characteristic improvement of forward angle spin observable predictions provided by the relativistic approach. The sensitivity to ambiguities in the Lorentz scalar and vector composition of the optical potential is displayed and discussed.
Relativistic (Dirac equation) effects in microscopic elastic scattering calculations
Hynes, M.V.; Picklesimer, A.; Tandy, P.C.; Thaler, R.M.
1985-04-01
A simple relativistic extension of the first-order multiple scattering mechanism for the optical potential is employed within the context of a Dirac equation description of elastic nucleon-nucleus scattering. A formulation of this problem in terms of a momentum-space integral equation displaying an identifiable nonrelativistic sector is described and applied. Extensive calculations are presented for proton scattering from /sup 40/Ca and /sup 16/O at energies between 100 and 500 MeV. Effects arising from the relativistic description of the propagation of the projectile are isolated and are shown to be responsible for most of the departures from typical nonrelativistic (Schroedinger) results. Off-shell and nonlocal effects are included and these, together with uncertainties in the nuclear densities, are shown not to compromise the characteristic improvement of forward angle spin observable predictions provided by the relativistic approach. The sensitivity to ambiguities in the Lorentz scalar and vector composition of the optical potential is displayed and discussed.
Schr\\"odinger-Pauli Equation for the Standard Model Extension CPT-Violating Dirac Equation
Gutierrez, Thomas D
2015-01-01
It is instructive to investigate the non-relativistic limit of the simplest Standard Model Extension (SME) CPT-violating Dirac-like equation but with minimal coupling to the electromagnetic fields. In this limit, it becomes an intuitive Schr\\"odinger-Pauli-like equation. This is comparable to the free particle treatment as explored by Kostelecky and Lane, but this exercise only considers the $a$ and $b$ CPT-violating terms and $\\vec{p}/m$ terms to first order. Several toy systems are discussed.
Dirac oscillators and quasi-exactly solvable operators
Brihaye, Y
2005-01-01
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a quite general spherically symmetric form for these potentials and we analyse some exactly and quasi exactly solvable properties of the underlying matricial linear operators.
Inferences about interactions: Fermions and the Dirac equation
Knuth, Kevin H.
2013-08-01
At a fundamental level every measurement process relies on an interaction where one entity influences another. The boundary of an interaction is given by a pair of events, which can be ordered by virtue of the interaction. This results in a partially ordered set (poset) of events often referred to as a causal set. In this framework, an observer can be represented by a chain of events. Quantification of events and pairs of events, referred to as intervals, can be performed by projecting them onto an observer chain, or even a pair of observer chains, which in specific situations leads to a Minkowski metric replete with Lorentz transformations. We illustrate how this framework of interaction events gives rise to some of the well-known properties of the Fermions, such as Zitterbewegung. We then take this further by making inferences about events, which is performed by employing the process calculus, which coincides with the Feynman path integral formulation of quantum mechanics. We show that in the 1+1 dimensional case this results in the Feynman checkerboard model of the Dirac equation describing a Fermion at rest.
The Dirac equation applied to graphene in the presence of topological defects
Cunha, Marcio de Moura; Ribeiro, Carlos Alberto de Lima [Universidade Estadual de Feira de Santana, BA (Brazil)
2011-07-01
Full text: The Dirac equation was proposed by Paul Dirac in 1928, in an attempt to get a relativistic wave equation for particles of spin 1/2, because the Schroedinger equation does not remain invariant under Lorentz transformations and the Klein-Gordon only serves for spin 0 particles . Since then, it has been able to describe various systems, in several areas of physics, such as Field Theory, Condensed Matter, among others. Recently, some researchers have use this equation to study the graphene, a very promising material, that consist essentially in a monolayer of carbon atoms, with interesting electronic and transport properties and several possibilities of applications in Material Science and Engineering, for instance. In this work, we study the application of the Dirac equation in graphene, more specifically in the presence of topological defects, that change the physical properties of the material. This is possible because in the formalism of the Dirac equation, we can replace the derivative usual term by a term of covariant derivative, capable of describing the geometry of the space considered. From the job of Vozmediano {sup a} and others found in the literature, we write the dirac equation for graphene in presence of a defect, making a modification in the usual Dirac equation. (author)
A Matter of Principle: The Principles of Quantum Theory, Dirac's Equation, and Quantum Information
Plotnitsky, Arkady
2015-01-01
This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be be addressed as well in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac's work, which, in particular Dirac's derivation of his relativistic equation for the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall, however, also consider Heisenberg's derivation of quantum mechanics, which inspired Dirac. I argue that Heisenberg's and Dirac's work alike was guided by their adherence to and confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by G. M. D' Ariano and his coworkers on the principles of quantum information theory, which extends quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac's equation from these principles alone...
张学骜; 陈柯; 段正路
2005-01-01
Solving the Klein-Gordon equation and Dirac equation with ring-shaped non-spherical oscillator gives the exact bound state wavefunction and energy equation, and the relations between non-relativistic Schrodinger equation, KleinGordon equation and Dirac equation with equal scalar and vector potentials.
On Charge Conjugation, Chirality and Helicity of the Dirac and Majorana Equation for Massive Leptons
Eckart Marsch
2015-04-01
Full Text Available We revisit the charge-conjugation operation for the Dirac equation in its chiral representation. A new decomposition of the Dirac spinor field is suggested and achieved by means of projection operators based on charge conjugation, which is discussed here in a non-standard way. Thus, two separate two-component Majorana-type field equations for the eigenfields of the charge-conjugation operator are obtained. The corresponding free fields are entirely separated without a gauge field, but remain mixed and coupled together through an electromagnetic field term. For fermions that are charged and, thus, subjected to the gauge field of electrodynamics, these two Majorana fields can be reassembled into a doublet, which is equivalent to a standard four-component Dirac spinor field. In this way, the Dirac equation is retained in a new guise, which is fully equivalent to that equation in its chiral form.
Exact solution to the one-dimensional Dirac equation of linear potential
Long Chao-Yun; Qin Shui-Jie
2007-01-01
In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.
Solution of the Dirac equation in a curved space with static metric
Alhaidari, A D
2015-01-01
Compatibility of symmetric quantization of the Dirac equation in a curved space with general covariance gives a special representation of the spin connections in which their dot product with the Dirac gamma matrices becomes equal to the "covariant divergence" of the latter. Requiring that the square of the equation gives the conventional Klein-Gordon equation in a curved space results in an operator algebra for the Dirac gamma matrices that involves the "covariant derivative" connections and the Riemann-Christoffel connections. In 1+1 space-time with static metric, we obtain exact solutions of this Dirac equation model for some examples. We also formulate the interacting theory of the model with various coupling modes and solve it in the same space for a given potential configuration.
A detailed study of nonperturbative solutions of two-body Dirac equations
Crater, H.W.; Becker, R.L.; Wong, C.Y.; Van Alstine, P.
1992-12-01
In quark model calculations of the meson spectrums fully covariant two-body Dirac equations dictated by Dirac's relativistic constraint mechanics gave a good fit to the entire meson mass spectrum for light quark mesons as well as heavy quark mesons with constituent world scalar and vector potentials depending on just one or two parameters. In this paper, we investigate the properties of these equations that made them work so well by solving them numerically for quantum electrodynamics (QED) and related field theories. The constraint formalism generates a relativistic quantum mechanics defined by two coupled Dirac equations on a sixteen component wave function which contain Lorentz covariant constituent potentials that are initially undetermined. An exact Pauli reduction leads to a second order relativistic Schroedinger-like equation for a reduced eight component wave function determined by an effective interaction -- the quasipotential. We first determine perturbatively to lowest order the relativistic quasipotential for the Schroedinger-like equation by comparing that form with one derived from the Bethe-Salpeter equation. Insertion of this perturbative information into the minimal interaction structures of the two-body Dirac equations then completely determines their interaction structures. Then we give a procedure for constructing the full sixteen component solution to our coupled first-order Dirac equations from a solution of the second order equation for the reduced wave function. Next, we show that a perturbative treatment of these equations yields the standard spectral results for QED and related interactions.
Supersymmetry and the Dirac Equation with Vector and Scalar Coupling Potentials
无
2000-01-01
This paper shows that one type of first-order Dirac equation with vector coupling and scalar coupling potentials can be brought into the framework of non-relativistic supersymmetric quantum mechanics. The conclusion is independent of the concrete forms of the vector and scalar coupling potentials because of the nilpotent matrix realization of supersymmetric quantum mechanical algebra. The supersymmetry of this kind of Dirac equation requires that a spin-orbit coupling term be introduced into the associated supersymmetric Hamiltonian.
Dirac equation in gauge and affine-metric gravitation theories
Giachetta, G
1995-01-01
We show that the covariant derivative of Dirac fermion fields in the presence of a general linear connection on a world manifold is universal for Einstein's, gauge and affine-metric gravitation theories.
P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation
Duran, Antonio J [Departamento de Analisis Matematico, Universidad de Sevilla, Apdo (PO BOX) 1160, 41080 Sevilla (Spain); Gruenbaum, F Alberto [Department of Mathematics, University of California, Berkeley, CA 94720 (United States)
2006-04-07
The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.
Rauscher, Elizabeth A
2011-01-01
The Maxwell, Einstein, Schrödinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal
Basic quantum mechanics for three Dirac equations in a curved spacetime
Arminjon, Mayeul
2008-01-01
We consider three versions of the Dirac equation in a curved spacetime: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed tensor representation of the Dirac field (TRD). These three equations differ in the covariant derivative D_mu. A common tool is the hermitizing matrix A. Having the current conservation for any solution of the Dirac equation is equivalent to D_mu (A gamma^mu)=0, where gamma^mu is the field of Dirac matrices. This condition is always verified for DFW with its restricted choice for the field gamma^mu. It similarly restricts the choice of the field gamma^mu for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. For the Dirac Hamiltonian, a positive definite scalar product is defined for all reference frames, and a hermiticity condition is derived in a general curved spacetime with minor restrictions on the coordinate system. For DFW, the hermiticity of t...
Exact Solution of the Curved Dirac Equation in Polar Coordinates: Master Function Approach
H. Panahi
2015-01-01
Full Text Available We show that the (2+1 curved Dirac equation in polar coordinates can be transformed into Schrodinger-like differential equation for upper spinor component. We compare this equation with the Schrodinger equation derived from shape invariance property of second order differential equations of mathematical physics. This formalism enables us to determine the electrostatic potential and relativistic energy in terms of master function and corresponding weight function. We also obtain the spinor wave function in terms of orthogonal polynomials.
Fukushima, Kimichika
2015-01-01
This paper presents analytical eigenenergies for a pair of confined fundamental fermion and antifermion under a linear potential derived from the Wilson loop for the non-Abelian Yang-Mills field. We use basis functions localized in spacetime, and the Hamiltonian matrix of the Dirac equation is analytically diagonalized. The squared system eigenenergies are proportional to the string tension and the absolute value of the Dirac's relativistic quantum number related to the total angular momentum, consistent with the expectation.
Single-cone real-space finite difference schemes for the Dirac von Neumann equation
Schreilechner, Magdalena
2015-01-01
Two finite difference schemes for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian are presented. Both utilize a single-cone staggered space-time grid which ensures a single-cone energy dispersion to formulate a numerical treatment of the mixed-state dynamics within the von Neumann equation. The first scheme executes the time-derivative according to the product rule for "bra" and "ket" indices of the density operator. It therefore directly inherits all the favorable properties of the difference scheme for the pure-state Dirac equation and conserves positivity. The second scheme proposed here performs the time-derivative in one sweep. This direct scheme is investigated regarding stability and convergence. Both schemes are tested numerically for elementary simulations using parameters which pertain to topological insulator surface states. Application of the schemes to a Dirac Lindblad equation and real-space-time Green's function formulations are discussed.
Extended Wronskian Determinant Approach and Iterative Solutions of One-Dimensional Dirac Equation
XU Ying; LU Meng; SU Ru-Keng
2004-01-01
An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation, which can be solved by iterative procedure to find the wave functions, is established. We employ this approach to study the one-dimensional Dirac equation with one-well potential,and give the energy levels and wave functions up to the first order iterative approximation. For double-well potential,the energy levels up to the first order approximation are given.
Solutions for the Klein-Gordon and Dirac equations on the lattice based on Chebyshev polynomials
Faustino, Nelson
2016-01-01
The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein-Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein-Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.
A Tale of Three Equations Breit, Eddington-Guant, and Two-Body Dirac
Van Alstine, P; Alstine, Peter Van; Crater, Horace W.
1997-01-01
G.Breit's original paper of 1929 postulates the Breit equation as a correction to an earlier defective equation due to Eddington and Gaunt, containing a form of interaction suggested by Heisenberg and Pauli. We observe that manifestly covariant electromagnetic Two-Body Dirac equations previously obtained by us in the framework of Relativistic Constraint Mechanics reproduce the spectral results of the Breit equation but through an interaction structure that contains that of Eddington and Gaunt. By repeating for our equation the analysis that Breit used to demonstrate the superiority of his equation to that of Eddington and Gaunt, we show that the historically unfamiliar interaction structures of Two-Body Dirac equations (in Breit-like form) are just what is needed to correct the covariant Eddington Gaunt equation without resorting to Breit's version of retardation.
Barut, A. O.
1990-04-01
By exact explicit solution it is shown that the Lorentz-Dirac equation with radiation reaction and proper initial conditions does not violate causality, even if the force is nonanalytic. We also show that if the equation is correctly renormalized there are no runaway solutions.
Analytic solution for Gauged Dirac-Weyl equation in $(2+1)$-dimensions
Ardenghi, Juan Sebastián; Sourrouille, Lucas
2016-01-01
A gauged Dirac-Weyl equation in (2+1)-dimension is considered. This equation describe relativistic matter. In particular we are interested in matter interacting with a Chern-Simons gauge fields. We show that exact self-dual solutions are admitted. These solutions are the same as those supported by nonrelativistic matter interacting with a Chern-Simons gauge field.
The relation between Maxwell, Dirac, and the Seiberg-Witten equations
Waldyr A. Rodrigues
2003-01-01
Full Text Available We discuss unsuspected relations between Maxwell, Dirac, and the Seiberg-Witten equations. First, we present the Maxwell-Dirac equivalence (MDE of the first kind. Crucial to that proposed equivalence is the possibility of solving for ψ (a representative on a given spinorial frame of a Dirac-Hestenes spinor field the equation F=ψγ21ψ˜, where F is a given electromagnetic field. Such task is presented and it permits to clarify some objections to the MDE which claim that no MDE may exist because F has six (real degrees of freedom and ψ has eight (real degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth, even if there is no evidence until now for magnetic monopoles, because there are at least two faithful field equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail associated with Maxwell theory and the other is a (nonlinear equation (of the generalized Maxwell type satisfied by the 2-form field part of a Dirac-Hestenes spinor field that solves the Dirac-Hestenes equation for a free electron. This is a new result which can also be called MDE of the second kind. Finally, we use the MDE of the first kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A physical interpretation for those equations is proposed.
Dirac equation from the Hamiltonian and the case with a gravitational field
Arminjon, M
2006-01-01
Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the correspondence. This derivation applies in the same form to a free particle, to one in an electromagnetic field, and to one subjected to geodesic motion in a static metric, and leads to the same, usual form of the Dirac equation--in special coordinates. To use the equation in the static-gravitational case, we need to rewrite it in more general coordinates. This can be done only if the usual, spinor transformation of the wave function is replaced by the 4-vector transformation. We show that the latter also makes the flat-space-time Dirac equation Lorentz-covariant, although the Dirac matrices are not invariant. Because the equation itself is left unchanged in the flat case, the 4-vector transformation does not alter the main physical consequences of that equation in that case. However, the equation derived in the ...
Reduced Dirac equation and Lamb shift as off-mass-shell effect in quantum electrodynamics
Ni, Guang-Jiong; Xu, Jian-Jun; Lou, Sen-Yue
2011-02-01
Based on the accurate experimental data of energy-level differences in hydrogen-like atoms, especially the 1S—2S transitions of hydrogen and deuterium, the necessity of introducing a reduced Dirac equation with reduced mass as the substitution of original electron mass is stressed. Based on new cognition about the essence of special relativity, we provide a reasonable argument for the reduced Dirac equation to have two symmetries, the invariance under the (newly defined) space-time inversion and that under the pure space inversion, in a noninertial frame. By using the reduced Dirac equation and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state—-a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Reduced Dirac equation and Lamb shift as off-mass-shell effect in quantum electrodynamics
Ni Guang-Jiong; Xu Jian-Jun; Lou Sea-Yue
2011-01-01
Based on the accurate experimental data of energy-level differences in hydrogen-like atoms, especially the 1S-2S transitions of hydrogen and deuterium, the necessity of introducing a reduced Dirac equation with reduced mass as the substitution of original electron mass is stressed. Based on new cognition about the essence of special relativity, we provide a reasonable argument for the reduced Dirac equation to have two symmetries, the invariance under the (newly defined) space-time inversion and that under the pure space inversion, in a noninertial frame. By using the reduced Dirac equation and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state-a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Dirac constraint analysis and symplectic structure of anti-self-dual Yang–Mills equations
U Camci; Z Can; Y Nutku; Y Sucu; D Yazici
2006-12-01
We present the explicit form of the symplectic structure of anti-self-dual Yang–Mills (ASDYM) equations in Yang's - and -gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac's theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang–Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both - and -gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten–Zuckerman formalism. We show that the appropriate component of the Witten–Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac's theory. Finally, we present the Bäcklund transformation between the - and -gauges in order to apply Magri's theorem to the respective two Hamiltonian structures.
Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups
Breev, A. I.; Mosman, E. A.
2016-12-01
The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.
Najafizade, S. A.; Hassanabadi, H.; Zarrinkamar, S.
2017-09-01
In this study, the information-theoretic measures of (1+1)-dimensional Dirac equation in both position and momentum spaces are investigated for the trigonometric Rosen-Morse and the Morse potentials. The solutions of the corresponding Dirac equation are obtained in an exact analytical manner in the first step. Next, using the Fourier transformation, the position and momentum Shannon information entropies are obtained and some features of the probability densities are analyzed. The consistency with Bialynicki-Birula-Mycielski inequality and Heisenberg uncertainty is checked.
Schroedinger's radial equation - Solution by extrapolation
Goorvitch, D.; Galant, D. C.
1992-01-01
A high-accuracy numerical method for the solution of a 1D Schroedinger equation that is suitable for a diatomic molecule, obtained by combining a finite-difference method with iterative extrapolation to the limit, is presently shown to have several advantages over more conventional methods. Initial guesses for the term values are obviated, and implementation of the algorithm is straightforward. The method is both less sensitive to round-off error, and faster than conventional methods for equivalent accuracy. These advantages are illustrated through the solution of Schroedinger's equation for a Morse potential function suited for HCl and a numerically derived Rydberg-Klein-Rees potential function for the X 1Sigma(+) state of CO.
Numerical Solution of Radial Biquaternion Klein-Gordon Equation
Christianto V.
2008-01-01
Full Text Available In the preceding article we argue that biquaternionic extension of Klein-Gordon equation has solution containing imaginary part, which differs appreciably from known solution of KGE. In the present article we present numerical/computer solution of radial biquaternionic KGE (radialBQKGE; which differs appreciably from conventional Yukawa potential. Further observation is of course recommended in order to refute or verify this proposition.
The Radially Symmetric Euler Equations as an Exterior Differential System
Baty, Roy; Ramsey, Scott; Schmidt, Joseph
2016-11-01
This work develops the Euler equations as an exterior differential system in radially symmetric coordinates. The Euler equations are studied for unsteady, compressible, inviscid fluids in one-dimensional, converging flow fields with a general equation of state. The basic geometrical constructions (for example, the differential forms, tangent planes, jet space, and differential ideal) used to define and analyze differential equations as systems of exterior forms are reviewed and discussed for converging flows. Application of the Frobenius theorem to the question of the existence of solutions to radially symmetric converging flows is also reviewed and discussed. The exterior differential system is further applied to derive and analyze the general family of characteristic vector fields associated with the one-dimensional inviscid flow equations.
Spin eigen-states of Dirac equation for quasi-two-dimensional electrons
Eremko, Alexander, E-mail: eremko@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); Brizhik, Larissa, E-mail: brizhik@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); Loktev, Vadim, E-mail: vloktev@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); National Technical University of Ukraine “KPI”, Peremohy av., 37, Kyiv, 03056 (Ukraine)
2015-10-15
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.
Bound states of the Dirac equation with some physical potentials by the Nikiforov-Uvarov method
Setare, Mohammad R; Haidari, S [Department of Physics, University of Kurdistan, Pasdaran Avenue, Sanandaj (Iran, Islamic Republic of)], E-mail: rezakord@ipm.ir, E-mail: heidary.somayeh@gmail.com
2010-01-15
Exact analytical solutions for the s-wave Dirac equation with the reflectionless-type, Rosen-Morse and Manning-Rosen potentials are obtained, under the condition of spin symmetry. We obtained bound state energy eigenvalues and corresponding spinor wave function in the framework of the Nikiforov-Uvarov (NU) method.
Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass
YANG Jin; XIANG An-Ping; YU Wan-Lun
2003-01-01
We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable forthe case with time-independence mass.
Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass
YANGJin; XIANGAn-Ping; YUWan-Lun
2003-01-01
We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable for the case with time-independence mass.
The exact solution for the Dirac equation with the Cornell potential
Trevisan, L A; Andrade, F M
2013-01-01
An analytical solution of the Dirac equation with a Cornell potential, with identical scalar and vectorial parts, is presented. The solution is obtained by using the linear potential solution, related to Airy functions, multiplied by another function to be determined. The energy levels are obtained and we notice that they obey a band structure.
Supersymmetry and Solution of Dirac Equation with Vector and Scalar Potentials
ZHANG Xiao-Long; JU Guo-Xing; FENG Mang; REN Zhong-Zhou; GAO Ke-Lin
2008-01-01
The Dirac equations with vector and scalar potentials of the Coulomb types in two and three dimensions are solved using the supersymmetric quantum mechanics method. For the system of such potentials, the analytical expressions of the matrix dements for both position and momentum operators are obtained.
Dirac equation and optical scalars in the Einstein-Cartan theory
Timofeev, Vladimir
2016-03-01
The article deals with the Dirac equation in the Newman-Penrose formalism within the framework of Einstein-Cartan theory and behavior of isotropic congruence of autoparallels, i. e. a congruence of the curves along which tangent null vector transferred in parallel.
An Introduction to Relativistic Quantum Mechanics. I. From Relativity to Dirac Equation
De Sanctis, M
2007-01-01
By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.
Soylu, A [Department of Physics, Nigde University, 51350, Nigde (Turkey); Bayrak, O; Boztosun, I [Department of Physics, Erciyes University, 38039, Kayseri (Turkey)
2008-02-15
For any spin-orbit coupling term {kappa}, the analytical solutions of the Dirac equation for the Eckart potential are presented by using the asymptotic iteration method within the framework of the spin and pseudospin symmetry concept. The energy eigenvalues are obtained in the closed form by applying an approximation to the spin-orbit coupling potential.
Spinor-electron wave guided modes in coupled quantum wells structures by solving the Dirac equation
Linares, Jesus [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)], E-mail: suso.linares.beiras@usc.es; Nistal, Maria C. [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)
2009-05-04
A quantum analysis based on the Dirac equation of the propagation of spinor-electron waves in coupled quantum wells, or equivalently coupled electron waveguides, is presented. The complete optical wave equations for Spin-Up (SU) and Spin-Down (SD) spinor-electron waves in these electron guides couplers are derived from the Dirac equation. The relativistic amplitudes and dispersion equations of the spinor-electron wave-guided modes in a planar quantum coupler formed by two coupled quantum wells, or equivalently by two coupled slab electron waveguides, are exactly derived. The main outcomes related to the spinor modal structure, such as the breaking of the non-relativistic degenerate spin states, the appearance of phase shifts associated with the spin polarization and so on, are shown.
A new propagation method for the radial Schroedinger equation
Devries, P. L.
1979-01-01
A new method for propagating the solution of the radial Schroedinger equation is derived from a Taylor series expansion of the wavefunction and partial re-summation of the infinite series. Truncation of the series yields an approximation to the exact propagator which is applied to a model calculation and found to be highly convergent.
NON-NEGATIVE RADIAL SOLUTION FOR AN ELLIPTIC EQUATION
Yang Guoying; Guo Zongming
2005-01-01
We study the structure and behavior of non-negative radial solution for the following elliptic equation △u = uv, x ∈ Rn with 0 ＜ v ＜ 1. We also obtain the detailed asymptotic expansion of u near infinity.
Large scale radial stability density of Hill's equation
Broer, Henk; Levi, Mark; Simo, Carles
2013-01-01
This paper deals with large scale aspects of Hill's equation (sic) + (a + bp(t)) x = 0, where p is periodic with a fixed period. In particular, the interest is the asymptotic radial density of the stability domain in the (a, b)-plane. It turns out that this density changes discontinuously in a certa
Existence of Majorana fermion mode and Dirac equation in cavity quantum electrodynamics
Sarkar, Sujit, E-mail: sujit.tifr@gmail.com
2015-10-15
We present the results of low lying collective mode of coupled optical cavity arrays. We derive the Dirac equation for this system and explain the existence of Majorana fermion mode in the system. We present quite a few analytical relations between the Rabi frequency oscillation and the atom–photon coupling strength to explain the different physical situation of our study and also the condition for massless collective mode in the system. We present several analytical relations between the Dirac spinor field, order and disorder operators for our systems. We also show that the Luttinger liquid physics is one of the intrinsic concepts in our system.
Global Solutions of Einstein—Dirac Equation on the Conformal Space
LUQi－Keng; WANGShi－Kun; 等
2001-01-01
The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.
Global Solutions of Einstein-Dirac Equation on the Conformal Space
LU Qi-Keng; WANG Shi-Kun; WU Ke
2001-01-01
The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic. Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group, which is isomorphic to SL(2, ). Hence the results and their interpretation coming from the two theories would be different. In this short note we study only the Lorentz spinor manifold and, especially, the solutions of Einstein-Dirac equations on the conformal space, which is closely related to the AdS/CFT correspondence.
Generalized Klein-Gordon and Dirac Equations from Nonlocal Kinetic Approach
El-Nabulsi, Rami Ahmad
2016-09-01
In this note, I generalized the Klein-Gordon and the Dirac equations by using Suykens's nonlocal-in-time kinetic energy approach, which is motivated from Feynman's kinetic energy functional formalism where the position differences are shifted with respect to one another. I proved that these generalized equations are similar to those obtained in literature in the presence of minimal length based on the Quesne-Tkachuk algebra.
Highly covariant quantum lattice gas model of the Dirac equation
Yepez, Jeffrey
2011-01-01
We revisit the quantum lattice gas model of a spinor quantum field theory-the smallest scale particle dynamics is partitioned into unitary collide and stream operations. The construction is covariant (on all scales down to a small length {\\ell} and small time {\\tau} = c {\\ell}) with respect to Lorentz transformations. The mass m and momentum p of the modeled Dirac particle depend on {\\ell} according to newfound relations m = mo cos (2{\\pi}{\\ell}/{\\lambda}) and p = (h/2{\\pi}{\\ell}) sin(2{\\pi}{\\ell}/{\\lambda}), respectively, where {\\lambda} is the Compton wavelength of the modeled particle. These relations represent departures from a relativistically invariant mass and the de Broglie relation-when taken as quantifying numerical errors the model is physically accurate when {\\ell} {\\ll} {\\lambda}. Calculating the vacuum energy in the special case of a massless spinor field, we find that it vanishes (or can have a small positive value) for a sufficiently large wave number cutoff. This is a marked departure from th...
Relativistic integro-differential form of the Lorentz-Dirac equation in 3D without runaways
Ibison, Michael; Puthoff, Harold E.
2001-04-01
It is well known that the third-order Lorentz-Dirac equation admits runaway solutions wherein the energy of the particle grows without limit, even when there is no external force. These solutions can be denied simply on physical grounds, and on the basis of careful analysis of the correspondence between classical and quantum theory. Nonetheless, one would prefer an equation that did not admit unphysical behavior at the outset. Such an equation - an integro-differential version of the Lorentz-Dirac equation - is currently available either in 1 dimension only, or in 3 dimensions only in the non-relativistic limit. It is shown herein how the Lorentz-Dirac equation may be integrated without approximation, and is thereby converted to a second-order integro-differential equation in 3D satisfying the above requirement. I.E., as a result, no additional constraints on the solutions are required because runaway solutions are intrinsically absent. The derivation is placed within the historical context established by standard works on classical electrodynamics by Rohrlich, and by Jackson.
Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation
Arij Bouzelmate
2007-05-01
Full Text Available This paper concerns the existence, uniqueness and asymptotic properties (as $r=|x|oinfty$ of radial self-similar solutions to the nonlinear Ornstein-Uhlenbeck equation [ v_t=Delta_p v+xcdot abla (|v|^{q-1}v ] in $mathbb{R}^Nimes (0, +infty$. Here $q>p-1>1$, $Ngeq 1$, and $Delta_p$ denotes the $p$-Laplacian operator. These solutions are of the form [ v(x,t=t^{-gamma} U(cxt^{-sigma}, ] where $gamma$ and $sigma$ are fixed powers given by the invariance properties of differential equation, while $U$ is a radial function, $U(y=u(r$, $r=|y|$. With the choice $c=(q-1^{-1/p}$, the radial profile $u$ satisfies the nonlinear ordinary differential equation $$ (|u'|^{p-2}u''+frac{N-1}r |u'|^{p-2}u'+frac{q+1-p}{p} r u'+(q-1 r(|u|^{q-1}u'+u=0 $$in $mathbb{R}_+$. We carry out a careful analysis of this equation anddeduce the corresponding consequences for the Ornstein-Uhlenbeck equation.
Generalized Lorentz-Dirac equation for a strongly coupled gauge theory.
Chernicoff, Mariano; García, J Antonio; Güijosa, Alberto
2009-06-19
We derive a semiclassical equation of motion for a "composite" quark in strongly coupled large-N_{c} N = 4 super Yang-Mills theory, making use of the anti-de Sitter space/conformal field theory correspondence. The resulting nonlinear equation incorporates radiation damping, and reduces to the standard Lorentz-Dirac equation for external forces that are small on the scale of the quark Compton wavelength, but has no self-accelerating or preaccelerating solutions. From this equation one can read off a nonstandard dispersion relation for the quark, as well as a Lorentz-covariant formula for its radiation rate.
Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues
Canada, Antonio
2011-01-01
This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \\ 1 \\leq p \\leq +\\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\\real^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
On the Existence of Additional (Hydrino) states in the Dirac equation
Khelashvili, Anzor
2016-01-01
In case of spinless particles there appear additional (singular) solutions in the framework of relativistic Klein-Gordon equation for Coulomb potential. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states (hydrino, small hydrogen) should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle (electron) in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions.
Dirac-Kahler equation in curved space-time, relation between spinor and tensor formulations
Red'kov, V M
2011-01-01
A common view is that generalization of a wave equation on Riemannian space-time is substantially determined by what a particle is - boson or fermion. As a rule, they say that tensor equations for bosons are extended in a simpler way then spinor equations for fermions. In that context, a very interesting problem is of extension a wave equation for Dirac--K\\"{a}hler field (Ivanenko--Landau field was historically first term, also the term a vector field of general type was used). The article relates a generally covariant tensor formalism to a spinor one when these both are applied to description of the Dirac-K\\"ahler field in a Rimannian space-time. Both methods are taken to be equivalent and the tensor equations are derived from spinor ones. It is shown that, for characterization of Dirac-K\\"ahler's tensor components, two alternative approaches are suitable: these are whether a tetrad-based pseudo tensor classification or a generally coordinate pseudo tensor one. By imposing definite restrictions on the the Di...
Fermion unification model based on the intrinsic SU(8 symmetry of a generalized Dirac equation
Eckart eMarsch
2015-10-01
Full Text Available A natural generalization of the original Dirac spinor into a multi-component spinor is achieved, which corresponds to the single lepton and the three quarks of the first family of the standard model of elementary particle physics. Different fermions result from similarity transformations of the Dirac equation, but apparently there can be no more fermions according to the maximal multiplicity revealed in this study. Rotations in the fermion state space are achieved by the unitary generators of the U(1 and the SU(3 groups, corresponding to quantum electrodynamics (QED based on electric charge and chromodynamics (QCD based on colour charge. In addition to hypercharge the dual degree of freedom of hyperspin emerges, which occurs due to the duplicity implied by the two related (Weyl and Dirac representations of the Dirac equation. This yields the SU(2 symmetry of the weak interaction, which can be married to U(1 to generate the unified electroweak interaction as in the standard model. Therefore, the symmetry group encompassing all the three groups mentioned above is SU(8, which can accommodate and unify the observed eight basic stable fermions.
QED2+1 in Graphene: Symmetries of Dirac Equation in 2+1 Dimensions
Kosiński, P.; Maślanka, P.; S´nska, J.; Zasada, I.
2012-10-01
It is well-known that the tight-binding Hamiltonian of graphene describes the low-energy excitations that appear to be massless chiral Dirac fermions. Thus, in the continuum limit one can analyze the crystal properties using the formalism of quantum electrodynamics in 2+1 dimensions (QED_{2+1}) which provides the opportunity to verify the high energy physics phenomena in the condensed matter system. We study the symmetry properties of 2+1-dimensional Dirac equation, both in the noninteracting case and in the case with constant uniform magnetic field included in the model. The maximal symmetry group of the massless Dirac equation is considered by putting it in the Jordan block form and determining the algebra of operators leaving invariant the subspace of solutions. It is shown that the resulting symmetry operators expressed in terms of Dirac matrices cannot be described exclusively in terms of γ matrices (and their products) entering the corresponding Dirac equation. It is a consequence of the reducibility of the considered representation in contrast to the 3+1-dimensional case. Symmetry algebra is demonstrated to be a direct sum of two gL(2,C) algebras plus an eight-dimensional abelian ideal. Since the matrix structure which determines the rotational symmetry has all required properties of the spin algebra, the pseudospin related to the sublattices (M. Mecklenburg and B. C. Regan, Phys. Rev. Lett. 106 (2011), 116803) gains the character of the real angular momentum, although the degrees of freedom connected with the electron's spin are not included in the model. This seems to be graphene's analogue of the phenomenon called ``spin from isospin'' in high energy physics.
Wells, J C; Eichler, J
1999-01-01
We discuss the two-center, time-dependent Dirac equation describing the dynamics of an electron during a peripheral, relativistic heavy-ion collision at extreme energies. We derive a factored form, which is exact in the high-energy limit, for the asymptotic channel solutions of the Dirac equation, and elucidate their close connection with gauge transformations which transform the dynamics into a representation in which the interaction between the electron and a distant ion is of short range. We describe the implications of this relationship for solving the time-dependent Dirac equation for extremely relativistic collisions.
Dirac equation of spin particles and tunneling radiation from a Kinnersly black hole
Li, Guo-Ping; Zu, Xiao-Tao [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Feng, Zhong-Wen [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); China West Normal University, College of Physics and Space Science, Nanchong (China); Li, Hui-Ling [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Shenyang Normal University, College of Physics Science and Technology, Shenyang (China)
2017-04-15
In curved space-time, the Hamilton-Jacobi equation is a semi-classical particle equation of motion, which plays an important role in the research of black hole physics. In this paper, starting from the Dirac equation of spin 1/2 fermions and the Rarita-Schwinger equation of spin 3/2 fermions, respectively, we derive a Hamilton-Jacobi equation for the non-stationary spherically symmetric gravitational field background. Furthermore, the quantum tunneling of a charged spherically symmetric Kinnersly black hole is investigated by using the Hamilton-Jacobi equation. The result shows that the Hamilton-Jacobi equation is helpful to understand the thermodynamic properties and the radiation characteristics of a black hole. (orig.)
Dispersion Equation of the Coaxial-Radial Line
WANG Wenxiang; YUE Lingna; YU Guofen; GONG Yubing; HUANG Minzhi
2004-01-01
An all-metal slow-wave structure,coaxial-radial line,which is suitable for application in broadband high power traveling wave tube (TWT) and relativistic TWT as a RF system is introduced.Making use of the field matching method and variational method together with the orthogonality of the Bessel function and the Floquet Theroem for the periodic system,the dispersion characteristic expression is derived.This equation is more rigorous than that of precious reports.
Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays
Koke, C; Angelakis, D G
2016-01-01
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. Gravitational fields can be incorporated as background spacetime if the back-action of matter on it can be neglected, yielding modified Dirac or Klein-Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many scenarios including expanding universe, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (the Unruh effect). In this work, we give a pedagogical introduction to the Dirac equation in a general 2D spacetime and show examples of spinor wave packet dynamics in some background spacetimes. In particular, we cover the phenomenon of particle creation in a time-dependent metric. Photonic analogs of these effects are then proposed, where classical light propagating in an array of coupled waveguides provides a visualis...
A Vlasov equation with Dirac potential used in fusion plasmas
Bardos, Claude [Universite Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 Place Jussieu, 75252 Paris Cedex 05 (France); Nouri, Anne [Laboratoire d' Analyse, Topologie et Probabilites (UMR 6632), Aix-Marseille Universite, 39 Rue Joliot-Curie, 13453 Marseille Cedex 13 (France)
2012-11-15
Well-posedness of the Cauchy problem is analyzed for a singular Vlasov equation governing the evolution of the ionic distribution function of a quasineutral fusion plasma. The Penrose criterium is adapted to the linearized problem around a time and space homogeneous distribution function showing (due to the singularity) more drastic differences between stable and unstable situations. This pathology appears on the full nonlinear problem, well-posed locally in time with analytic initial data, but generally ill-posed in the Hadamard sense. Eventually with a very different class of solutions, mono-kinetic, which constrains the structure of the density distribution, the problem becomes locally in time well-posed.
On exact solutions of the Dirac equation in a homogeneous magnetic field in the Lobachevsky space
Ovsiyuk, E M; Red'kov, V M
2010-01-01
There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Lobachevsky space geometry, has been obtained.
The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization
Reifler, Frank; Morris, Randall
1992-01-01
Currently, there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors.
Scattering states of Dirac particle equation with position dependent mass under the cusp potential
Chabab, M; Hassanabadi, H; Oulne, M; Zare, S
2016-01-01
We solved the one-dimensional position-dependent mass Dirac equation in the presence of the cusp potential and reported the solutions in terms of the Whittaker functions. We have derived the reflection and transmission coefficients by making use of the matching conditions on the wave functions. The effect of position dependent mass on the reflection and transmission coefficients of the system is duly investigated.
Scattering states of Dirac particle equation with position-dependent mass under the cusp potential
Chabab, M.; El Batoul, A.; Hassanabadi, H.; Oulne, M.; Zare, S.
2016-11-01
We solved the one-dimensional position-dependent mass Dirac equation in the presence of the cusp potential and reported the solutions in terms of the Whittaker functions. We have derived the reflection and transmission coefficients by making use of the matching conditions on the wave functions. The effect of the position-dependent mass on the reflection and transmission coefficients of the system is duly investigated.
Analytic Continuation in the Coupling Constant Method for the Dirac Equation
张时声; 郭建友; 张双全; 孟杰
2004-01-01
On the basis of the Dirac equation, the analytic continuation in the coupling constant method is employed to investigate the energies and widths of single-particle resonant in square-well, harmonic-oscillator, and Woodsconvergent energies and widths of single-particle resonant states can be obtained, which makes the application of the analytic continuation in the coupling constant for the relativistic mean field theory possible.
Single-site Green function of the Dirac equation for full-potential electron scattering
Kordt, Pascal
2012-05-30
I present an elaborated analytical examination of the Green function of an electron scattered at a single-site potential, for both the Schroedinger and the Dirac equation, followed by an efficient numerical solution, in both cases for potentials of arbitrary shape without an atomic sphere approximation. A numerically stable way to calculate the corresponding regular and irregular wave functions and the Green function is via the angular Lippmann-Schwinger integral equations. These are solved based on an expansion in Chebyshev polynomials and their recursion relations, allowing to rewrite the Lippmann-Schwinger equations into a system of algebraic linear equations. Gonzales et al. developed this method for the Schroedinger equation, where it gives a much higher accuracy compared to previous perturbation methods, with only modest increase in computational effort. In order to apply it to the Dirac equation, I developed relativistic Lippmann-Schwinger equations, based on a decomposition of the potential matrix into spin spherical harmonics, exploiting certain properties of this matrix. The resulting method was embedded into a Korringa-Kohn-Rostoker code for density functional calculations. As an example, the method is applied by calculating phase shifts and the Mott scattering of a tungsten impurity. (orig.)
M. R. Setare; O. Hatami
2008-01-01
Based on the shape invariance property we obtain exact solutions of the Dirac equation for an electron moving in the presence of a certain varying magnetic field, then we also show its non-relativistic limit.
Energy Loss by Radiation in Many-Particle Numerical Simulation With Lorentz-Dirac Equation
Žáček, Martin
2006-01-01
We studied the possibilities for numerical integration of Lorentz-Dirac equation that is the equation describing the motion of a charged point particle when radiation reaction is taken into account. In numerical modelling based on particle models usually the equations of motion without radiation force are used and the corrections for radiation are used consequently, expressed by laws given by averaged particle parameters as the temperature or particle density. If the complete equation of motion concluding the radiation would be used, the corrections for radiation reaction force could be used for every charged particle individually from more fundamental laws. Thus the model could be able to describe more physical phenomena. However from theory of Lorentz-Dirac equation there are known various problems with non-physical solutions and nonuniqueness that are often solved and tested by various methods. One way to eliminate the non-physical solutions is to use integro-differential equation, which is used here. The leap-frog method is used for numerical integrating and accuracy is verified for electron in magnetic field. This approach is proposed to be used for PIC (particle-in-cell) integration method, which is often used as an effective method of simulation in plasma physics for many charged particles interactinge with electromagnetic field.
Numeric Solutions of Dirac-Gursey Spinor Field Equation Under External Gaussian White Noise
Aydogmus, Fatma
2016-06-01
In this paper, we consider the Dirac-Gursey spinor field equation that has particle-like solutions derived classical field equations so-called instantons, formed by using Heisenberg ansatz, under the effect of an additional Gaussian white noise term. Our purpose is to understand how the behavior of spinor-type excited instantons in four dimensions can be affected by noise. Thus, we simulate the phase portraits and Poincaré sections of the obtained system numerically both with and without noise. Recurrence plots are also given for more detailed information regarding the system.
Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays
Koke, Christian, E-mail: christian.koke@stud.uni-heidelberg.de [Institut für theoretische Physik, Philosophenweg 16, D-69120 Heidelberg (Germany); Noh, Changsuk, E-mail: changsuk@kias.re.kr [Korea Institute for Advanced Study, 85 Hoegiro, Seoul 130-722 (Korea, Republic of); Angelakis, Dimitris G., E-mail: dimitris.angelakis@gmail.com [Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117542 (Singapore); School of Electronic and Computer Engineering, Technical University of Crete, Chania, Crete, 73100 (Greece)
2016-11-15
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. A gravitational field can be incorporated as a background spacetime if the back-action of matter on the field can be neglected, resulting in modifications of the Dirac or Klein–Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many situations, including the expanding-universe scenario, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (the Unruh effect). In this work, we give a pedagogical introduction to the Dirac equation in a general 2D spacetime and show examples of spinor wave packet dynamics in flat and curved background spacetimes. In particular, we cover the phenomenon of particle creation in a time-dependent metric. Photonic analogs of these effects are then proposed, where classical light propagating in an array of coupled waveguides provides a visualisation of the Dirac spinor propagating in a curved 2D spacetime background. The extent to which such a single-particle description can be said to mimic particle creation is discussed.
Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays
Koke, Christian; Noh, Changsuk; Angelakis, Dimitris G.
2016-11-01
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. A gravitational field can be incorporated as a background spacetime if the back-action of matter on the field can be neglected, resulting in modifications of the Dirac or Klein-Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many situations, including the expanding-universe scenario, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (the Unruh effect). In this work, we give a pedagogical introduction to the Dirac equation in a general 2D spacetime and show examples of spinor wave packet dynamics in flat and curved background spacetimes. In particular, we cover the phenomenon of particle creation in a time-dependent metric. Photonic analogs of these effects are then proposed, where classical light propagating in an array of coupled waveguides provides a visualisation of the Dirac spinor propagating in a curved 2D spacetime background. The extent to which such a single-particle description can be said to mimic particle creation is discussed.
A Conceptual Shift to Rectify a Defect in the Lorentz-Dirac Equation
Oliver, M A
2013-01-01
In his analysis of the Classical Theory of Radiating Electrons, Dirac (1938) draws attention to the characteristic instability of solutions to the third order equation of motion. He remarks that changing the sign of the self-force eliminates the runaway solutions and gives `reasonable behaviour'. Dirac rejects such a change and proceeds with an ad hoc modification to the solutions of the initial value problem that is not consistent with the principle of causality. We argue that his reasons for rejecting the change of sign are invalid on both physical and mathematical grounds. The conceptual shift is to treat the physical particle as a composite of the source particle and the energy-momentum that is reversibly generated in its self-field by its motion. The reversibly generated energy in the self-field is interpreted as kinetic energy, and the changes that follow result in Dirac's change of sign. Several exact solutions to the new equation of motion and its linearisation are given. For a particle in orbital mot...
Improvement of the basis for the solution of the Dirac equation in Cassini coordinates
Hahn, W.; Artemyev, A. N.; Surzhykov, A.
2017-08-01
We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in our earlier article [J. Phys. B 43, 235207 (2010)]. For the calculations in the above article, we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-splines only. Therefore, we include basis functions which are defined using functions with step-like behavior instead of B-splines. Thereby, we achieve a significant increase of accuracy of results.
Improvement of the Basis for the Solution of the Dirac Equation in Cassini Coordinates
Hahn, Walter; Surzhykov, Andrey
2016-01-01
We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in [1]. For the calculations in [1], we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-spines only. Therefore, we include basis functions which are defined using functions with step-like behaviour instead of B-splines. Thereby, we achieve a significant increase of accuracy of results as compared to [1].
Erdoğan, M. Burak; Green, William R.
2017-06-01
We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a t -1 decay rate as an operator from the Hardy space H 1 to BMO, the space of functions of bounded mean oscillation. This estimate, along with the L 2 conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold s-wave resonance does not destroy the t -1 decay rate. As a consequence of our analysis we obtain a limiting absorption principle in the neighborhood of the threshold, and show that there are only finitely many eigenvalues in the spectral gap.
Bound States of the Klein-Gordon and Dirac equations for potential V0 tanh2(r/d)
Qiang Wen-Chao
2004-01-01
The exact bound state wavefunctions and energy equations of Klein-Gordon and Dirac equations are given with equal scalar and vector potential s(r) = v(r) = V(r)/2 = V0 tanh2(r/d). The relation between the energy equation and that of relativistic harmonic is discussed.
Existence of infinitely many radial solutions for quasilinear Schrodinger equations
Gui Bao
2014-10-01
Full Text Available In this article we prove the existence of radial solutions with arbitrarily many sign changes for quasilinear Schrodinger equation $$ -\\sum_{i,j=1}^{N}\\partial_j(a_{ij}(u\\partial_iu +\\frac{1}{2}\\sum_{i,j=1}^{N}a'_{ij}(u\\partial_iu\\partial_ju+V(xu =|u|^{p-1}u,~x\\in\\mathbb{R}^N, $$ where $N\\geq3$, $p\\in(1,\\frac{3N+2}{N-2}$. The proof is accomplished by using minimization under a constraint.
Chicurel-Uziel, Enrique
2007-08-01
A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.
Feshbach P -Q partitioning technique and the two-component Dirac equation
Luo, Da-Wei; Pyshkin, P. V.; Yu, Ting; Lin, Hai-Qing; You, J. Q.; Wu, Lian-Ao
2016-09-01
We provide an alternative approach to relativistic dynamics based on the Feshbach projection technique. Instead of directly studying the Dirac equation, we derive a two-component equation for the upper spinor. This approach allows one to investigate the underlying physics in a different perspective. For particles with small mass such as the neutrino, the leading-order equation has a Hermitian effective Hamiltonian, implying there is no leakage between the upper and lower spinors. In the weak relativistic regime, the leading order corresponds to a non-Hermitian correction to the Pauli equation, which takes into account the nonzero possibility of finding the lower-spinor state and offers a more precise description.
The Lorentz-Dirac equation and the physical meaning of the Maxwell's fields
De Souza, M M
1995-01-01
Classical Electrodynamics is not a consistent theory because of the inadequate behaviour of its fields in the vicinity of their sources. Its problems with the electron equation of motion and with non-integrable singularity of the electron self field and of its stress tensor are well known. These inconsistencies are eliminated if the discrete and localized (classical photons) character of the electromagnetic interaction is anticipatively recognized already in a classical context. This is possible, in a manifestly covariant way, with a new model of spacetime structure, shown in a previous paper \\cite{hep-th/9505169}, that invalidates the Lorentz-Dirac equation. For the new equation of motion of a point classical electron there is no field singularity, no causality violation and no conflict with energy conservation in the electron equation of motion. The electromagnetic field must be re-interpreted in terms of average flux of classical photons. Implications of a singularity-free formalism to field theory are dis...
Radially Symmetric Solutions of a Nonlinear Elliptic Equation
Edward P. Krisner
2011-01-01
Full Text Available We investigate the existence and asymptotic behavior of positive, radially symmetric singular solutions of +((−1/−||−1=0, >0. We focus on the parameter regime >2 and 10. Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval (min,max. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions of the equation. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval (0,∞, and which satisfy 00. An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of “super singular” solutions which lie entirely above 1(.
Batic, D
2005-01-01
In this paper we compute the square root of the generalized squared total angular momentum operator $J$ for a Dirac particle in the Kerr-Newman metric. The separation constant $\\lambda$ arising from the Chandrasekahr separation ansatz turns out to be the eigenvalue of $J$. After proving that $J$ is a symmetry operator, we show the completeness of Chandrasekhar Ansatz for the Dirac equation in oblate spheroidal coordinates and derive an explicit formula for the propagator $e^{-itH}$.
A new CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
Casana, R; Silva, E O; Passos, E; Santos, F E P dos
2013-01-01
In this work, we propose a CPT-even and Lorentz-violating dimension-five nonminimal coupling between fermionic and gauge fields, involving the CTP-even and Lorentz-violating gauge tensor of the SME. This nonminimal coupling modifies the Dirac equation, whose nonrelativistic regime is governed by a Hamiltonian which induces new effects, such as an electric-Zeeman-like spectrum splitting and an anomalous-like contribution to the electron magnetic moment, between others. Some of these new effects allows to constrain the magnitude of this nonminimal coupling in 1 part in 10^16.
New CPT-even and Lorentz-violating nonminimal coupling in the Dirac equation
Casana, R.; Ferreira, M. M., Jr.; Passos, E.; dos Santos, F. E. P.; Silva, E. O.
2013-02-01
In this work, we propose a CPT-even and Lorentz-violating dimension-five nonminimal coupling between fermionic and gauge fields, involving the CPT-even and Lorentz-violating gauge tensor of the Standard Model extension. This nonminimal coupling modifies the Dirac equation, whose nonrelativistic regime is governed by a Hamiltonian which induces new effects, such as an electric-Zeeman-like spectrum splitting and an anomalouslike contribution to the electron magnetic moment, among others. Some of these new effects allow one to constrain the magnitude of this nonminimal coupling in 1 part in 1016(eV)-1.
Solutions of the Dirac Equation in a Magnetic Field and Intertwining Operators
Alonso Contreras-Astorga
2012-10-01
Full Text Available The intertwining technique has been widely used to study the Schrödinger equation and to generate new Hamiltonians with known spectra. This technique can be adapted to find the bound states of certain Dirac Hamiltonians. In this paper the system to be solved is a relativistic particle placed in a magnetic field with cylindrical symmetry whose intensity decreases as the distance to the symmetry axis grows and its field lines are parallel to the x−y plane. It will be shown that the Hamiltonian under study turns out to be shape invariant.
M Hamzavi; S M Ikhdair
2014-07-01
The Hellmann potential is simply a superposition of an attractive Coulomb potential $−a/r$ plus a Yukawa potential e${}^{−δr} /r$. The generalized parametric Nikiforov–Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number in the presence of exact spin and pseudospin (p-spin) symmetries. As a particular case, we obtain the energy eigenvalues of the pure Coulomb potential in the non-relativistic limit.
Goncalves, Bruno; Dias Junior, Mario Marcio [Instituto Federal de Educacacao, Ciencia e Tecnologia Sudeste de Minas Gerais, Juiz de Fora, MG (Brazil)
2013-07-01
Full text: The discussion of experimental manifestations of torsion at low energies is mainly related to the torsion-spin interaction. In this respect the behavior of Dirac field and the spinning particle in an external torsion field deserves and received very special attention. In this work, we consider the combined action of torsion and magnetic field on the massive spinor field. In this case, the Dirac equation is not straightforward solved. We suppose that the spinor has two components. The equations have mixed terms between the two components. The electromagnetic field is introduced in the action by the usual gauge transformation. The torsion field is described by the field S{sub μ}. The main purpose of the work is to get an explicit form to the equation of motion that shows the possible interactions between the external fields and the spinor in a Hamiltonian that is independent to each component. We consider that S{sub 0} is constant and is the unique non-vanishing term of S{sub μ}. This simplification is taken just to simplify the algebra, as our main point is not to describe the torsion field itself. In order to get physical analysis of the problem, we consider the non-relativistic approximation. The final result is a Hamiltonian that describes a half spin field in the presence of electromagnetic and torsion external fields. (author)
Dirac equation for the Hulthén potential within the Yukawa-type tensor interaction
Oktay Aydo(g)du; Elham Maghsoodi; Hassan Hassanabadi
2013-01-01
Using the Nikiforov-Uvarov (NU) method,pseudospin and spin symmetric solutions of the Dirac equation for the scalar and vector Hulthén potentials with the Yukawa-type tensor potential are obtained for an arbitrary spin-orbit coupling quantum number κ.We deduce the energy eigenvalue equations and corresponding upper-and lower-spinor wave functions in both the pseudospin and spin symmetry cases.Numerical results of the energy eigenvalue equations and the upper-and lower-spinor wave functions are presented to show the effects of the external potential and particle mass parameters as well as pseudospin and spin symmetric constants on the bound-state energies and wave functions in the absence and presence of the tensor interaction.
Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity
无
2006-01-01
Based on unified theory of electromagnetic interactions and gravitational interactions, the non-relativistic limit of the equation of motion of a charged Dirac particle in gravitational field is studied. From the Schrodinger equation obtained from this non-relativistic limit, we can see that the classical Newtonian gravitational potential appears as a part of the potential in the Schrodinger equation, which can explain the gravitational phase effects found in COW experiments.And because of this Newtonian gravitational potential, a quantum particle in the earth's gravitational field may form a gravitationally bound quantized state, which has already been detected in experiments. Three different kinds of phase effects related to gravitational interactions are studied in this paper, and these phase effects should be observable in some astrophysical processes. Besides, there exists direct coupling between gravitomagnetic field and quantum spin, and radiation caused by this coupling can be used to directly determine the gravitomagnetic field on the surface of a star.
Invariance properties of the Dirac equation with external electro-magnetic ﬁeld
N D Sen Gupta
2003-01-01
In this paper, we attempt to obtain the nature of the external ﬁeld such that the Dirac equation with external electro-magnetic ﬁeld is invariant. The Poincar´e group, which is the maximal symmetry group for ﬁeld free case, is constrained by the presence of the external ﬁeld. Introducing inﬁnitesimal transformation of x andψ, we apply Lie’s extended group method to obtain the class of external ﬁeld which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic ﬁeld, though only potentials explicity appears in the equation.
Chen Gang; Chen Zi-Dong; Lou Zhi-Mei
2004-01-01
The exact bound state solutions of the Klein-Gordon equation and Dirac equation with scalar and vector pseudoharmonic oscillator potentials are obtained in this paper. Furthermore, we have used the supersymmetric quantum mechanics, shape invariance and alternative method to obtain the required results.
Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism
Steane, Andrew M
2014-01-01
It is widely believed that classical electromagnetism is either unphysical or inconsistent, owing to pathological behaviour when self-force and radiation reaction are non-negligible. We argue that there is no inconsistency as long as it is recognized that certain types of charge distribution are simply impossible, such as, for example, a point particle with finite charge and finite inertia. This is owing to the fact that negative inertial mass is an unphysical concept in classical physics. It remains useful to obtain an equation of motion for small charged objects that describes their motion to good approximation without requiring knowledge of the charge distribution within the object. We give a simple method to achieve this, leading to a reduced-order form of the Abraham-Lorentz-Dirac equation, essentially as proposed by Eliezer, Landau and Lifshitz.
Staggered grid leap-frog scheme for the (2+1)D Dirac equation
Hammer, René
2013-01-01
A numerical scheme utilizing a grid which is staggered in both space and time is proposed for the numerical solution of the (2+1)D Dirac equation in presence of an external electromagnetic potential. It preserves the linear dispersion relation of the free Weyl equation for wave vectors aligned with the grid and facilitates the implementation of open (absorbing) boundary conditions via an imaginary potential term. This explicit scheme has second order accuracy in space and time. A functional for the norm is derived and shown to be conserved. Stability conditions are derived. Several numerical examples, ranging from generic to specific to textured topological insulator surfaces, demonstrate the properties of the scheme which can handle general electromagnetic potential landscapes.
Kazinski, P. O.; Shipulya, M. A.
2011-06-01
We present a study of planar physical solutions to the Lorentz-Dirac equation in a constant electromagnetic field. In this case, we reduced the Lorentz-Dirac equation to one second-order differential equation. We obtained the asymptotics of physical solutions to this equation at large proper times. It turns out that, in a crossed constant uniform electromagnetic field with vanishing invariants, a charged particle enters a universal regime at large times. We found that the ratios of momentum components that tend to constants are determined only by the external field. This effect is essentially due to a radiation reaction. There is no such effect for the Lorentz equation in this field.
Kazinski, P O
2010-01-01
We present a study of planar physical solutions to the Lorentz-Dirac equation in a constant electromagnetic field. In this case, we reduced the Lorentz-Dirac equation to the one second order differential equation. We found the asymptotics of physical solutions to this equation at large proper times. It turns out that, in the crossed constant uniform electromagnetic field with vanishing invariants, a charged particle goes to a universal regime at large times. We found the ratio of momentum components which tends to a constant determined only by the external field. This effect is essentially due to a radiation reaction. There is no such an effect for the Lorentz equation in this field.
Reduced Dirac Equation and Lamb Shift as an Off-mass-shell effect in Quantum Electrodynamics
Ni, G; Xu, J; Lou, Senyue; Ni, Guang-jiong; Xu, Jianjun
2005-01-01
Based on the precision experimental data of energy-level differences in hydrogenlike atoms, especially the 1S-2S transition of hydrogen and deuterium, the necessity of establishing a reduced Dirac equation (RDE) with reduced mass as the substitution of original electron mass is stressed. The theoretical basis of RDE lies on two symmetries, the invariance under the space-time inversion and that under the pure space inversion. Based on RDE and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state--a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Global existence for an L^2 critical Nonlinear Dirac equation in one dimension
Candy, Timothy
2011-01-01
We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the Thirring model. Local existence in $H^s$ for $s>0$, and global existence for $s>1/2$, has recently been proven by Selberg and Tesfahun by using $X^{s, b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, we first prove local existence in $L^2$ by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of $L^2$ norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all $s>0$.
Analytical solutions of the two-dimensional Dirac equation for a topological channel intersection
Anglin, J. R.; Schulz, A.
2017-01-01
Numerical simulations in a tight-binding model have shown that an intersection of topologically protected one-dimensional chiral channels can function as a beam splitter for noninteracting fermions on a two-dimensional lattice [Qiao, Jung, and MacDonald, Nano Lett. 11, 3453 (2011), 10.1021/nl201941f; Qiao et al., Phys. Rev. Lett. 112, 206601 (2014), 10.1103/PhysRevLett.112.206601]. Here we confirm this result analytically in the corresponding continuum k .p model, by solving the associated two-dimensional Dirac equation, in the presence of a "checkerboard" potential that provides a right-angled intersection between two zero-line modes. The method by which we obtain our analytical solutions is systematic and potentially generalizable to similar problems involving intersections of one-dimensional systems.
Bagci, A
2016-01-01
The author in his previous works were presented a numerical integration method, namely, global-adaptive with the Gauss-Kronrod numerical integration extension in order to accurate calculation of molecular auxiliary functions integrals involve power functions with non-integer exponents. They are constitute elements of molecular integrals arising in Dirac equation when Slater-type orbitals with non-integer principal quantum numbers are used. Binomial series representation of power functions method, so far, is used for analytical evaluation of the molecular auxiliary function integrals however, intervals of integration cover areas beyond the condition of convergence. In the present study, analytical evaluation of these integrals is re-examined. They are expressed via prolate spheroidal coordinates. An alternative analytical approximation are derived. Slowly convergent binomial series representation formulae for power functions is investigated through nonlinear sequence transformations for the acceleration of con...
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).
A CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
Ferreira Junior, Manoel; Casana, M.R.; Santos, Frederico E.P. dos; Silva, E.O. [UFMA, Sao Luis (Brazil); Passos, E. [UFCG, Campina Grande, PB (Brazil)
2013-07-01
Full text: The Standard Model Extension (SME) has been the usual framework for investigating signals of Lorentz violation in physical systems. It is the natural framework for studying properties of physical systems with Lorentz-violation since it includes Lorentz-violating terms in all sectors of the minimal standard model. The Lorentz-violating (LV) terms are generated as vacuum expectation values of tensors defined in a high energy scale. This framework has inspired a great deal of investigation in recent years. Such works encompass several distinct aspects involving fermion systems and radiative corrections, CPT- probing experiments, the electromagnetic CPT- and Lorentz-odd term, the 19 electromagnetic CPT-even coefficients. Recently, some studies involving higher dimensional operators have also been reported with great interest, including nonminimal interactions. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients. In the present work, we propose a new CPT-even, dimension-five, nonminimal coupling linking the fermionic and gauge fields in the context of the Dirac equation, involving the CPT-even tensor of the gauge term of the SME. By considering the nonrelativistic limit of the modified Dirac equation, we explicitly evaluate the new contributions to the nonrelativistic Hamiltonian. These new terms imply a direct correction on the anomalous magnetic moment, a kind of electrical Zeeman-like effect on the atomic spectrum, and a Rashba-like coupling term. These effects are then used to impose upper bounds on the magnitude of the non minimally coupled LV coefficients at the level of 1 part in 10{sub 16}. (author)
FFT-split-operator code for solving the Dirac equation in 2+1 dimensions
Mocken, Guido R.; Keitel, Christoph H.
2008-06-01
provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 474 937 No. of bytes in distributed program, including test data, etc.: 4 128 347 Distribution format: tar.gz Programming language: C++ Computer: Any, but SMP systems are preferred Operating system: Linux and MacOS X are actively supported by the current version. Earlier versions were also tested successfully on IRIX and AIX Number of processors used: Generally unlimited, but best scaling with 2-4 processors for typical problems RAM: 160 Megabytes minimum for the examples given here Classification: 2.7 External routines: FFTW Library [3,4], Gnu Scientific Library [5], bzip2, bunzip2 Nature of problem: The relativistic time evolution of wave functions according to the Dirac equation is a challenging numerical task. Especially for an electron in the presence of high intensity laser beams and/or highly charged ions, this type of problem is of considerable interest to atomic physicists. Solution method: The code employs the split-operator method [1,2], combined with fast Fourier transforms (FFT) for calculating any occurring spatial derivatives, to solve the given problem. An autocorrelation spectral method [6] is provided to generate a bound state for use as the initial wave function of further dynamical studies. Restrictions: The code in its current form is restricted to problems in two spatial dimensions. Otherwise it is only limited by CPU time and memory that one can afford to spend on a particular problem. Unusual features: The code features dynamically adapting position and momentum space grids to keep execution time and memory requirements as small as possible. It employs an object-oriented approach, and it relies on a Clifford algebra class library to represent the mathematical objects of the Dirac formalism which we employ. Besides that it includes a feature (typically called "checkpointing") which allows the resumption of an
Leviatan, A
2004-01-01
We show that the Dirac equation in 3+1 dimensions gives rise to supersymmetric patterns when the scalar and vector potentials are (i) Coulombic with arbitrary strengths or (ii) when their sum or difference is a constant, leading to relativistic pseudospin and spin symmetries. The conserved quantities and the common intertwining relation responsible for such patterns are discussed.
Leviatan, A
2004-05-21
We show that the Dirac equation in (3+1) dimensions gives rise to supersymmetric patterns when the scalar and vector potentials are (i). Coulombic with arbitrary strengths or (ii). when their sum or difference is a constant, leading to relativistic pseudospin and spin symmetries. The conserved quantities and the common intertwining relation responsible for such patterns are discussed.
Koga, James
2004-10-01
Usually the motion of an electron under the influence of electromagnetic fields is influenced to a small extent by radiation damping. With the advent of high power high irradiance lasers it has become possible to generate focused laser irradiances where electrons interacting with the laser become highly relativistic over very short time and spatial scales. By focusing petawatt class lasers to very small spot sizes the amount of radiation emitted by electrons can become very large. Resultingly, the damping of the electron motion by the emission of this radiation can become large. In order to study this problem a code is written to solve a set of equations describing the evolution of a strong electromagnetic wave interacting with a single electron. Usually the equation of motion of an electron including radiation damping under the influence of electromagnetic fields is derived from the Lorentz-Dirac equation treating the damping as a perturbation. We use this equation to integrate forward in time and use the Lorentz-Dirac equation to integrate backward in time. We show that for very short wavelength electromagnetic radiation deep in the quantum regime at high irradiances differences between the perturbation equation and Lorentz-Dirac can be seen. However, for electron motion in the classical regime the differences are negligible. For electron motion in the classical regime the first order damping equation is found to be very adequate.
Nakhleh, Charles
2012-01-01
In this pedagogical note, I revisit the problem of the equation of motion of a relativistic classical electron coupled to the electromagnetic field, a problem that is not so much addressed in the education of the typical physics student as put aside en route to the more difficult problems that arise in quantum field theory. The equations governing the motion of a classical electron under the influence of its electromagnetic field have been discussed for a century, and continue to be actively investigated in the current literature, but it appears that a consistent approach to the problem from the point of view of modern renormalization theory has not previously been reported. I show that the methods of modern renormalization theory applied to the full Maxwell-Lorentz system provide a natural and intuitive derivation of the Lorentz-Dirac equation as an effective description of the electron motion valid for distances large compared to the classical electron radius. Moreover, a consistent treatment of the Lorentz...
Fillion-Gourdeau, F; Bandrauk, A D
2015-01-01
A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron-molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.
Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis
Fillion-Gourdeau, F., E-mail: filliong@CRM.UMontreal.ca [Université du Québec, INRS – Énergie, Matériaux et Télécommunications, Varennes, J3X 1S2 (Canada); Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4 (Canada); Lorin, E., E-mail: elorin@math.carleton.ca [School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6 (Canada); Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4 (Canada); Bandrauk, A.D., E-mail: andre.bandrauk@usherbrooke.ca [Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, J1K 2R1 (Canada); Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4 (Canada)
2016-02-15
A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron–molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.
Ghoumaid, A.; Benamira, F.; Guechi, L. [Laboratoire de Physique Théorique, Département de Physique, Faculté des Sciences Exactes, Université des Frères Mentouri, Constantine, Route d’Ain El Bey, Constantine (Algeria)
2016-02-15
It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given is incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions {sub 2}F{sub 1}(a, b, c; z). The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function.
A Proof Of Existence Of Particle-like Solutions Of Einstein Dirac Equations
Bird, E J
2005-01-01
We prove existence of a ground state particle-like solution to the Einstein-Dirac equations: a b ′= 1&parl0;r Ar&parr0; -wTr +m&parl0;A r&parr0; wT r- m&parl0; Ar&parr0; -1&parl0;rA r&parr0; ˙a b A′r= 1- Ar r- 16pwr T2r a2r+b 2r T′r=T rA r-1 2rAr -16pwT3r &parl0;a2r +b2r&parr0; 2rAr +32pT2r ab2r2 Ar +16pmT2r a2r -b2r 2rAr By a ground state particle-like solution we mean a solution of the above equations that satisfies the constraints: limr→∞ r1-Ar <∞lim r- ∞Tr <∞0∞ &parl0;a2r +b2r&parr0; TAdr<∞ .
Radial coherent and intelligent states of paraxial wave equation
Karimi, Ebrahim; 10.1364/OL.37.002484
2012-01-01
Ladder operators for the radial index of the paraxial optical modes in the cylindrical coordinates are calculated. The operators obey the su(1,1) algebra commutation relations. Based on this Lie algebra, we found that coherent modes constructed as eigenstates of the destruction operator or resulting from the action of the displacement operator on the fundamental mode are different. Some properties of these two kinds of radial coherent modes are studied in detail.
The many faces of Maxwell, Dirac and Einstein equations. A Clifford bundle approach
Rodrigues, W.A. Jr.; Oliveira, E.C. de
2007-07-01
This book is a thoughtful exposition of the algebra and calculus of differential forms, the Clifford and Spin-Clifford bundles formalisms with emphasis in calculation procedures, and vistas to a formulation of some important concepts of differential geometry necessary for a deep understanding of spacetime physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields, which were originally considered objects of a very different mathematical nature, are shown to have representatives as objects of the same mathematical nature, i.e. as sections of an appropriate Clifford bundle. This approach reveals unity in the diversity and also the many faces of the equations satisfied by those fields. Moreover, it suggests relationships which are hidden in the standard formalisms and new paths for research. Some foundational issues of relativistic field theories, in particular the one concerning the conditions for the existence of the conservation laws of energy-momentum and angular momentum in spacetime theories and many misconceptions concerning this issue is analyzed in details. The book will be useful as reference book for researchers and advanced students of theoretical physics and mathematics. Calculation procedures are illustrated by many exercises solved in detail, using the ''tricks of the trade''. Furthermore the readers will appreciate the comprehensive list of mathematical symbols as well as a list of acronyms and abbreviations. (orig.)
General solution of the Dirac equation for quasi-two-dimensional electrons
Eremko, Alexander, E-mail: eremko@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); Brizhik, Larissa, E-mail: brizhik@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); Loktev, Vadim, E-mail: vloktev@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); National Technical University of Ukraine “KPI”, Peremohy av., 37, Kyiv, 03056 (Ukraine)
2016-06-15
The general solution of the Dirac equation for quasi-two-dimensional electrons confined in an asymmetric quantum well, is found. The energy spectrum of such a system is exactly calculated using special unitary operator and is shown to depend on the electron spin polarization. This solution contains free parameters, whose variation continuously transforms one known particular solution into another. As an example, two different cases are considered in detail: electron in a deep and in a strongly asymmetric shallow quantum well. The effective mass renormalized by relativistic corrections and Bychkov–Rashba coefficients are analytically obtained for both cases. It is demonstrated that the general solution transforms to the particular solutions, found previously (Eremko et al., 2015) with the use of spin invariants. The general solution allows to establish conditions at which a specific (accompanied or non-accompanied by Rashba splitting) spin state can be realized. These results can prompt the ways to control the spin degree of freedom via the synthesis of spintronic heterostructures with the required properties.
Rivasseau, Vincent; Fuchs, Jean-Nöel
2017-01-01
This fifteenth volume of the Poincare Seminar Series, Dirac Matter, describes the surprising resurgence, as a low-energy effective theory of conducting electrons in many condensed matter systems, including graphene and topological insulators, of the famous equation originally invented by P.A.M. Dirac for relativistic quantum mechanics. In five highly pedagogical articles, as befits their origin in lectures to a broad scientific audience, this book explains why Dirac matters. Highlights include the detailed "Graphene and Relativistic Quantum Physics", written by the experimental pioneer, Philip Kim, and devoted to graphene, a form of carbon crystallized in a two-dimensional hexagonal lattice, from its discovery in 2004-2005 by the future Nobel prize winners Kostya Novoselov and Andre Geim to the so-called relativistic quantum Hall effect; the review entitled "Dirac Fermions in Condensed Matter and Beyond", written by two prominent theoreticians, Mark Goerbig and Gilles Montambaux, who consider many other mater...
Ovsiyuk, E M; Red'kov, V M
2010-01-01
There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in the space of constant positive curvature, spherical Riemann space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Riemann space geometry, is obtained.
Debergh, N M; Samsonov, B F; Van den Bossche, B
2002-01-01
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined through the knowledge of only two eigenfunctions of the first Dirac Hamiltonian. Moreover this operator together with its adjoint and the two Hamiltonians generate a quadratic deformation of the superalgebra subtending the usual supersymmetric quantum mechanics. Our developments are illustrated on the free particle case and the generalized Coulomb interaction. In the latter case, a relativistic counterpart of shape-invariance is observed.
Bednarcyk, Brett A.; Aboudi, Jacob; Arnold, Steven M.
2008-02-01
The radial return method is a well-known algorithm for integrating the classical plasticity equations. Mendelson presented an alternative method for integrating these equations in terms of the so-called plastic strain—total strain plasticity relations. In the present communication, it is shown that, although the two methods appear to be unrelated, they are actually equivalent. A table is provided demonstrating the step by step correspondence of the radial return and Mendelson algorithms in the case of isotropic hardening.
Mohammad Mehdi Mazarei
2012-01-01
Full Text Available This paper presents numerical solution of elliptic partial differential equations (Poisson's equation using a combination of logarithmic and multiquadric radial basis function networks. This method uses a special combination between logarithmic and multiquadric radial basis functions with a parameter r. Further, the condition number which arises in the process is discussed, and a comparison is made between them with our earlier studies and previously known ones. It is shown that the system is stable.
Altuğ Arda
2017-01-01
Full Text Available We find the exact bound state solutions and normalization constant for the Dirac equation with scalar-vector-pseudoscalar interaction terms for the generalized Hulthén potential in the case where we have a particular mass function m(x. We also search the solutions for the constant mass where the obtained results correspond to the ones when the Dirac equation has spin and pseudospin symmetry, respectively. After giving the obtained results for the nonrelativistic case, we search then the energy spectra and corresponding upper and lower components of Dirac spinor for the case of PT-symmetric forms of the present potential.
Hu, Huayu
2015-01-01
Nonperturbative calculation of QED processes participated by a strong electromagnetic field, especially provided by strong laser facilities at present and in the near future, generally resorts to the Furry picture with the usage of analytical solutions of the particle dynamical equation, such as the Klein-Gordon equation and Dirac equation. However only for limited field configurations such as a plane-wave field could the equations be solved analytically. Studies have shown significant interests in QED processes in a strong field composed of two counter-propagating laser waves, but the exact solutions in such a field is out of reach. In this paper, inspired by the observation of the structure of the solutions in a plane-wave field, we develop a new method and obtain the analytical solution for the Klein-Gordon equation and equivalently the action function of the solution for the Dirac equation in this field, under a largest dynamical parameter condition that there exists an inertial frame in which the particl...
Spin-orbit splittings in heavy-light mesons and Dirac equation
Riazuddin, [Quaid-i-Azam University Campus, National Centre for Physics, Islamabad (Pakistan); Shafiq, Sidra [National University of Science and Technology, Centre for Advance Mathematics and Physics, Islamabad (Pakistan)
2012-03-15
The spin-orbit splitting in heavy-light mesons is seen to be suppressed experimentally, which may be due to a relativistic dynamical symmetry for the Dirac Hamiltonian. An alternative derivation of such a symmetry is given. Furthermore, the dynamics necessary for a qualitative understanding of the spin-orbit splitting seen experimentally is discussed. (orig.)
A test of Lorentz-Dirac and Lienard-Wiechert equations
Comay, E.
1987-12-01
Gedanken experiments of two charges moving uniformly along a circle are used for testing both the Lorentz-Dirac radiation reaction force and the Lienard-Wiechert formulas of retarted potentials. It is shown that if some additional postulates hold then these expressions are acceptable only as low order approximations.
SINGULAR POSITIVE RADIAL SOLUTIONS FOR A GENERAL SEMILINEAR ELLIPTIC EQUATION
Yang Fen
2012-01-01
The existence and uniqueness of singular solutions decaying like r-m (see (1.4)) of the equation △u+ kΣi=1ci|x|liupi =0,x∈Rn (0.1) are obtained,where n≥3,ci ＞0,li ＞-2,i=1,2,…,k,pi＞1,i=1,2,…,k and the separation structure of singular solutions decaying like r-(n-2) of eq.(0.1) are discussed.moreover,we obtain thc explicit critical exponent ps(l) (see (1.9)).
Dirac equation with spin symmetry for the modiﬁed Pöschl–Teller potential in dimensions
D Agboola
2011-06-01
We present solutions of the Dirac equation with spin symmetry for vector and scalar modiﬁed Pöschl–Teller potentials within the framework of an approximation of the centrifugal term. The relativistic energy spectrum is obtained using the Nikiforov–Uvarov method and the two-component spinor wave functions obtained are in terms of the Jacobi polynomials. It is found that there exist only positive energy states for bound states under spin symmetry, and the energy of a level with ﬁxed value of , increases with increase in dimension of space time and the potential range parameter .
Gromes, Dieter
2015-01-01
We calculate the energy radiated coherently by a system of $N$ charged non relativistic particles. It disagrees with the energy loss which is obtained if one employs the Lorentz Abraham Dirac (LAD) equation for each particle, and sums up the contributions. This fact was already clearly stated in the classical literature long ago. The reason for the discrepancy is the omission of the mixing terms in the Poynting vector. For some simple systems we present a generalized equation for the radiation reaction force which cures this defect. The counter examples show that the LAD equation cannot be generally valid and that all "proofs" must fail somewhere. We demonstrate this failure for some popular examples in the literature.
The many faces of Maxwell, Dirac and Einstein equations a Clifford bundle approach
Rodrigues, Jr, Waldyr A
2016-01-01
This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research. This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields. The exercises with solut...
Consistency of multi-time Dirac equations with general interaction potentials
Deckert, Dirk-André; Nickel, Lukas
2016-07-01
In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac's multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.
A Partially-ordered-set Based Approach to the Dirac Equation in 3+1 space-time
Earle, Keith; Knuth, Kevin
2012-02-01
Recent work by Knuth and co-workers has shown how insights into Einstein's Theory of Special Relativity may be obtained by careful reasoning about consistent quantification of a poset. The Feynman Chessboard problem in 1+1 spacetime can be treated from this perspective, for example. Alternative methods of solution based on techniques borrowed from statistical mechanics have also been developed over the years to solve the Feynman Chessboard model in 1+1 spacetime. One particularly intriguing solution is based on a master-equation approach developed by McKeon and Ord for 1+1 spacetime. We will show how this model may be extended to 3+1 spacetime using techniques developed by Bialynicki-Birula, thus providing an alternative derivation of the Dirac equation. An external electromagnetic field can be accommodated very naturally in the formalism from which a pleasing pictorial representation of electromagnetic interactions in the lattice picture emerges.
A novel method for analytically solving a radial advection-dispersion equation
Lai, Keng-Hsin; Liu, Chen-Wuing; Liang, Ching-Ping; Chen, Jui-Sheng; Sie, Bing-Ruei
2016-11-01
An analytical solution for solute transport in a radial flow field has a variety of practical applications in the study of the transport in push-pull/divergent/convergent flow tracer tests, aquifer remediation by pumping and aquifer storage and recovery. However, an analytical solution for radial advective-dispersive transport has been proven very difficult to develop and relatively few in subsurface hydrology have made efforts to do so, because variable coefficients in the governing partial differential equations. Most of the solutions for radial advective-dispersive transport presented in the literature have generally been solved semi-analytically with the final concentration values being obtained with the help of a numerical Laplace inversion. This study presents a novel solution strategy for analytically solving the radial advective-dispersive transport problem. A Laplace transform with respect to the time variable and a generalized integral transform technique with respect to the spatial variable are first performed to convert the transient governing partial differential equations into an algebraic equation. Subsequently, the algebraic equation is solved using simple algebraic manipulations, easily yielding the solution in the transformed domain. The solution in the original domain is ultimately obtained by successive applications of the Laplace and corresponding generalized integral transform inversions. A convergent flow tracer test is used to demonstrate the robustness of the proposed method for deriving an exact analytical solution to the radial advective-dispersive transport problem. The developed analytical solution is verified against a semi-analytical solution taken from the literature. The results show perfect agreement between our exact analytical solution and the semi-analytical solution. The solution method presented in this study can be applied to create more comprehensive analytical models for a great variety of radial advective
Ozturk, Okkes; Yilmazer, Resat
2017-07-01
One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus (DFC) has also an important position in the fractional calculus. The nabla operator in DFC is practical for the singular differential equations. The purpose of this study is to obtain particular solutions of the radial Schrödinger equation (that is, the most important equation of quantum physics) via nabla DFC operator. These solutions were obtained in the forms of discrete fractional.
Crater, Horace; Yang, Dujiu
1991-09-01
A semirelativistic expansion in powers of 1/c2 is canonically matched through order (1/c4) of the two-particle total Hamiltonian of Wheeler-Feynman vector and scalar electrodynamics to a similar expansion of the center of momentum (c.m.) total energy of two interacting particles obtained from covariant generalized mass shell constraints derived with the use of the classical Todorov equation and Dirac's Hamiltonian constraint mechanics. This determines through order 1/c4 the direct interaction used in the covariant Todorov constraint equation. We show that these interactions are momentum independent in spite of the extensive and complicated momentum dependence of the potential energy terms in the Wheeler-Feynman Hamiltonian. The invariant expressions for the relativistic reduced mass and energy of the fictitious particle of relative motion used in the Todorov equation are also dynamically determined through this order by this same procedure. The resultant covariant Todorov equation then not only reproduces the noncovariant Wheeler-Feynman dynamics through order 1/c4 but also implicitly provides a rather simple covariant extrapolation of it to all orders of 1/c2.
Fang, Zhi; Shi, Min; Guo, Jian-You; Niu, Zhong-Ming; Liang, Haozhao; Zhang, Shi-Sheng
2017-02-01
Resonances play critical roles in the formation of many physical phenomena, and many techniques have been developed for the exploration of resonances. In a recent letter [Phys. Rev. Lett. 117, 062502 (2016), 10.1103/PhysRevLett.117.062502], we proposed a new method for probing single-particle resonances by solving the Dirac equation in complex momentum representation for spherical nuclei. Here, we develop the theoretical formalism of this method for deformed nuclei. We elaborate numerical details and calculate the bound and resonant states in 37Mg. The results are compared with those from the coordinate representation calculations with a satisfactory agreement. In particular, the present method can expose clearly the resonant states in a complex momentum plane and determine precisely the resonance parameters for not only narrow resonances but also broad resonances that were difficult to obtain before.
Hall, Richard L.; Zorin, Petr
2016-04-01
The classic comparison theorem of quantum mechanics states that if two potentials are ordered then the corresponding energy eigenvalues are similarly ordered, that is to say if Va≤ Vb, then Ea≤ Eb. Such theorems have recently been established for relativistic problems even though the discrete spectra are not easily characterized variationally. In this paper we improve on the basic comparison theorem for the Dirac equation with spin and pseudo-spin symmetry in d≥ 1 dimensions. The graphs of two comparison potentials may now cross each other in a prescribed manner implying that the energy values are still ordered. The refined comparison theorems are valid for the ground state in one dimension and for the bottom of an angular momentum subspace in d>1 dimensions. For instance in a simplest case in one dimension, the condition Va≤ Vb is replaced by Ua≤ Ub, where Ui(x)=int0x Vi(t)dt, xin[0,∞), and i=a or b.
Mapping curved spacetimes into Dirac spinors
Sabín, Carlos
2016-01-01
We show how to transform a Dirac equation in curved spacetime into a Dirac equation in flat spacetime. In particular, we show that any solution of the free massless Dirac equation in a 1+1 dimensional flat spacetime can be transformed via a local phase transformation into a solution of the corresponding Dirac equation in a curved background, where the spacetime metric is encoded into the phase. In this way, the existing quantum simulators of the Dirac equation can naturally incorporate curved spacetimes. As a first example we use our technique to obtain solutions of the Dirac equation in a particular family of interesting spacetimes in 1+1 dimensions.
Benchmark values for molecular three-center integrals arising in Dirac equation
Bagci, A
2015-01-01
The authors in their previous papers obtained compact, arbitrarily accurate expressions for two center one and two electron Dirac integrals expressed over Slater orbitals with noninteger principal quantum numbers. In this present study, the accuracy limits of given expressions is examined for three-center nuclear attraction integrals, which are the first integral set do not have analytically closed form relations. They are expressed through new molecular auxiliary functions obtained via Neumann expansion of Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrarily values of orbital parameters, principal quantum numbers in Slater orbitals. Several methods, such as Laplace expansion of Coulomb interaction, single center expansions, Fourier transformation methods, have been developed for evaluation of three-center integrals when integer principal quantum number is used. This is the first attempts to study the three-center integrals with noninteger Slater orbita...
Quiney, H. M.; Glushkov, V. N.; Wilson, S.; Sabin,; Brandas, E
2001-01-01
A comparison is made of the accuracy achieved in finite difference and finite basis set approximations to the Dirac equation for the ground state of the hydrogen molecular ion. The finite basis set calculations are carried out using a distributed basis set of Gaussian functions the exponents and pos
Quiney, H. M.; Glushkov, V. N.; Wilson, S.; Sabin,; Brandas, E
2001-01-01
A comparison is made of the accuracy achieved in finite difference and finite basis set approximations to the Dirac equation for the ground state of the hydrogen molecular ion. The finite basis set calculations are carried out using a distributed basis set of Gaussian functions the exponents and
ON THE RADIAL GROUND STATE OFP-LAPLACIAN EQUATION WITH GRADIENT TERM PERTURBATION
无
2000-01-01
In this paper,authors consider the existence,uniqueness and nonexistence of the radial ground state to the following p-Laplacian equation:△pu+uq-|Dulσ=0,x ∈Rn,where 2≤p
ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA
Wu Haigen; Zhang Junyong
2012-01-01
In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d ≥ 3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0 ＜ γ ≤ 4 and γ＜ d with Hartree potential V(x) =|x|-γ.
Quantum-classical correspondence of the Dirac equation with a scalar-like potential
Mai-Lin Liang; Shun-Lin Shu; Bing Yuan
2009-05-01
Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein–Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.
Photoconductivity in Dirac materials
J. M. Shao
2015-11-01
Full Text Available Two-dimensional (2D Dirac materials including graphene and the surface of a three-dimensional (3D topological insulator, and 3D Dirac materials including 3D Dirac semimetal and Weyl semimetal have attracted great attention due to their linear Dirac nodes and exotic properties. Here, we use the Fermi’s golden rule and Boltzmann equation within the relaxation time approximation to study and compare the photoconductivity of Dirac materials under different far- or mid-infrared irradiation. Theoretical results show that the photoconductivity exhibits the anisotropic property under the polarized irradiation, but the anisotropic strength is different between 2D and 3D Dirac materials. The photoconductivity depends strongly on the relaxation time for different scattering mechanism, just like the dark conductivity.
Photoconductivity in Dirac materials
Shao, J. M.; Yang, G. W., E-mail: stsygw@mail.sysu.edu.cn [State Key Laboratory of Optoelectronic Materials and Technologies, Nanotechnology Research Center, School of Materials & Engineering, School of Physics & Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong (China)
2015-11-15
Two-dimensional (2D) Dirac materials including graphene and the surface of a three-dimensional (3D) topological insulator, and 3D Dirac materials including 3D Dirac semimetal and Weyl semimetal have attracted great attention due to their linear Dirac nodes and exotic properties. Here, we use the Fermi’s golden rule and Boltzmann equation within the relaxation time approximation to study and compare the photoconductivity of Dirac materials under different far- or mid-infrared irradiation. Theoretical results show that the photoconductivity exhibits the anisotropic property under the polarized irradiation, but the anisotropic strength is different between 2D and 3D Dirac materials. The photoconductivity depends strongly on the relaxation time for different scattering mechanism, just like the dark conductivity.
Point-particle effective field theory III: relativistic fermions and the Dirac equation
Burgess, C. P.; Hayman, Peter; Rummel, Markus; Zalavári, László
2017-09-01
We formulate point-particle effective field theory (PPEFT) for relativistic spin-half fermions interacting with a massive, charged finite-sized source using a first-quantized effective field theory for the heavy compact object and a second-quantized language for the lighter fermion with which it interacts. This description shows how to determine the near-source boundary condition for the Dirac field in terms of the relevant physical properties of the source, and reduces to the standard choices in the limit of a point source. Using a first-quantized effective description is appropriate when the compact object is sufficiently heavy, and is simpler than (though equivalent to) the effective theory that treats the compact source in a second-quantized way. As an application we use the PPEFT to parameterize the leading energy shift for the bound energy levels due to finite-sized source effects in a model-independent way, allowing these effects to be fit in precision measurements. Besides capturing finite-source-size effects, the PPEFT treatment also efficiently captures how other short-distance source interactions can shift bound-state energy levels, such as due to vacuum polarization (through the Uehling potential) or strong interactions for Coulomb bound states of hadrons, or any hypothetical new short-range forces sourced by nuclei.
A Simple General Solution of the Radial Schrodinger Equation for Spherically Symmetric Potentials
Erbil, H H
2003-01-01
By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
Sharma, Anushrut
2014-01-01
It is well-known that in the Newman-Penrose formalism the Riemann tensor can be expressed as a set of eighteen complex first-order equations, in terms of the twelve spin coefficients, known as Ricci identities. The Ricci tensor herein is determined via the Einstein equations. It is also known that the Dirac equation in a curved spacetime can be written in the Newman-Penrose formalism as a set of four first-order coupled equations for the spinor components of the wave-function. In the present article we suggest that it might be possible to think of the Dirac equations in the N-P formalism as a special case of the Ricci identities, after an appropriate identification of the four Dirac spinor components with four of the spin coefficients, provided torsion is included in the connection, and after a suitable generalization of the energy-momentum tensor. We briefly comment on similarities with the Einstein-Cartan-Sciama-Kibble theory. The motivation for this study is to take some very preliminary steps towards deve...
The HEUN-SCHRÖDINGER Radial Equation for Dh-Atoms
Tarasov, V. F.
This article deals with the connection between Schrödinger's multidimensional equation for DH-atoms (D≥1) and the confluent Heun equation with two auxiliary parameters ν and τ, where |1-ν| = o(1) and τ∈ℚ+, which influence the spectrum of eigenvalues, the Coulomb potential and the radial function. The case τ = ν = 1 and D = 3 corresponds to the "standard" form of Schrödinger's equation for a 3H-atom. With the help of parameter ν, e.g., some "quantum corrections" may be considered. The cases 01, but â = (n-l-1)τ≥0 is an integer, change the "geometry" of the electron cloud in the atom, i.e. the so-called "exotic" 3H-like atoms arise, where Kummer's function 1F1(-â c; z) has â zeros and the discrete spectrum depends only on Z/(νn) but not on l and τ. Diagrams of the radial functions hat Pnl(r;τ ,ν ) as n≤3 are given.
Hawking radiation of charged Dirac particles in Vaidya-Bonner space-time
朱建阳; 张建华; 赵峥
1995-01-01
The dynamical properties of charged Dirac spinor particles in the Vaidya-Bonner space-time are investigated. The asymptotic solution to the radial part of the charged Dirac equation near the event horizon of the black hole is obtained. The Hawking temperature and the event horizon of the charged evaporating black hole, as well as the spectrum of the Hawking radiation of the Dirac particles, are exactly shown. Thereby, a new approach to the back-reaction of radiation from the non-stationary black holes is established.
Kononets, Yu. V.
2016-12-01
The presented enhanced version of Eriksen's theorem defines an universal transform of the Foldy-Wouthuysen type and in any external static electromagnetic field (ESEMF) reveals a discrete symmetry of Dirac's equation (DE), responsible for existence of a highly influential conserved quantum number—the charge index distinguishing two branches of DE spectrum. It launches the charge-index formalism (CIF) obeying the charge-index conservation law (CICL). Via its unique ability to manipulate each spectrum branch independently, the CIF creates a perfect charge-symmetric architecture of Dirac's quantum mechanics (DQM), which resolves all the riddles of the standard DE theory (SDET). Besides the abstract CIF algebra, the paper discusses: (1) the novel accurate charge-symmetric definition of the electric-current density; (2) DE in the true-particle representation, where electrons and positrons coexist on equal footing; (3) flawless "natural" scheme of second quantization; and (4) new physical grounds for the Fermi-Dirac statistics. As a fundamental quantum law, the CICL originates from the kinetic-energy sign conservation and leads to a novel single-particle physics in strong-field situations. Prohibiting Klein's tunneling (KT) in Klein's zone via the CICL, the precise CIF algebra defines a new class of weakly singular DE solutions, strictly confined in the coordinate space and experiencing the total reflection from the potential barrier.
Yuan Hongjun; Jin Yang
2005-01-01
The aim of this paper is to discuss the existence and uniqueness of solutions for the porous medium equation ut - (um)xx = μ(x) in (x,t) ∈ R × (0, +∞) with initial condition u(x, 0) = uo(x) x ∈ (-∞, +∞),whereμ(x) is a nonnegative finite Radon measure, u0 ∈ L1 (R)∩L∞ (R) is a nonnegative function, and m ＞ 1, and R ≡ (-∞, +∞).
Radial pulsations and stability of anisotropic stars with quasi-local equation of state
Horvat, Dubravko; Marunovic, Anja
2010-01-01
Quasi-local variables, i.e. quantities whose values can be derived from physics accessible within an arbitrarily small neighborhood of a spacetime point, are used to construct the equation of state for the anisotropic fluid in spherical symmetry. One parameter families of equilibrium solutions are obtained making it possible to assess stability properties by means of the standard M(R) method. Normal modes of radial pulsation are computed as well and are found to confirm the onset of instability as predicted by the M(R) method. As an example, a stable configuration with outwardly increasing energy density in the core is obtained with a simple quasi-local extension of the polytropic equation of state. It is also found that the loss of stability occurs at higher surface compactness when the anisotropy of pressures is present.
Numerical solution of differential equations using multiquadric radial basis functions networks.
Mai-Duy, N; Tran-Cong, T
2001-03-01
This paper presents mesh-free procedures for solving linear differential equations (ODEs and elliptic PDEs) based on multiquadric (MQ) radial basis function networks (RBFNs). Based on our study of approximation of function and its derivatives using RBFNs that was reported in an earlier paper (Mai-Duy, N. & Tran-Cong, T. (1999). Approximation of function and its derivatives using radial basis function networks. Neural networks, submitted), new RBFN approximation procedures are developed in this paper for solving DEs, which can also be classified into two types: a direct (DRBFN) and an indirect (IRBFN) RBFN procedure. In the present procedures, the width of the RBFs is the only adjustable parameter according to a(i) = betad(i), where d(i) is the distance from the ith centre to the nearest centre. The IRBFN method is more accurate than the DRBFN one and experience so far shows that beta can be chosen in the range 7 < or = beta 10 for the former. Different combinations of RBF centres and collocation points (uniformly and randomly distributed) are tested on both regularly and irregularly shaped domains. The results for a 1D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0 x 10(-4)) and O(1.0 x 10(-8)), respectively, with a centre density of 50. Similarly, the results for a 2D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0 x 10(-3)) and O(1.0 x10(-6)) respectively, with a centre density of 12 X 12.
Anderson's absolute objects and constant timelike vector hidden in Dirac matrices
2001-01-01
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute objects of the Dirac equation. There are two ways of the Dirac matrices transformation: (1) Dirac matrices form a 4-vector and wave function is a scalar, (2) Dirac matrices are scalars and the wave function is a spinor. In the first case the Dirac equation ...
The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations
Wei, Changhua
2017-10-01
This paper investigates the lower bound of the lifespan of three-dimensional spherically symmetric solutions to the non-isentropic relativistic Euler equations, when the initial data are prescribed as a small perturbation with compact support to a constant state. Based on the structure of the hyperbolic system, we show the almost global existence of the smooth solutions to Eulerian flows (polytropic gases and generalized Chaplygin gases) with genuinely nonlinear characteristics. While for the Eulerian flows (Chaplygin gas and stiff matter) with mild linearly degenerate characteristics, we show the global existence of the radial solutions, moreover, for the non-strictly hyperbolic system (pressureless perfect fluid) satisfying the mild linearly degenerate condition, we prove the blowup phenomenon of the radial solutions and show that the lifespan of the solutions is of order O(ɛ ^{-1}), where ɛ denotes the width of the perturbation. This work can be seen as a complement of our work (Lei and Wei in Math Ann 367:1363-1401, 2017) for relativistic Chaplygin gas and can also be seen as a generalization of the classical Eulerian fluids (Godin in Arch Ration Mech Anal 177:497-511, 2005, J Math Pures Appl 87:91-117, 2007) to the relativistic Eulerian fluids.
量子与经典对应：Dirac方程中的速度算符%Quantum and classical correspondence for velocity operator in Dirac equation
张治国; 封文江; 郑伟; 陈皓; 崔崧
2016-01-01
Due to Bohr correspondence principle in the quantum mechanics,quantum mechanics go to classical mechanics in the case of large quantum number. Based upon Heisenberg correspondence principle, quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Using Heisenberg correspondence principle, quantum-classical correspondence of the relativistic free particle and the 1/2 spin charged particles in a constant magnetic field are studied.Applying Heisenberg correspondence principle to Dirac equation in relativistic realm,the operator of free particle in the Dirac theory and quantum-classical correspondence are obtained,and 1/2 spin charged particle in a constant magnetic field in Dirac equation is also studied.For the relativistic free particle or the charged particle in a constant magnetic field,the velocity operator in the Dirac theory will reduce to the classical velocity.%由量子力学中的Bohr对应原理可知,在大量子数情形下,量子力学应过渡到经典力学。在经典极限下,由 Heisenberg 对应原理可知,厄密算符的量子矩阵元对应经典物理量的Fourier展开系数。利用 Heisenberg对应原理研究相对论效应的自由粒子和在匀磁场中的带电粒子的量子经典对应问题。将 Heisenberg对应原理应用到相对论领域的 Dirac 方程,计算出自由粒子的Dirac方程中的α算符及其经典近似,并且研究自旋1/2的带电粒子在匀磁场中的Dirac 方程情形。对于相对论效应的自由粒子和在匀磁场中的带电粒子,Dirac理论中的α算符将对应经典的速度。
Positive radially symmetric solution for a system of quasilinear biharmonic equations in the plane
Joshua Barrow
2015-01-01
Full Text Available We study the boundary value system for the two-dimensional quasilinear biharmonic equations $$\\displaylines{ \\Delta (|\\Delta u_i|^{p-2}\\Delta u_i=\\lambda_iw_i(xf_i(u_1,\\ldots,u_m,\\quad x\\in B_1,\\cr u_i=\\Delta u_i=0,\\quad x\\in\\partial B_1,\\quad i=1,\\ldots,m, }$$ where $B_1=\\{x\\in\\mathbb{R}^2:|x|<1\\}$. Under some suitable conditions on $w_i$ and $f_i$, we discuss the existence, uniqueness, and dependence of positive radially symmetric solutions on the parameters $\\lambda_1,\\ldots,\\lambda_m$. Moreover, two sequences are constructed so that they converge uniformly to the unique solution of the problem. An application to a special problem is also presented.
A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential
Craig Haile
2000-01-01
Full Text Available We study the time-independent Schrodinger equation with radially symmetric potential $k|x|^alpha$, $k ge 0$, $k in mathbb{R}, alpha ge 2$ on a bounded domain $Omega$ in $mathbb{R}^n$, $(n ge 2$ with Dirichlet boundary conditions. In particular, we compare the eigenvalue $lambda_2(Omega$ of the operator $-Delta + k |x|^alpha $ on $Omega$ with the eigenvalue $lambda_2(S_1$ of the same operator $-Delta +kr^alpha$ on a ball $S_1$, where $S_1$ has radius such that the first eigenvalues are the same ($lambda_1(Omega = lambda_1(S_1$. The main result is to show $lambda_2(Omega le lambda_2(S_1$. We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain $Omega$ with Dirichlet boundary conditions.
Barik, N.; Das, M.
1983-12-01
The effect of confinement on the magnetic moment of a quark has been studied in a simple independent-quark model based on the Dirac equation with a power-law potential. The magnetic moments so obtained for the constituent quarks, which are found to be significantly different from their corresponding Dirac moments, are used in predicting the magnetic moments of baryons in the nucleon octet as well as those in the charmed and b-flavored sectors. We not only get an improved result for the proton magnetic moment, but the calculation for the rest of the nucleon octet also turns out to be in reasonable agreement with experiment. The overall predictions for the charmed and b-flavored baryons are also comparable with other model predictions.
Bohr-Sommerfeld quantization condition for Dirac states derived from an Ermakov-type invariant
Thylwe, Karl-Erik [KTH-Mechanics, Royal Institute of Technology, S-10044 Stockholm (Sweden); McCabe, Patrick [CCDC, 12 Union Road, CB2 1EZ Cambridge (United Kingdom)
2013-05-15
It is shown that solutions of the second-order decoupled radial Dirac equations satisfy Ermakov-type invariants. These invariants lead to amplitude-phase-type representations of the radial spinor solutions, with exact relations between their amplitudes and phases. Implications leading to a Bohr-Sommerfeld quantization condition for bound states, and a few particular atomic/ionic and nuclear/hadronic bound-state situations are discussed.
Pseudospin symmetry in the Dirac phenomenology
Marcos, S.; Niembro, R. [Universidad de Cantabria, Departamento de Fisica Moderna, Santander (Spain); Lopez-Quelle, M. [Universidad de Cantabria, Departamento de Fisica Aplicada, Santander (Spain); Savushkin, L.N. [St. Petersburg University for Telecommunications, Department of Physics, St. Petersburg (Russian Federation)
2007-12-15
In the phenomenological relativistic framework of the Dirac equation for spherical nuclei, we use different kinds of single-particle central potentials ({sigma}{sub S}+{sigma}{sub 0}) to investigate certain aspects of the spin and pseudospin (PS) symmetries. Neither the splitting of PS doublets (PSDs) nor the similarity of the radial parts of the small components (F/r) of the corresponding Dirac spinors have been found related with the magnitude of {sigma}{sub S}+{sigma}{sub 0}, in the sense predicted by several authors in the last decade. This conclusion is shown to be valid, in particular, for a potential of Coulomb type. We give a simple explanation for the strong correlation established in the relativistic calculations between the similarity of the radial parts of the big (small) components of the Dirac spinors of two spin (pseudospin) partners and the number of their nodes. The direct effects of the so-called PS symmetry-breaking term (and its singularity point) on the F functions of the PSDs are also analysed. (orig.)
ON THE RADIAL GROUND STATE OF P-LAPLACIAN EQUATION INVOLVING SUPER-CRITICAL OR CRITICAL EXPONENTS
Xuan Benjin; Chen Zuchi
2000-01-01
In this paper, we consider the existence and uniqueness of the radial ground state to the following p-Laplacian equation involving super-critical or critical exponents: Δpu + uq - |Du|σ = 0, x ∈ Rn, 2 ＜ p ＜ n, q _＞ [n(p - 1) + p]/(n - p), σ ＞ 0. Applying the shooting argument, the Schauder's fixed point theorem and some delicate estimates of auxiliary functions, we study the influence of the parameters n, p, q, σ on the existence and uniqueness of the radial ground state to the above p-Laplacian equation.
Wang, Zhiheng
2014-12-10
A meshless local radial basis function method is developed for two-dimensional incompressible Navier-Stokes equations. The distributed nodes used to store the variables are obtained by the philosophy of an unstructured mesh, which results in two main advantages of the method. One is that the unstructured nodes generation in the computational domain is quite simple, without much concern about the mesh quality; the other is that the localization of the obtained collocations for the discretization of equations is performed conveniently with the supporting nodes. The algebraic system is solved by a semi-implicit pseudo-time method, in which the convective and source terms are explicitly marched by the Runge-Kutta method, and the diffusive terms are implicitly solved. The proposed method is validated by several benchmark problems, including natural convection in a square cavity, the lid-driven cavity flow, and the natural convection in a square cavity containing a circular cylinder, and very good agreement with the existing results are obtained.
Tunnelling of scalar and Dirac particles from squashed charged rotating Kaluza-Klein black holes
Stetsko, M.M. [Ivan Franko National University of Lviv, Department of Theoretical Physics, Lviv (Ukraine)
2016-02-15
The thermal radiation of scalar particles and Dirac fermions from squashed charged rotating five-dimensional black holes is considered. To obtain the temperature of the black holes we use the tunnelling method. In the case of scalar particles we make use of the Hamilton-Jacobi equation. To consider tunnelling of fermions the Dirac equation was investigated. The examination shows that the radial parts of the action for scalar particles and fermions in the quasi-classical limit in the vicinity of horizon are almost the same and as a consequence it gives rise to identical expressions for the temperature in the two cases. (orig.)
Tunnelling of scalar and Dirac particles from squashed charged rotating Kaluza-Klein black holes
Stetsko, M. M.
2016-02-01
The thermal radiation of scalar particles and Dirac fermions from squashed charged rotating five-dimensional black holes is considered. To obtain the temperature of the black holes we use the tunnelling method. In the case of scalar particles we make use of the Hamilton-Jacobi equation. To consider tunnelling of fermions the Dirac equation was investigated. The examination shows that the radial parts of the action for scalar particles and fermions in the quasi-classical limit in the vicinity of horizon are almost the same and as a consequence it gives rise to identical expressions for the temperature in the two cases.
Menculini, L; Roy, P
2013-01-01
We study the (2+1) dimensional Dirac equation in an homogeneous magnetic field (relativistic Landau problem) within a minimal length, or generalized uncertainty principle -GUP-, scenario. We derive exact solutions for a given explicit representation of the GUP and provide expressions of the wave functions in the momentum representation. We find that in the minimal length case the degeneracy of the states is modified and that there are states that do not exist in the ordinary quantum mechanics limit (\\beta -->0). We also discuss the mass-less case which may find application in describing the behavior of charged fermions in new materials like Graphene.
Sochichiu, Corneliu
2011-01-01
We study the emergence of Dirac fermionic field in the low energy description of non-relativistic dynamical network models. The Dirac fermionic field appears as the effective field describing the excitations above point-like Fermi levels. Together with the Dirac fermionic field an effective space-time metric is also emerging. We analyze the conditions for such Fermi points to appear in general, paying special attention to the case of two and three spacial dimensions.
赵静; 曲晓英
2007-01-01
精确求解了N-维无限深球势阱中的Klein-Gordon方程和Dirac方程,结果表明:在N-维无限深球势阱中,Klein-Gordon方程和Dirac方程的径向方程在形式上与非相对论中的三维中心场的径向方程一致,均为贝塞尔方程.通过求解Bessel方程,任意束缚态的本征函数已被获得,其解可用通常的球贝塞尔函数表示.利用径向波函数在r=a处的连续性条件,其相应的能谱公式也被发现.对于Klein-Gordon方程:E2nr,l'=m2+x2nr,l'/a2,而对于Dirac方程,则E2nr,l'=-m2+(√m2a2+x2nr,l'/a2).%The Klein-Gordon equation and Dirac equation of spherical potential well of infinite depth have been solved exactly in N-dimensions spaces.The results show that for spherical potential well of infinite depth,the radial equations of Klein-Gordon equation and Dirac equation are similar to the radial equation of three-dimensional schrdinger equation with spherical potential well of infinite depth.Through solving the Bessel equation,the radial wave function that can be expressed by the usual Bessel spherical function has been obtained for all bound states.The corresponding energy spectrum formulas (namely,E2nr,l'=m2+x2nr,l'/a2 and E2nr,l'=-m2+(√m2a2+x2nr,l'/a2)) also has been found by making use of the continuity of the radial wave function at r=a.
Maxwell and Dirac theories as an already unified theory
1995-01-01
In this paper we formulate Maxwell and Dirac theories as an already unified theory (in the sense of Misner and Wheeler). We introduce Dirac spinors as "Dirac square root" of the Faraday bivector, and use this in order to find a spinorial representation of Maxwell equations. Then we show that under certain circunstances this spinor equation reduces to an equation formally identical to Dirac equation. Finally we discuss certain conditions under which this equation can be really interpreted as D...
The Dirac particle on central backgrounds and the anti-de Sitter oscillator
Cotaescu, I I
1998-01-01
It is shown that, for spherically symmetric static backgrounds, a simple reduced Dirac equation can be obtained by using the Cartesian tetrad gauge in Cartesian holonomic coordinates. This equation is manifestly covariant under rotations so that the spherical coordinates can be separated in terms of angular spinors like in special relativity, obtaining a pair of radial equations and a specific form of the radial scalar product. As an example, we analytically solve the anti-de Sitter oscillator giving the formula of the energy levels and the form of the corresponding eigenspinors.
Strength and equation of state of NaCl from radial x-ray diffraction
Xiong, Lun; Bai, Ligang; Liu, Jing
2014-01-01
The strength and equation of state of NaCl were determined under nonhydrostatic compression up to 27 GPa using an energy-dispersive radial x-ray diffraction technique in a diamond-anvil cell using the lattice strain theory. Together with estimation of the high-pressure shear modulus, it is suggested that NaCl could support a maximum differential stress of 0.980 GPa at 22.6 GPa under uniaxial compression. The differential stress rapidly drops at 27.2 GPa due to the phase transition from B1 phase to B2 phase for NaCl. The hydrostatic compression data of B1 phase yield a bulk modulus K0 = 25.6(8) GPa and its pressure derivative K0' = 5.16(20) using Pt pressure scale. In addition, a comparative study of the observed pressures from Pt scale and ruby-fluorescence scale shows that the ruby-fluorescence pressures may reflect the lower stress state under nonhydrostatic compression compared with hydrostatic compression.
Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations
Flyer, Natasha; Barnett, Gregory A.; Wicker, Louis J.
2016-07-01
Polynomials are used together with polyharmonic spline (PHS) radial basis functions (RBFs) to create local RBF-finite-difference (RBF-FD) weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences (FD). The presented method replicates the convergence properties of FD but for arbitrary node layouts. It is tested on the 2D compressible Navier-Stokes equations at low Mach number, relevant to atmospheric flows. Test cases are taken from the numerical weather prediction community and solved on bounded domains. Thus, attention is given on how to handle boundaries with the RBF-FD method, as well as a novel implementation for hyperviscosity. Comparisons are done on Cartesian, hexagonal, and quasi-uniform node layouts. Consideration and guidelines are given on PHS order, polynomial degree and stencil size. The main advantages of the present method are: 1) capturing the basic physics of the problem surprisingly well, even at very coarse resolutions, 2) high-order accuracy without the need of tuning a shape parameter, and 3) the inclusion of polynomials eliminates stagnation (saturation) errors. A MATLAB code is given to calculate the differentiation weights for this novel approach.
Strength and equation of state of NaCl from radial x-ray diffraction
Xiong, Lun; Liu, Jing, E-mail: liuj@ihep.ac.cn [Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 (China); Bai, Ligang [Department of Physics and Astronomy, University of Nevada Las Vegas and High Pressure Science and Engineering Center (HiPSEC), Las Vegas, Nevada 89154-4002 (United States)
2014-01-21
The strength and equation of state of NaCl were determined under nonhydrostatic compression up to 27 GPa using an energy-dispersive radial x-ray diffraction technique in a diamond-anvil cell using the lattice strain theory. Together with estimation of the high-pressure shear modulus, it is suggested that NaCl could support a maximum differential stress of 0.980 GPa at 22.6 GPa under uniaxial compression. The differential stress rapidly drops at 27.2 GPa due to the phase transition from B1 phase to B2 phase for NaCl. The hydrostatic compression data of B1 phase yield a bulk modulus K{sub 0} = 25.6(8) GPa and its pressure derivative K{sub 0}′ = 5.16(20) using Pt pressure scale. In addition, a comparative study of the observed pressures from Pt scale and ruby-fluorescence scale shows that the ruby-fluorescence pressures may reflect the lower stress state under nonhydrostatic compression compared with hydrostatic compression.
Parameter estimation for stiff equations of biosystems using radial basis function networks
Sugimoto Masahiro
2006-04-01
Full Text Available Abstract Background The modeling of dynamic systems requires estimating kinetic parameters from experimentally measured time-courses. Conventional global optimization methods used for parameter estimation, e.g. genetic algorithms (GA, consume enormous computational time because they require iterative numerical integrations for differential equations. When the target model is stiff, the computational time for reaching a solution increases further. Results In an attempt to solve this problem, we explored a learning technique that uses radial basis function networks (RBFN to achieve a parameter estimation for biochemical models. RBFN reduce the number of numerical integrations by replacing derivatives with slopes derived from the distribution of searching points. To introduce a slight search bias, we implemented additional data selection using a GA that searches data-sparse areas at low computational cost. In addition, we adopted logarithmic transformation that smoothes the fitness surface to obtain a solution simply. We conducted numerical experiments to validate our methods and compared the results with those obtained by GA. We found that the calculation time decreased by more than 50% and the convergence rate increased from 60% to 90%. Conclusion In this work, our RBFN technique was effective for parameter optimization of stiff biochemical models.
Moduli Space of Integrable Dirac Structures
Milani, Vida
2009-01-01
In this paper we introduce the notion of integrable Dirac structures on Hermitian modules. The moduli space of the space of integrable Dirac structures is studied. Then a necessary and sufficient condition for the integrability of a Dirac structure is obtained as the solution of a certain partial differential equation.
Hnizdo, V.
1988-06-01
Refutation is given of a recent claim that both the Lorentz-Dirac radiation reaction force and Lienard-Wiechert retarded potentials satisfy energy conservation only to a low order of approximation in a system of two charges which move uniformly along a circle. When correctly calculated, the power radiated by such a system equals exactly the rate at which work is done on the system by external forces.
Jentsch, V.
1984-03-01
The steady state proton flux in the earth's radiation belt is analyzed in detail based on a first-order partial differential equation which is equivalent to the radial diffusion equation with charge exchange and energy degradation included. It is found that for the most part of invariant space, the diffusion flux is directed inward. However, it is directed outward in a narrow L range centered on L about two, when charge exchange and energy loss are of comparable importance. Radial diffusion and losses strongly modify the proton flux's spectral shape, with the spectra exponentially decreasing at the outer boundary, becoming flat around L = 3.5, and assuming large positive gradients further downward. Proton fluxes gain anisotropy in the course of diffusion; the diffusion coefficient governs both the magnitude and the shape of the proton flux. External effects are important in the diffusion-dominated zone, but are relatively unimportant in the loss-dominated region.
Harrabi, Abdellaziz; Rebhi, Salem; Selmi, Abdelbaki
In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in R, N⩾3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.
A Short Biography of Paul A. M. Dirac and Historical Development of Dirac Delta Function
Debnath, Lokenath
2013-01-01
This paper deals with a short biography of Paul Dirac, his first celebrated work on quantum mechanics, his first formal systematic use of the Dirac delta function and his famous work on quantum electrodynamics and quantum statistics. Included are his first discovery of the Dirac relativistic wave equation, existence of positron and the intrinsic…
Cari, C.; Suparmi, A.; Yunianto, M.; Pratiwi, B. N.
2016-08-01
The Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.
Mohammadi, Vahid; Chenaghlou, Alireza
2017-09-01
The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum E3‧, anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang-Coulomb monopole (YCM) superintegrable system (a SU(2) non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.
Hammer, René; Arnold, Anton
2013-01-01
A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly. Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in $l_2$ which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stabi...
Casajús Ramo, A
2006-01-01
DIRAC is the LHCb Workload and Data Management System. Based on a service-oriented architecture, it enables generic distributed computing with lightweight Agents and Clients for job execution and data transfers. DIRAC implements a client-server architecture exposing server methods through XML Remote Procedure Call (XML-RPC) protocol. DIRAC is mostly coded in python. DIRAC security infrastructure has been designed to be a completely generic XML-RPC transport over a SSL tunnel. This new security layer is able to handle standard X509 certificates as well as grid-proxies to authenticate both sides of the connection. Serve and client authentication relies over OpenSSL and py-Open SSL, but to be able to handle grid proxies some modifications have been added to those libraries. DIRAC security infrastructure handles authorization and authorization as well as provides extended capabilities like secure connection tunneling and file transfer. Using this new security infrastructure all LHCb users can safely make use o...
Solutions to the -dimensional radial Schrödinger equation for the potential $ar^2 + br − c/r$
Ramesh Kumar; Fakir Chand
2014-07-01
Approximate solutions to the -dimensional radial Schrödinger equation for the potential $ar^2 + br − c/r$ are obtained by employing the formulation described in Ciftci et al, J. Phys. A 43, 415206 (2010). The problem, for some special cases, is solved numerically. Using this analysis, the energy spectra of a two-dimensional two-electron quantum dot (QD) in a magnetic field are also obtained. The results of this study are in good agreement with the other studies.
Cheng, Xing; Miao, Changxing; Zhao, Lifeng
2016-09-01
We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in H1 (Rd) and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in H1 (Rd) below the threshold for radial data when d ≤ 4.
A Radial Basis Function (RBF) Method for the Fully Nonlinear 1D Serre Green-Naghdi Equations
Fabien, Maurice S
2014-01-01
In this paper, we present a spectral method based on Radial Basis Functions (RBFs) for numerically solving the fully nonlinear 1D Serre Green-Naghdi equations. The approximation uses an RBF discretization in space and finite differences in time; the full discretization is obtained by the method of lines technique. For select test cases (see Bonnenton et al. [2] and Kim [11]) the approximation achieves spectral (exponential) accuracy. Complete \\textsc{matlab} code of the numerical implementation is included in this paper (the logic is easy to follow, and the code is under 100 lines).
Zabihi, F.; Saffarian, M.
2016-07-01
The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.
Héctor Torres-Silva
2008-11-01
Full Text Available In the present article we propose a simple equality involving the Dirac operator and the Maxwell operators from a chiral approach. This equality establishes a direct connection between solutions of the two systems. Moreover, we show that the connection is valid when a fairly natural relationship between the frequency of the electromagnetic wave and the energy of the Dirac particle is fulfilled, if the electric field is parallel to the magnetic field . Our analysis is based on the quaternionic form of the Dirac equation and on the quaternionic form of the Maxwell equations. In both cases the quaternionic reformulations are completely equivalent to the traditional form of the Dirac and Maxwell systems. This theory is a new quantum mechanics (QM interpretation. The research below shows that the QM represents the electrodynamics of the curvilinear closed chiral waves. This concords entirely with the modern interpretation and results of the quantum field theory.En el presente artículo se propone un simple igualdad que considera el operador de Dirac y los operadores de Maxwell bajo un enfoque quiral. Esta igualdad establece una conexión directa entre las soluciones de los dos sistemas. Además, se muestra que es válida cuando una relación muy natural se cumple entre la frecuencia de la onda electromagnética y la energía de la partícula Dirac, si el campo eléctrico es paralelo al campo magnético . Este análisis se basa en la forma cuaterniónica de la ecuación de Dirac y la forma cuaterniónica de las ecuaciones de Maxwell. En ambos casos las reformulaciones con cuaterniones son completamente equivalentes a la forma tradicional de los sistemas de Dirac y Maxwell. Esta teoría es una nueva interpretación de la mecánica cuántica. Este trabajo prueba que la mecánica cuántica representa la electrodinámica de ondas quirales curvilíneas cerradas. Esto está enteramente de acuerdo con la moderna interpretación y resultados de la teoría cu
Wigner function for the Dirac oscillator in spinor space
MA Kai; WANG Jian-Hua; YUAN Yi
2011-01-01
The Wigner function for the Dirac oscillator in spinor space is studied in this paper.Firstly,since the Dirac equation is described as a matrix equation in phase space,it is necessary to define the Wigner function as a matrix function in spinor space.Secondly,the matrix form of the Wigner function is proven to support the Dirac equation.Thirdly,by solving the Dirac equation,energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.
The Dirac operator and gamma matrices for quantum Minkowski spaces
1997-01-01
Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.
Kambe, Takahide; Saito, Koichi
2016-01-01
As the interior density of a neutron star can become very high, it has been expected and discussed that quark matter may exist inside it. To describe the transition from hadron to quark phases (and vice versa), there are mainly two methods; one is the first-order phase transition, and the other is the crossover phenomenon. In the present study, using the flavor-SU (3) NJL model with the vector coupling interaction, we have calculated the equation of state for the quark phase at high density. Furthermore, for the hadron phase at low density, we have used two kinds of the equations of state; one is a relatively soft one by the QHD model, and the other is a stiff one calculated with relativistic Brueckner-Hartree-Fock approximation. Using those equations of state for the two phases, we have investigated the influence of various choices of parameters concerning the crossover region on the mass and radius of a neutron star.
Trapping Dirac fermions in tubes generated by two scalar fields
Casana, R.; Gomes, A. R.; Martins, G. V.; Simas, F. C.
2014-04-01
In this work we consider (1,1)-dimensional resonant Dirac fermionic states on tubelike topological defects. The defects are formed by rings in (2,1) dimensions, constructed with two scalar fields ϕ and χ, and embedded in the (3,1)-dimensional Minkowski spacetime. The tubelike defects are attained from a Lagrangian density explicitly dependent with the radial distance r relative to the ring axis and the radius and thickness of its cross section are related to the energy density. For our purposes we analyze a general Yukawa-like coupling between the topological defect and the fermionic field ηF(ϕ ,χ)ψ¯ψ. With a convenient decomposition of the fermionic fields in left and right components, we establish a coupled set of first-order differential equations for the amplitudes of the left and right components of the Dirac field. After decoupling and decomposing the amplitudes in polar coordinates, the radial modes satisfy Schrödinger-like equations whose eigenvalues are the masses of the fermionic states. With F(ϕ ,χ)=ϕχ the Schrödinger-like equations are numerically solved with appropriated boundary conditions. Several resonance peaks for both components are obtained, and the results are confronted with the qualitative analysis of the Schrödinger-like potentials.
Field Equations and Radial Solution in a Noncommutative Spherically Symmetric Geometry
Yazdani, Aref
2014-01-01
We study a noncommutative theory of gravity in the framework of torsional spacetime. This theory is based on a Lagrangian obtained by applying the technique of dimensional reduction of noncommutative gauge theory and that the yielded diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity. Field equations are derived in the framework of teleparallel gravity through Weitzenbock geometry. We solve these field equations by considering a mass that is distributed spherically symmetrically in a stationary static spacetime in order to obtain a noncommutative line element.This new line element interestingly reaffirms the coherent state theory for a noncommutative Schwarzschild black hole. For the first time, we derive the Newtonian gravitational force equation in the commutative relativity framework, and this result could provide the possibility to investigate examples in various topics in quantum and ordinary theories of gravity.
Gómez, F; Afanasev, L; Benayoun, M; Brekhovskikh, V; Caragheorgheopol, G; Cechák, T; Chiba, M; Constantinescu, S; Doudarev, A; Dreossi, D; Drijard, Daniel; Ferro-Luzzi, M; Gallas, M V; Gerndt, J; Giacomich, R; Gianotti, P; Goldin, D; Gorin, A; Gortchakov, O; Guaraldo, C; Hansroul, M; Hosek, R; Iliescu, M; Jabitski, M; Kalinina, N; Karpoukhine, V; Kluson, J; Kobayshi, M; Kokkas, P; Komarov, V; Koulikov, A; Kouptsov, A; Krouglov, V; Krouglova, L; Kuroda, K I; Lanaro, A; Lapshine, B; Lednicky, R; Leruste, P; Levisandri, P; López-Aguera, A; Lucherini, V; Mäki, T; Manuilov, I; Montanet, L; Narjoux, J L; Nemenov, L; Nikitin, M; Nunez Pardo, T; Okada, K; Olchevskii, V; Pazos, A; Pentia, M; Penzo, Aldo L; Perreau, J M; Petrascu, C; Pló, M; Ponta, T; Pop, D; Riazantsev, A; Rodríguez, J M; Rodriguez Fernandez, A; Rykaline, V; Santamarina, C; Saborido, J; Schacher, J; Sidorov, A; Smolik, J; Takeutchi, F; Tarasov, A; Tauscher, L; Tobar, M J; Trusov, S; Vasquez, P; Vlachos, S; Yazkov, V; Yoshimura, Y; Zrelov, P
2001-01-01
The main objective of DIRAC experiment is the measurement of the lifetime tau of the exotic hadronic atom consisting of pi /sup +/ and pi /sup -/ mesons. The lifetime of this atom is determined by the decay mode pi /sup +/ pi /sup -/ to pi /sup 0/ pi /sup 0/ due to the strong interaction. Through the precise relationship between the lifetime and the S-wave pion-pion scattering length difference a/sub 0/-a/sub 2/ for isospin 0 and 2 (respectively), a measurement of tau with an accuracy of 10% will allow a determination of a/sub 0/-a/sub 2/at a 5% precision level. Pion-pion scattering lengths have been calculated in the framework of chiral perturbation theory with an accuracy below 5%. In this way DIRAC experiment will provide a crucial test of the chiral symmetry breaking scheme in QCD effective theories at low energies. (19 refs).
Quiney, HM; Glushkov, VN; Wilson, S
2002-01-01
Using basis sets of distributed s-type Gaussian functions with positions and exponents optimized so as to support Hartree-Fock total energies with an accuracy approaching the sub-muHartree level, Dirac-Hartree-Fock-Coulomb calculations are reported for the ground states of the H-2, LiH, and BH molec
E. Ghanbari Adivi
2007-09-01
Full Text Available A method is presented to reduce the singular Lippmann-Schwinger integral equation to a simple matrix equation. This method is applied to calculate the matrix elements of the reaction and transition operators, respectively, on the real axis and on the complex plane. The phase shifts and the differential scattering amplitudes are computable as well as the differential cross sections if the R- and/or T-matrix elements on the energy-shell are known. The method is applicable by using the Gaussian quadratures based on the Legenre, Laguer Chebyshev and shifted Chebyshev polynomials. Choosing the nodal points and weight functions depends on the aspects of the problem.
Dirac-Kahler Theory and Massless Fields
Pletyukhov, V A
2010-01-01
Three massless limits of the Dirac-Kahler theory are considered. It is shown that the Dirac-Kahler equation for massive particles can be represented as a result of the gauge-invariant mixture (topological interaction) of the above massless fields.
On localization of Dirac fermions by disorder
Medvedyeva, Mariya Vyacheslavivna
2011-01-01
This thesis is devoted to the effects of disorder on two-dimensional systems of Dirac fermions. Disorder localizes the usual electron system governed by the Schroedinger equation. The influence of disorder on Dirac fermions is qualitevely different. We concentrate on a random mass term in the Dira
Dirac and Weyl semimetals with holographic interactions
Jacobs, V.P.J.
2015-01-01
Dirac and Weyl semimetals are states of matter exhibiting the relativistic physics of, respectively, the Dirac and Weyl equation in a three-dimensional bulk material. These three-dimensional semimetals have recently been realized experimentally in various crystals. Theoretically, especially the noni
Piret, Cécile
2012-05-01
Much work has been done on reconstructing arbitrary surfaces using the radial basis function (RBF) method, but one can hardly find any work done on the use of RBFs to solve partial differential equations (PDEs) on arbitrary surfaces. In this paper, we investigate methods to solve PDEs on arbitrary stationary surfaces embedded in . R3 using the RBF method. We present three RBF-based methods that easily discretize surface differential operators. We take advantage of the meshfree character of RBFs, which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension. Two out of the three methods, which we call the orthogonal gradients (OGr) methods are the result of our work and are hereby presented for the first time. © 2012 Elsevier Inc.
Le Roy, Robert J.
2017-01-01
This paper describes program LEVEL, which can solve the radial or one-dimensional Schrödinger equation and automatically locate either all of, or a selected number of, the bound and/or quasibound levels of any smooth single- or double-minimum potential, and calculate inertial rotation and centrifugal distortion constants and various expectation values for those levels. It can also calculate Franck-Condon factors and other off-diagonal matrix elements, either between levels of a single potential or between levels of two different potentials. The potential energy function may be defined by any one of a number of analytic functions, or by a set of input potential function values which the code will interpolate over and extrapolate beyond to span the desired range.
Anikeev, A. A.; Bogdanova, Yu. A.; Gubin, S. A.
Multicomponent hypernetted-chain/soft core mean spherical approximation (HMSA) was shown to be successfully applied for the problem of ambidextrous attractive/repulsive interaction simulation in dense fluids like shock compression products of CxNyOz liquid systems. This approximation provides high numerical accuracy for thermodynamic quantities due to its self-consistency. In addition, distribution function integral equation theory (DFIET) doesn't require chemical equilibrium for simulated systems. Reproducible shock Hugoniot curves verify the macroscopic properties such as pressure and internal energy. Radial distribution function analysis, proposed in this paper, approves macroscopic and microscopic/structural short-range order properties both by molecular Monte-Carlo (MC) method for multicomponent dissociation products of liquid CO2 up to 160 GPa.
Massless Dirac particles in the vacuum C-metric
Bini, Donato; Geralico, Andrea
2015-01-01
We study the behavior of massless Dirac particles in the vacuum C-metric spacetime, representing the nonlinear superposition of the Schwarzschild black hole solution and the Rindler flat spacetime associated with uniformly accelerated observers. Under certain conditions, the C-metric can be considered as a unique laboratory to test the coupling between intrinsic properties of particles and fields with the background acceleration in the full (exact) strong-field regime. The Dirac equation is separable by using, e.g., a spherical-like coordinate system, reducing the problem to one-dimensional radial and angular parts. Both radial and angular equations can be solved exactly in terms of general Heun functions. We also provide perturbative solutions to first-order in a suitably defined acceleration parameter, and compute the acceleration-induced corrections to the particle absorption rate as well as to the angle-averaged cross section of the associated scattering problem in the low-frequency limit. Furthermore, we...
Trapping Dirac fermions in tubes generated by two scalar fields
Casana, R; Martins, G V; Simas, F C
2013-01-01
In this work we consider $(1,1)-$dimensional resonant Dirac fermionic states on tube-like topological defects. The defects are formed by rings in $(2,1)$ dimensions, constructed with two scalar field $\\phi$ and $\\chi$, and embedded in the $(3,1)-$dimensional Minkowski spacetime. The tube-like defects are attained from a lagrangian density explicitly dependent with the radial distance $r$ relative to the ring axis and the radius and thickness of the its cross-section are related to the energy density. For our purposes we analyze a general Yukawa-like coupling between the topological defect and the fermionic field $\\eta F(\\phi,\\chi)\\bar\\psi\\psi$. With a convenient decomposition of the fermionic fields in left- and right- chiralities, we establish a coupled set of first order differential equations for the amplitudes of the left- and right- components of the Dirac field. After decoupling and decomposing the amplitudes in polar coordinates, the radial modes satisfy Schr\\"odinger-like equations whose eigenvalues a...
Sadykov, N. R.
2011-03-01
It is suggested to extend the results obtained for Maxwell's equations in Majorana form (spin-1 particles) for spin particles with a half-integer spin and a nonzero mass. It is shown that in an unbounded "chiral medium" (twisted media) the degeneration existing between particles of different helicities is removed. For ultrarelativistic particles, an analog to the inverse optical Magnus effect follows where the effect is determined by the chirality of the medium. From the inverse scattering problem for the transforms under consideration it follows that the amplitude of the wave function of a particle in a chiral medium can vary with time according to a linear law (for example, the process of neutrino (antineutrino) production or annihilation), and the parameters of the medium satisfy the evolution equation.
Anderson's absolute objects and constant timelike vector hidden in Dirac matrices
Rylov, Yu A
2001-01-01
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute objects of the Dirac equation. There are two ways of the Dirac matrices transformation: (1) Dirac matrices form a 4-vector and wave function is a scalar, (2) Dirac matrices are scalars and the wave function is a spinor. In the first case the Dirac equation is nonrelativistic, in the second one it is relativistic. Transforming Dirac equation to another scalar-vector variables, one shows that the first way of transformation is valid, and the Dirac equation is not relativistic
On Huygens' principle for Dirac operators associated to electromagnetic fields
CHALUB FABIO A.C.C.
2001-01-01
Full Text Available We study the behavior of massless Dirac particles, i.e., solutions of the Dirac equation with m = 0 in the presence of an electromagnetic field. Our main result (Theorem 1 is that for purely real or imaginary fields any Huygens type (in Hadamard's sense Dirac operators is equivalent to the free Dirac operator, equivalence given by changes of variables and multiplication (right and left by nonzero functions.
Dirac particle in gravitational quantum mechanics
Pedram, Pouria
2011-08-01
In this Letter, we consider the effects of the Generalized (Gravitational) Uncertainty Principle (GUP) on the eigenvalues and the eigenfunctions of the Dirac equation. This form of GUP is consistent with various candidates of quantum gravity such as string theory, loop quantum gravity, doubly special relativity and black hole physics and predicts both a minimum measurable length and a maximum measurable momentum. The modified Hamiltonian contains two additional terms proportional to a( and a( where αi are Dirac matrices and a∼1/MPlc is the GUP parameter. For the case of the Dirac free particle and the Dirac particle in a box, we solve the generalized Dirac equation and find the modified energy eigenvalues and eigenfunctions.
Torsion Gravity for Dirac Fields
Fabbri, Luca
2016-01-01
In this article we will take into account the most complete back-ground with torsion and curvature, providing the most exhaustive coupling for the Dirac field: we will discuss the integrability of the interaction of the matter field and the reduction of the matter field equations.
Nakatsuji, Hiroshi
2012-09-18
Just as Newtonian law governs classical physics, the Schrödinger equation (SE) and the relativistic Dirac equation (DE) rule the world of chemistry. So, if we can solve these equations accurately, we can use computation to predict chemistry precisely. However, for approximately 80 years after the discovery of these equations, chemists believed that they could not solve SE and DE for atoms and molecules that included many electrons. This Account reviews ideas developed over the past decade to further the goal of predictive quantum chemistry. Between 2000 and 2005, I discovered a general method of solving the SE and DE accurately. As a first inspiration, I formulated the structure of the exact wave function of the SE in a compact mathematical form. The explicit inclusion of the exact wave function's structure within the variational space allows for the calculation of the exact wave function as a solution of the variational method. Although this process sounds almost impossible, it is indeed possible, and I have published several formulations and applied them to solve the full configuration interaction (CI) with a very small number of variables. However, when I examined analytical solutions for atoms and molecules, the Hamiltonian integrals in their secular equations diverged. This singularity problem occurred in all atoms and molecules because it originates from the singularity of the Coulomb potential in their Hamiltonians. To overcome this problem, I first introduced the inverse SE and then the scaled SE. The latter simpler idea led to immediate and surprisingly accurate solution for the SEs of the hydrogen atom, helium atom, and hydrogen molecule. The free complement (FC) method, also called the free iterative CI (free ICI) method, was efficient for solving the SEs. In the FC method, the basis functions that span the exact wave function are produced by the Hamiltonian of the system and the zeroth-order wave function. These basis functions are called complement
Dirac dynamical resonance states around Schwarzschild black holes
Zhou, Xiang-Nan; Yang, Ke; Liu, Yu-Xiao
2013-01-01
Recently, a novel kind of scalar wigs around Schwarzschild black holes---scalar dynamical resonance states were introduced in [Phys. Rev. D 84, 083008 (2011)] and [Phys. Rev. Lett. 109, 081102 (2012)]. In this paper, we investigate the existence and evolution of Dirac dynamical resonance states. First we look for stationary resonance states of a Dirac field around a Schwarzchild black hole by using the Schrodinger-like equations reduced from the Dirac equation in Schwarzschild spacetime. Then Dirac pseudo-stationary configurations are constructed from the stationary resonance states. We use these configurations as initial data and investigate their numerical evolutions and energy decay. These dynamical solutions are the so-called "Dirac dynamical resonance states". It is found that the energy of the Dirac dynamical resonance states shows an exponential decay. The decay rate of energy is affected by the resonant frequency, the mass of Dirac field, the total angular momentum, and the spin-orbit interaction. In ...
Optical analogue of relativistic Dirac solitons in binary waveguide arrays
Tran, Truong X., E-mail: truong.tran@mpl.mpg.de [Department of Physics, Le Quy Don University, 236 Hoang Quoc Viet str., 10000 Hanoi (Viet Nam); Max Planck Institute for the Science of Light, Günther-Scharowsky str. 1, 91058 Erlangen (Germany); Longhi, Stefano [Department of Physics, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano (Italy); Biancalana, Fabio [Max Planck Institute for the Science of Light, Günther-Scharowsky str. 1, 91058 Erlangen (Germany); School of Engineering and Physical Sciences, Heriot-Watt University, EH14 4AS Edinburgh (United Kingdom)
2014-01-15
We study analytically and numerically an optical analogue of Dirac solitons in binary waveguide arrays in the presence of Kerr nonlinearity. Pseudo-relativistic soliton solutions of the coupled-mode equations describing dynamics in the array are analytically derived. We demonstrate that with the found soliton solutions, the coupled mode equations can be converted into the nonlinear relativistic 1D Dirac equation. This paves the way for using binary waveguide arrays as a classical simulator of quantum nonlinear effects arising from the Dirac equation, something that is thought to be impossible to achieve in conventional (i.e. linear) quantum field theory. -- Highlights: •An optical analogue of Dirac solitons in nonlinear binary waveguide arrays is suggested. •Analytical solutions to pseudo-relativistic solitons are presented. •A correspondence of optical coupled-mode equations with the nonlinear relativistic Dirac equation is established.
Wigner function for the Dirac oscillator in spinor space
马凯; 王剑华; 袁毅
2011-01-01
The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space.
The Dirac oscillator in a rotating frame of reference
Strange, P.; Ryder, L. H.
2016-10-01
The Dirac equation in a rotating frame of reference is derived from first principles within a linear approximation. This equation is employed to exhibit an equivalence between a particle in a Dirac oscillator potential and a free particle in a rotating frame of reference. A zero-point contribution to the energy of the particle, resulting from its spin, is also noted.
Quantum game interpretation of Dirac spinor field
Zhi, Haizhao
2011-01-01
This paper introduced the classical prisoner dilemma with the character and structure of quantum prisoner dilemma's strategy space. Associate with the Dirac spinor field, apply the basic quantum game strategy to the translation of the dynamics of Dirac equation. Decompose the real space and time to lattice we found that the basic interaction of spinor could be translated into quantum game theory. At the same time, we gained the new dynamics of quantized spacial evolutionary game.
Klein-Gordon and Dirac gyroscopes
SadurnI, E [Instituto de Ciencias Fisicas, Universidad Nacional Autonoma de Mexico, Cuernavaca, Morelos (Mexico)], E-mail: sadurni@fis.unam.mx
2009-01-09
The formulation of a rigid body in relativistic quantum mechanics is studied. Departing from an alternate approach at the relativistic classical level, the corresponding Klein-Gordon and Dirac operators for the rigid body are obtained in covariant form. The resulting wave equations are shown to be consistent, by construction, with earlier definitions of a relativistic rigid body by Aldinger et al (1983 Phys. Rev. D 28 3020). Wavefunctions and spectra for both cases are obtained explicitly, including the Dirac gyroscope with asymmetries.
Gravitational Repulsion and Dirac Antimatter
Kowitt, Mark E.
1996-03-01
Based on an analogy with electron and hole dynamics in semiconductors, Dirac's relativistic electron equation is generalized to include a gravitational interaction using an electromagnetic-type approximation of the gravitational potential. With gravitational and inertial masses decoupled, the equation serves to extend Dirac's deduction of antimatter parameters to include the possibility of gravitational repulsion between matter and antimatter. Consequences for general relativity and related “antigravity” issues are considered, including the nature and gravitational behavior of virtual photons, virtual pairs, and negative-energy particles. Basic cosmological implications of antigravity are explored—in particular, potential contributions to inflation, expansion, and the general absence of detectable antimatter. Experimental and observational tests are noted, and new ones suggested.
Spherically Symmetric Solution of the Weyl-Dirac Theory of Gravitation and its Consequences
Babourova, O. V.; Frolov, B. N.; Kudlaev, P. E.; Romanova, E. V.
2016-12-01
The Poincaré and Poincaré-Weyl gauge theories of gravitation with Lagrangians quadratic on curvature and torsion in post-Riemannian spaces with the Dirac scalar field is discussed in a historical aspect. The various hypotheses concerning the models of a dark matter with the help of a scalar field are considered. The new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl spacetime with the Dirac scalar field representing the dark matter model. A static spherically symmetric solution of the field equations in vacuum for a central compact mass is obtained as the metrics conformal to the Yilmaz-Rosen metrics. On the base of this solution one considers a radial movement of an interplanetary spacecraft starting from the Earth. Using the Newton approximation one obtains that the asymptotic line-of-sight velocity in this case depends on the parameters of the solution, and therefore one can obtain, on basis of the observable data, the values of these parameters and then the value of a rest mass of the Dirac scalar field.
Sun, Jiu-Xun; Wu, Qiang; Cai, Ling-Cang; Jin, Ke
2013-11-01
A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.
Hydrodynamics of the Dirac spectrum
Liu, Yizhuang, E-mail: yizhuang.liu@stonybrook.edu [Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800 (United States); Warchoł, Piotr, E-mail: piotr.warchol@uj.edu.pl [M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krakow (Poland); Zahed, Ismail, E-mail: ismail.zahed@stonybrook.edu [Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800 (United States)
2016-02-10
We discuss a hydrodynamical description of the eigenvalues of the Dirac spectrum in even dimensions in the vacuum and in the large N (volume) limit. The linearized hydrodynamics supports sound waves. The hydrodynamical relaxation of the eigenvalues is captured by a hydrodynamical (tunneling) minimum configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of unbroken chiral symmetry to a phase of broken chiral symmetry occurs over a time set by the speed of sound.
The Dirac-Electron Vacuum Wave
Daywitt W. C.
2016-07-01
Full Text Available This paper argues that the Dirac equation can be interpreted as an interaction between the electron core and the Planck vacuum state, where the positive and negative solutions represent respectively the dynamics of the electron core and a vacuum wave propagating within the vacuum state. Results show that the nonrelativistic positive solution reduces to the Schrödinger wave equation
The Clifford algebra of physical space and Dirac theory
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Dirac Field in FRW Spacetime: Current and Energy Momentum
Dhungel, P R
2011-01-01
The behaviour of the Dirac field in FRW space-time is investigated. The relevant equations are solved to determine the particle and energy distribution. The angular and radial parts are solved in terms of Jacobi polynomials. The time dependence of the massive field is solved in terms of known function only for the radiation filled flat space. WKB method is used for approximate solution in general Friedmann-Le Maitre space. The negative energy solution is found decay in time as the Universe expands, while the positive energy solution grows. This could be the source of the local particle current. The behaviour of the particle number and energy density are also investigated. It is found that the particles arrange themselves in a number and density distribution pattern that produces a constant Newtonian potential as required for the flat rotation curves of galaxies. Further, density contrast is found to grow with the expansion.
Integrated optical Dirac physics via inversion symmetry breaking
Collins, Matthew J.; Zhang, Fan; Bojko, Richard; Chrostowski, Lukas; Rechtsman, Mikael C.
2016-12-01
Graphene and boron nitride are two-dimensional materials whose atoms are arranged in a honeycomb lattice. Their unique properties arise because their electrons behave like relativistic particles (without and with mass, respectively)—namely, they obey the Dirac equation. Here, we use a photonic analog of boron nitride to observe Dirac physics in a silicon integrated optical platform. This will allow for photonic applications of Dirac dispersions (gapped and ungapped) to be realized in an on-chip, integrated nanophotonic platform.
DIRAC distributed secure framework
Casajus, A.; Graciani, R.; LHCb DIRAC Team
2010-04-01
DIRAC, the LHCb community Grid solution, provides access to a vast amount of computing and storage resources to a large number of users. In DIRAC users are organized in groups with different needs and permissions. In order to ensure that only allowed users can access the resources and to enforce that there are no abuses, security is mandatory. All DIRAC services and clients use secure connections that are authenticated using certificates and grid proxies. Once a client has been authenticated, authorization rules are applied to the requested action based on the presented credentials. These authorization rules and the list of users and groups are centrally managed in the DIRAC Configuration Service. Users submit jobs to DIRAC using their local credentials. From then on, DIRAC has to interact with different Grid services on behalf of this user. DIRAC has a proxy management service where users upload short-lived proxies to be used when DIRAC needs to act on behalf of them. Long duration proxies are uploaded by users to a MyProxy service, and DIRAC retrieves new short delegated proxies when necessary. This contribution discusses the details of the implementation of this security infrastructure in DIRAC.
On the spring and mass of the Dirac oscillator
Crawford, James P.
1993-01-01
The Dirac oscillator is a relativistic generalization of the quantum harmonic oscillator. In particular, the square of the Hamiltonian for the Dirac oscillator yields the Klein-Gordon equation with a potential of the form: (ar(sub 2) + b(L x S)), where a and b are constants. To obtain the Dirac oscillator, a 'minimal substitution' is made in the Dirac equation, where the ordinary derivative is replaced with a covariant derivative. However, an unusual feature of the covariant derivative in this case is that the potential is a non-trivial element of the Clifford algebra. A theory which naturally gives rise to gage potentials which are non-trivial elements of the Clifford algebra is that based on local automorphism invariance. An exact solution of the automorphism gage field equations which reproduces both the potential term and the mass term of the Dirac oscillator is presented.
The right inverse of Dirac operator in octonionic space
Wang, Haiyan; Bian, Xiaoli
2017-09-01
The octonion Dirac equation also called wave equation is an important equation which formulates the localization spaces for subluminal and superluminal particles. The purpose of this paper is to look for the right inverse operator of octonion Dirac operator in Hölder space. However, some difficulties will arise in noncommutative and nonassociative setting. We note that the associator is available to overcome the difficulties.
Dirac spinor in a nonstationary Godel-type cosmological Universe
Villalba, Victor M
2015-01-01
In the present article we solve, via separation of variables, the massless Dirac equation in a nonstationary rotating, causal G\\"odel-type cosmological universe, having a constant rotational speed in all the points of the space. We compute the frequency spectrum. We show that the spectrum of massless Dirac particles is discrete and unbounded.
Tunneling of Dirac Particles from Kaluza-Klein Black Hole
ZENG Xiao-Xiong; LI Qiang
2009-01-01
Applying the fermions tunneling method, proposed by Kerner and Mann recently, we discuss the tunneling characteristics of Dirac particles from the stationary Kaluza-Klein black hole. To choose Gamma matrix conveniently and avoid the ergosphere dragging effect, we perform it in the dragging coordinate frame. The result shows that Hawking temperature in this case also can be reproduced by the general Dirac equation.
Parand, K; Kazem, S; Rezaei, A R; 10.1016/j.cnsns.2010.07.011
2010-01-01
In this paper two common collocation approaches based on radial basis functions have been considered; one be computed through the integration process (IRBF) and one be computed through the differentiation process (DRBF). We investigated the two approaches on natural convection heat transfer equations embedded in porous medium which are of great importance in the design of canisters for nuclear wastes disposal. Numerical results show that the IRBF be performed much better than the common DRBF, and show good accuracy and high rate of convergence of IRBF process.
Deconstructing non-dissipative non-Dirac-hermitian relativistic quantum systems
Ghosh, Pijush K
2011-01-01
A method to construct non-dissipative non-Dirac-hermitian relativistic quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-hermitian operators, which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry
Deconstructing non-dissipative non-Dirac-Hermitian relativistic quantum systems
Ghosh, Pijush K.
2011-08-01
A method to construct non-dissipative non-Dirac-Hermitian relativistic quantum system that is isospectral with a Dirac-Hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-Hermitian operators, which are Hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-Hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry.
Dirac particle in a pseudoscalar potential
Moreno, M. [Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Ap. Postal 20-364, 01000 (Mexico), D.F.; Zentella-Dehesa, A. [Departamento de Fisicoquimica, Intituto de Quimica, UNAM Ap. Postal 70-213, 04510 (Mexico), D.F.
1996-02-01
We study the problem of a Dirac particle with a pseudoscalar interaction in the potential approximation. It is shown how nonperturbative relativistic solutions arise. The case of the central pseudoscalar potential is explicitly worked out also in a closed form. The angular functions are worked out in general for this central case. Finally for the special case of the spherical well the radial solutions are shown to behave like Bessel-type functions. {copyright} {ital 1996 American Institute of Physics.}
Scarring of Dirac fermions in chaotic billiards.
Ni, Xuan; Huang, Liang; Lai, Ying-Cheng; Grebogi, Celso
2012-07-01
Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrödinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions.
Dirac-Coulomb scattering with plane wave energy eigenspinors on de Sitter expanding universe
Cotaescu, Ion I
2007-01-01
The lowest order contribution of the amplitude of Dirac-Coulomb scattering in de Sitter spacetime is calculated assuming that the initial and final states of the Dirac field are described by exact solutions of the free Dirac equation on de Sitter spacetime with a given energy and helicity. We find that the total energy is conserved in the scattering process.
Rodrigues, R. de Lima [Universidade Federal de Campina Grande (UFCG), Cuite, PB (Brazil). Centro de Tecnologia. Unidade Academica de Educacao]. E-mail: rafael@df.ufcg.edu.br; rafaelr@cbpf.br
2007-07-01
In the present work we obtain a new representation for the Dirac oscillator based on the Clifford algebra C 7. The symmetry breaking and the energy eigenvalues for our model of the Dirac oscillator are studied in the non-relativistic limit. (author)
Daudé, Thierry
2017-01-01
In this paper, the authors study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, they establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. The authors also use the miraculous property (quoting Chandrasekhar)-that the Dirac equation can be separated into radial and angular ordinary differential equations-to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, the authors use this expression of ...
Consequences of Dirac Theory of the Positron
Heisenberg, W K
1936-01-01
According to Dirac's theory of the positron, an electromagnetic field tends to create pairs of particles which leads to a change of Maxwell's equations in the vacuum. These changes are calculated in the special case that no real electrons or positrons are present and the field varies little over a Compton wavelength.
Spin 1/2 particle in the field of the Dirac string on the background of de Sitter space-time
Red'kov, V M; Veko, O V
2011-01-01
The Dirac monopole string is specified for de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables, detailed analysis of the radial equations is performed; four types of solutions, singular, regular, in- and out- running waves, are constructed in terms of hypergeometric functions. The complete set of spinor wave solutions \\Psi_{\\epsilon, j,m, \\lambda}(t,r, \\theta, \\phi) has been constructed, special attention is given to treating the states of minimal values of the total angular moment j_{\\min}.
Monti, Dalida
1996-01-01
Relativamente poco noto al gran pubblico, il premio Nobel Paul Adrien Maurice Dirac appartiene a quel gruppo di uomini di ingegno che nei primi decenni del secolo contribuirono a dare alla nostra concezione del mondo fisico la sua impronta attuale. Assolutamente cruciali, per una valutazione dell'opera di Dirac, sono gli anni compresi tra il 1925 e il 1931: un periodo in cui il fisico fornisce la prima spiegazione chiara e coerente delle proprietà di spin dell'elettrone (equazione di Dirac) e perviene, in forza della pura deduzione matematica, alla scoperta dell'esistenza dell'elettrone positivo o positrone.
张爱萍; 强稳朝
2007-01-01
In this paper, the relativistic Rosen-Morse Ⅱ potential is investigated by solving the Klein-Gordon and the Dirac equations with equal attractive scalar s(r) and repulsive vector v(r) potentials. The exact energy equations of the bound state are obtained by the method of supersymmetric and shape invariance. Finally, a kind of special potential about Rosen-Morse Ⅱ potential is discussed.%在标量势和矢量势相等的情形,研究了Rosen-MorseⅡ势的相对论效应,应用超对称和形状不变势方法通过求解Klein-Gordon方程和Dirac方程得到了束缚态能量本征值,最后,讨论了Rosen-MorseⅡ势的一种特殊情况.
Sun, Jiu-Xun; Jin, Ke; Cai, Ling-Cang; Wu, Qiang
2014-08-01
The equation of state (EOS) for hard-sphere fluid derived from compressibility routes of Percus-Yevick theory (PYC) is extended. The two parameters are determined by fitting well-known virial coefficients of pure fluid. The extended cubic EOS can be directly extended to multi-component mixtures, merely demanding the EOS of mixtures also is cubic and combining two physical conditions for the radial distribution functions at contact (RDFC) of mixtures. The calculated virial coefficients of pure fluid and predicted compressibility factors and RDFC for both pure fluid and mixtures are excellent as compared with the simulation data. The values of RDFC for mixtures with extremely large size ratio 10 are far better than the BGHLL expressions in literature.
Sun, Jiu-Xun; Wu, Qiang; Cai, Ling-Cang; Jin, Ke
2014-06-01
A generalized cubic (GC) equation of state (EOS) with two independent parameters is proposed. The GC EOS can include EOS from both virial and compressibility routes of Percus-Yevick theory in it as special cases. The two parameters are determined by fitting well-known virial coefficients of pure fluid. The generalized cubic EOS can be directly and consistently extended to multi-component mixtures merely demanding of the EOS of mixtures also is cubic, and combining two strict physical conditions for the radial distribution functions at contact (RDFC) of mixtures. The calculated virial coefficients of pure fluid and predicted compressibility factors and RDFC for both pure fluid and mixtures are excellent as compared with the simulation data. The values of RDFC for mixtures with extremely large size ratio are far better than the expressions in literature.
Three dimensional Dirac semimetals
Zaheer, Saad
We extend the physics of graphene to three dimensional systems by showing that Dirac points can exist on the Fermi surface of realistic materials in three dimensions. Many of the exotic electronic properties of graphene can be ascribed to the pseudorelativistic behavior of its charge carriers due to two dimensional Dirac points on the Fermi surface. We show that certain nonsymmorphic spacegroups exhibit Dirac points among the irreducible representations of the appropriate little group at high symmetry points on the surface of the Brillouin zone. We provide a list of all Brillouin zone momenta in the 230 spacegroups that can host Dirac points. We describe microscopic considerations necessary to design materials in one of the candidate spacegroups such that the Dirac point appears at the Fermi energy without any additional non-Dirac-like Fermi pockets. We use density functional theory based methods to propose six new Dirac semimetals: BiO 2 and SbO2 in the beta-cristobalite lattice (spacegroup 227), and BiCaSiO4, BiMgSiO4, BiAlInO 4, and BiZnSiO4 in the distorted spinels lattice (spacegroup 74). Additionally we derive effective Dirac Hamiltonians given group representative operators as well as tight binding models incorporating spin-orbit coupling. Finally we study the Fermi surface of zincblende (spacegroup 216) HgTe which is effectively point-like at Gamma in the Brillouin zone and exhibits accidental degeneracies along a threefold rotation axis. Whereas compressive strain gaps the band structure into a topological insulator, tensile strain shifts the accidental degeneracies away from Gamma and enlarges the Fermi surface. States on the Fermi surface exhibit nontrivial spin texture marked by winding of spins around the threefold rotation axis and by spin vortices indicating a change in the winding number. This is confirmed by microscopic calculations performed in tensile strained HgTe and Hg0.5Zn 0.5 Te as well as k.p theory. We conclude with a summary of recent
Two Qubits in the Dirac Representation
Rajagopal, A K
2000-01-01
A general two qubit system expressed in terms of the complete set of unit and fifteen traceless, Hermitian Dirac matrices, is shown to exhibit novel features of this system. The well-known physical interpretations associated with the relativistic Dirac equation involving the symmetry operations of time-reversal T, charge conjugation C, parity P, and their products are reinterpreted here by examining their action on the basic Bell states. The transformation properties of the Bell basis states under these symmetry operations also reveal that C is the only operator that does not mix the Bell states whereas all others do. In a similar fashion, expressing the various logic gates introduced in the subject of quantum computers in terms of the Dirac matrices shows for example, that the NOT gate is related to the product of time-reversal and parity operators.
DIRAC Workload Management System
Paterson, S
2007-01-01
DIRAC (Distributed Infrastructure with Remote Agent Control) is the Workload and Data Management system (WMS) for the LHCb experiment. The DIRAC WMS offers a transparent way for LHCb users to submit jobs to the EGEE Grid as well as local clusters and individual PCs. This paper will describe workload management optimizations, which ensure high job efficiency and minimized job start times. The computing requirements of the LHCb experiment can only be fulfilled through the use of many distributed compute resources. DIRAC provides a robust platform to run data productions on all the resources available to LHCb including the EGEE Grid. More recently, user support was added to DIRAC that greatly simplifies the procedure of submitting, monitoring and retrieving output of Grid jobs for the LHCb user community. DIRAC submits Pilot Agents to the EGEE Grid via the gLite WMS as normal jobs. Pilot Agents then request jobs from the DIRAC Workload Management System after the local environment has been checked. Therefore DIR...
Anaguano, L.
2005-07-01
According to the theory of Quantum Electrodynamics (QED) the vacuum state will change in the presence of very strong electromagnetic fields. If the external field (in the simplest case purely electrostatic) exceeds a certain critical value the creation of electron-positron pairs will ensue, resulting the the formation of a charged vacuum. This process is characterized by the emergence of electron states with a binding energy larger than twice the electron rest mass. The effect up to now usually was studied for spherically symmetric systems, in particular for the Coulomb potential of a heavy nucleus. In the present thesis we investigate, how this phenomenon changes when passing from spherical to cylindrical geometry. For this, we derive the solutions of the Dirac equation for electrons in the electrostatic potential of a long, thin charged cylinder (a ''charged string'') and study the ensuing supercritical effects. Since the logarithmic potential of an infinitely long string rises indefinitely with growing distance, all electron states should be supercritical (i.e., electrons should be able to tunnel through the particle-antiparticle gap of the Dirac equation). Therefore on may expect that the central charge will surround itself with an oppositely charged sheath of vacuum electrons, leading to neutralization of the string. To develop a quantitative description of this process, we investigate the solutions of the Poisson equation and the Dirac equation in cylindrical symmetry. In the first step a series expansion of the electrostatic potential in the central plane of a homogeneously charge cylinder of finite length and finite radius is derived. Subsequently, we employ the tetrad (vierbein) formalism to separate the Dirac equation in cylindrical coordinates. The resulting radial Dirac equation is transformed to Schroedinger type. The bound states are evaluated using the method of uniform approximation (a version of the WKB approximation). We study
Red'kov, V M
2011-01-01
Tetrad based equation for Dirac-K\\"{a}hler particle is solved in spherical coordinates in the flat Minkocski space-time. Spherical solutions of boson type (J =0,1,2,...) are constructed. After performing a special transformation over spherical boson solutions of the Dirac-K\\"{a}hler equation, 4 \\times 4-matrices U(x) \\Longrightarrow V(x), simple linear expansions of the four rows of new representativeof the Dirac--K\\"{a}hler field V(x) in terms of spherical fermion solutions \\Psi_{i}(x) of the four ordinary Dirac equations have been derived. However, this fact cannot be interpreted as the possibility not to distinguish between the Dirac-K\\"{a}hler field and the system four Dirac fermions. The main formal argument is that the special transformation (I \\otimes S(x)) involved does not belong to the group of tetrad local gauge transformation for Dirac-K\\"{a}hler field, 2-rank bispinor under the Lorentz group. Therefore, the linear expansions between boson and fermion functions are not gauge invariant under the gr...
McConnell, Sean; Fritzsche, Stephan; Surzhykov, Andrey
2010-03-01
During recent years, the DIRAC package has proved to be an efficient tool for studying the structural properties and dynamic behavior of hydrogen-like ions. Originally designed as a set of MAPLE procedures, this package provides interactive access to the wave and Green's functions in the non-relativistic and relativistic frameworks and supports analytical evaluation of a large number of radial integrals that are required for the construction of transition amplitudes and interaction cross sections. We provide here a new version of the DIRAC program which is developed within the framework of MATHEMATICA (version 6.0). This new version aims to cater to a wider community of researchers that use the MATHEMATICA platform and to take advantage of the generally faster processing times therein. Moreover, the addition of new procedures, a more convenient and detailed help system, as well as source code revisions to overcome identified shortcomings should ensure expanded use of the new DIRAC program over its predecessor. New version program summaryProgram title: DIRAC Catalogue identifier: ADUQ_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUQ_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 45 073 No. of bytes in distributed program, including test data, etc.: 285 828 Distribution format: tar.gz Programming language: Mathematica 6.0 or higher Computer: All computers with a license for the computer algebra package Mathematica (version 6.0 or higher) Operating system: Mathematica is O/S independent Classification: 2.1 Catalogue identifier of previous version: ADUQ_v1_0 Journal reference of previous version: Comput. Phys. Comm. 165 (2005) 139 Does the new version supersede the previous version?: Yes Nature of problem: Since the early days of quantum mechanics, the
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
LI Zi-Ping; LI Ai-Min; JIANG Jin-Huan; WANG Yong-Long
2005-01-01
The extended canonical Noether identities and canonical first Noether theorem derived from an extended action in phase space for a system with a singular Lagrangian are formulated. Using these canonical Noether identities,it can be shown that the constraint multipliers connected with the first-class constraints may not be independent, so a query to a conjecture of Dirac is presented. Based on the symmetry properties of the constrained Hamiltonian system in phase space, a counterexample to a conjecture of Dirac is given to show that Dirac's conjecture fails in such a system.We present here a different way rather than Cawley's examples and other's ones in that there is no linearization of constraints in the problem. This example has a feature that neither the primary first-class constraints nor secondary first-class constraints are generators of the gauge transformation.
Dirac and Weyl Materials: Fundamental Aspects and Some Spintronics Applications
Yang, Shengyuan A.
2016-09-01
Dirac and Weyl materials refer to a class of solid materials which host low-energy quasiparticle excitations that can be described by the Dirac and Weyl equations in relativistic quantum mechanics. Starting with the advent of graphene as the first prominent example, these materials have been attracting tremendous interest owing to their novel fundamental properties as well as the great potential for applications. Here we introduce the basic concepts and notions related to Dirac and Weyl materials and briefly review some recent works in this field, particularly on the conceptual development and the possible spintronics/pseudospintronics applications.
Abel, Steven [Durham Univ. (United Kingdom). Inst. for Particle Physics Phenomenology; CERN, Geneva (Switzerland); Goodsell, Mark [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2011-02-15
A simple and natural model is presented that gives Dirac gauginos. The configuration is related to ''deconstructed gaugino mediation''. A high energy completion is provided based on existing ISS-like models of deconstructed gaugino mediation. This provides a complete picture of Dirac gauginos that includes the necessary extra adjoint fermions (generated as magnetic quarks of the ISS theory) and supersymmetry breaking (via the ISS mechanism). Moreover the screening of the scalar masses means that they can similar to or less than the gaugino masses, even though the supersymmetry breaking is driven by F-terms. (orig.)
Weyl, Majorana and Dirac fields from a unified perspective
Aste, Andreas
2016-01-01
A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory which are commonly not included in introductory courses treating fermionic fields via the Dirac equation in the first place.
Weyl, Majorana and Dirac Fields from a Unified Perspective
Andreas Aste
2016-08-01
Full Text Available A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory, which are commonly not included in introductory courses when treating fermionic fields via the Dirac equation in the first place.
Dirac Induction for loop groups
Posthuma, H.
2011-01-01
Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl-Kac character formula and show compatibility with Dirac induc
Classical electromagnetic radiation of the Dirac electron
Lanyi, G.
1973-01-01
A wave-function-dependent four-vector potential is added to the Dirac equation in order to achieve conservation of energy and momentum for a Dirac electron and its emitted electromagnetic field. The resultant equation contains solutions which describe transitions between different energy states of the electron. As a consequence it is possible to follow the space-time evolution of such a process. This evolution is shown in the case of the spontaneous emission of an electromagnetic field by an electron bound in a hydrogen-like atom. The intensity of the radiation and the spectral distribution are calculated for transitions between two eigenstates. The theory gives a self-consistent deterministic description of some simple radiation processes without using quantum electrodynamics or the correspondence principle.
Gravity, torsion, Dirac field and computer algebra using MAPLE and REDUCE
Vulcanov, D N
2002-01-01
The article presents computer algebra procedures and routines applied to the study of the Dirac field on curved spacetimes. The main part of the procedures is devoted to the construction of Pauli and Dirac matrices algebra on an anholonomic orthonormal reference frame. Then these procedures are used to compute the Dirac equation on curved spacetimes in a sequence of special dedicated routines. A comparative review of such procedures obtained for two computer algebra platforms (REDUCE + EXCALC and MAPLE + GRTensorII) is carried out. Applications for the calculus of Dirac equation on specific examples of spacetimes with or without torsion are pointed out.
One real function instead of the Dirac spinor function
Akhmeteli, Andrey
2010-01-01
Schr\\"{o}dinger (Nature, v.169, p.538(1952)) noted that for each solution of the equations of scalar electrodynamics (the Klein-Gordon-Maxwell electrodynamics) there is a physically equivalent (i.e. coinciding with it up to a gauge transform) solution with a real matter field, despite the widespread belief about charged fields requiring complex representation. Surprisingly, the same result is true for spinor electrodynamics (the Dirac-Maxwell electrodynamics): the Dirac equation for the four complex components of the spinor function can be replaced by a fourth-order equation for one of those components, and this component can be made real by a gauge transform.
Shankar, Varun; Wright, Grady B.; Fogelson, Aaron L.; Kirby, Robert M.
2014-05-01
We present a computational method for solving the coupled problem of chemical transport in a fluid (blood) with binding/unbinding of the chemical to/from cellular (platelet) surfaces in contact with the fluid, and with transport of the chemical on the cellular surfaces. The overall framework is the Augmented Forcing Point Method (AFM) (\\emph{L. Yao and A.L. Fogelson, Simulations of chemical transport and reaction in a suspension of cells I: An augmented forcing point method for the stationary case, IJNMF (2012) 69, 1736-52.}) for solving fluid-phase transport in a region outside of a collection of cells suspended in the fluid. We introduce a novel Radial Basis Function-Finite Difference (RBF-FD) method to solve reaction-diffusion equations on the surface of each of a collection of 2D stationary platelets suspended in blood. Parametric RBFs are used to represent the geometry of the platelets and give accurate geometric information needed for the RBF-FD method. Symmetric Hermite-RBF interpolants are used for enforcing the boundary conditions on the fluid-phase chemical concentration, and their use removes a significant limitation of the original AFM. The efficacy of the new methods are shown through a series of numerical experiments; in particular, second order convergence for the coupled problem is demonstrated.
P. G. L. Leach
2014-04-01
Full Text Available Dirac devised his theory of Quantum Mechanics and recognised that his operators resembled the canonical coordinates of Hamiltonian Mechanics. This gave the latter a new lease of life. We look at what happens to Dirac’s Quantum Mechanics if one starts from Hamiltonian Mechanics.
Trzetrzelewski, Maciej
2011-01-01
In c=1 units the product (mass x radius) for the neutron and the proton is about 4.7\\hbar assuming their radii equal to 1fm. We show that the corresponding products for the Dirac neutral and charged membrane coincide and are equal 1.6\\hbar.
Guignard, G
2005-01-01
The DIRAC project aims to the design and development of one of the key aspects of the international Facility for Antiproton and Ion Research (FAIR) planned for construction at GSI in Darmstadt, Germany: the broad implementation and optimization of ion storage/cooler rings and of in-ring experimentation with internal targets and secondary beams.
SU Ting; WANG Zhi-Wei
2010-01-01
@@ By using the generalized version of the dressing method,we consider a Dirac system.The types of nonlinear evo-lution equations discussed involve the integrable variable-coefficient Dirac equation and the defocusing nonlinear Schrodinger equation.As an application,their explicit solutions and Lax pairs are given.
Geometrization of the Dirac theory of the electron
Fock, V.
1977-01-01
Using the concept of parallel displacement of a half vector, the Dirac equations are generally written in invariant form. The energy tensor is formed and both the macroscopic and quantum mechanic equations of motion are set up. The former have the usual form: divergence of the energy tensor equals the Lorentz force and the latter are essentially identical with those of the geodesic line.
Wilson, B G; Sonnad, V
2011-02-14
Precise electronic structure calculations of ions in plasmas benefit from optimized numerical radial meshes. A new closed form expression for obtaining non-linear parameters for the efficient generation of analytic log-linear radial meshes is presented. In conjunction with the (very simple) algorithm for the rapid high precision evaluation of Lambert's W-function, the above identity allows the precise construction of generalized log-linear radial meshes adapted to various constraints.
A non-uniqueness problem of the Dirac theory in a curved spacetime
Arminjon, Mayeul
2009-01-01
The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. We find that, for each among the three versions of the equation, the changes that lead to an equivalent operator H are determined by initial data. Thus, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, whence a lack of uniqueness. In particular, the Dirac energy spectrum is not unique.
Dirac-Point Solitons in Nonlinear Optical Lattices
Xie, Kang; Boardman, Allan D; Guo, Qi; Shi, Zhiwei; Jiang, Haiming; Hu, Zhijia; Zhang, Wei; Mao, Qiuping; Hu, Lei; Yang, Tianyu; Wen, Fei; Wang, Erlei
2015-01-01
The discovery of a new type of solitons occuring in periodic systems without photonic bandgaps is reported. Solitons are nonlinear self-trapped wave packets. They have been extensively studied in many branches of physics. Solitons in periodic systems, which have become the mainstream of soliton research in the past decade, are localized states supported by photonic bandgaps. In this Letter, we report the discovery of a new type of solitons located at the Dirac point beyond photonic bandgaps. The Dirac point is a conical singularity of a photonic band structure where wave motion obeys the famous Dirac equation. These new solitons are sustained by the Dirac point rather than photonic bandgaps, thus provides a sort of advance in conceptual understanding over the traditional gap solitons. Apart from their theoretical impact within soliton theory, they have many potential uses because such solitons have dramatic stability characteristics and are possible in both Kerr material and photorefractive crystals that poss...
Time-dependent massless Dirac fermions in graphene
Khantoul, Boubakeur, E-mail: bobphys@gmail.com [Department of Mathematics, City University London, Northampton Square, London EC1V 0HB (United Kingdom); Department of Physics, University of Jijel, BP 98, Ouled Aissa, 18000 Jijel (Algeria); Fring, Andreas, E-mail: a.fring@city.ac.uk [Department of Mathematics, City University London, Northampton Square, London EC1V 0HB (United Kingdom)
2015-10-30
Using the Lewis–Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasi-particles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a time-dependent vector potential. The eigenvalue equations for the two spinor components of the Lewis–Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians. - Highlights: • An explicit analytical solution for a massless 2+1 dimensional time-dependent Dirac equation is found. • All steps of the Lewis–Riesenfeld method have been carried out.
陈少伟; 肖利琴
2016-01-01
研究了一类带大参数的周期Thomas-Fermi-Dirac-von Weizs(a)cker方程非零解的存在性问题,运用一个新的无穷维环绕定理,证明了当参数充分大时,该方程存在非零解.
Pramono, Subur; Cari, Cari
2016-01-01
In this work, we study the exact solution of Dirac equation in the hyper-spherical coordinate under influence of separable q-Deformed quantum potentials. The q-deformed hyperbolic Rosen-Morse potential is perturbed by q-deformed non-central trigonometric Scarf potentials, where whole of them can be solved by using Asymptotic Iteration Method (AIM). This work is limited to spin symmetry case. The relativistic energy equation and orbital quantum number equation lD-1 have been obtained using Asymptotic Iteration Method. The upper radial wave function equations and angular wave function equations are also obtained by using this method. The relativistic energy levels are numerically calculated using Mat Lab, the increase of radial quantum number n causes the increase of bound state relativistic energy level both in dimension D = 5 and D = 3. The bound state relativistic energy level decreases with increasing of both deformation parameter q and orbital quantum number nl.
DIRAC Workload Management System
Garonne, V; Stokes-Rees, I
2005-01-01
The Workload Management System is the core component of the DIRAC distributed MC production and analysis grid environment of the CERN LHCb experiment. This paper discusses the architecture, implementation and performance of this system. The WMS is a community scheduler, realizing a pull paradigm, particulary for the high troughput computing context. It has recently been used for an intensive physics simulation production involving more than 60 sites, 65 TB of data, and over 1000-GHz processor-years.
Quantum transport through 3D Dirac materials
Salehi, M. [Department of Physics, Sharif University of Technology, Tehran 11155-9161 (Iran, Islamic Republic of); Jafari, S.A., E-mail: jafari@physics.sharif.edu [Department of Physics, Sharif University of Technology, Tehran 11155-9161 (Iran, Islamic Republic of); Center of Excellence for Complex Systems and Condensed Matter (CSCM), Sharif University of Technology, Tehran 1458889694 (Iran, Islamic Republic of)
2015-08-15
Bismuth and its alloys provide a paradigm to realize three dimensional materials whose low-energy effective theory is given by Dirac equation in 3+1 dimensions. We study the quantum transport properties of three dimensional Dirac materials within the framework of Landauer–Büttiker formalism. Charge carriers in normal metal satisfying the Schrödinger equation, can be split into four-component with appropriate matching conditions at the boundary with the three dimensional Dirac material (3DDM). We calculate the conductance and the Fano factor of an interface separating 3DDM from a normal metal, as well as the conductance through a slab of 3DDM. Under certain circumstances the 3DDM appears transparent to electrons hitting the 3DDM. We find that electrons hitting the metal-3DDM interface from metallic side can enter 3DDM in a reversed spin state as soon as their angle of incidence deviates from the direction perpendicular to interface. However the presence of a second interface completely cancels this effect.
Zero Point Energy and the Dirac Equation
Forouzbakhsh, Farshid
2007-01-01
Zero Point Energy (ZPE) describes the random electromagnetic oscillations that are left in the vacuum after all other energy has been removed. One way to explain this is by means of the uncertainty principle of quantum physics, which implies that it is impossible to have a zero energy condition. I...... this article, the ZPE is explained by using a novel description of the graviton. This is based on the behavior of photons in gravitational field, leading to a new definition of the graviton. In effect, gravitons behave as if they have charge and magnetic effects. These are referred to as negative color charge...
Classical behaviour of the Dirac bispinor
Bell, S B M; Díaz, B M; Bell, Sarah B. M.; Cullerne, John P.; Diaz, Bernard M.
2000-01-01
It is usually supposed that the Dirac and radiation equations predict that the phase of a fermion will rotate through half the angle through which the fermion is rotated, which means, via the measured dynamical and geometrical phase factors, that the fermion must have a half-integral spin. We demonstrate that this is not the case and that the identical relativistic quantum mechanics can also be derived with the phase of the fermion rotating through the same angle as does the fermion itself. Under spatial rotation and Lorentz transformation the bispinor transforms as a four-vector like the potential and Dirac current. Previous attempts to provide this form of transformational behaviour have foundered because a satisfactory current could not be derived.(14)
Octonion generalization of Pauli and Dirac matrices
Chanyal, B. C.
2015-10-01
Starting with octonion algebra and its 4 × 4 matrix representation, we have made an attempt to write the extension of Pauli's matrices in terms of division algebra (octonion). The octonion generalization of Pauli's matrices shows the counterpart of Pauli's spin and isospin matrices. In this paper, we also have obtained the relationship between Clifford algebras and the division algebras, i.e. a relation between octonion basis elements with Dirac (gamma), Weyl and Majorana representations. The division algebra structure leads to nice representations of the corresponding Clifford algebras. We have made an attempt to investigate the octonion formulation of Dirac wave equations, conserved current and weak isospin in simple, compact, consistent and manifestly covariant manner.
Aloisi, A.M.; Nali, P. F.
2016-01-01
In 1931, Dirac advanced a startling prediction about the existence of a new elementary particle, characterized by a magnetic charge of a single polarity: the magnetic monopole. This prediction, that was not based on experimental reasons but on mathematical consistency considerations and the generalization of the formalism of quantum mechanics, illustrates emblematically the Dirac conception of the relationship between physics and mathematics. ----- Nel 1931 Dirac avanz\\`o una sorprendente pre...
Exact analytic solutions for a Dirac electron moving in graphene under magnetic fields.
Kuru, S; Negro, J; Nieto, L M
2009-11-11
Exact analytical solutions for the bound states of a graphene Dirac electron in various magnetic fields with translational symmetry are obtained. In order to solve the time-independent Dirac-Weyl equation the factorization method used in supersymmetric quantum mechanics is adapted to this problem. The behavior of the discrete spectrum, probability and current densities are discussed.
Unified Description of Dirac Electrons on a Curved Surface of Topological Insulators
Takane, Yositake; Imura, Ken-Ichiro
2013-07-01
Existence of a protected surface state described by a massless Dirac equation is a defining property of the topological insulator. Though this statement can be explicitly verified on an idealized flat surface, it remains to be addressed to what extent it could be general. On a curved surface, the surface Dirac equation is modified by the spin connection terms. Here, in the light of the differential geometry, we give a general framework for constructing the surface Dirac equation starting from the Hamiltonian for bulk topological insulators. The obtained unified description clarifies the physical meaning of the spin connection.
Darboux partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
Schulze-Halberg, Axel, E-mail: xbataxel@gmail.com [Department of Mathematics and Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary, IN 46408 (United States); Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, IN 46408 (United States); Roy, Barnana, E-mail: barnana@isical.ac.in [Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108 (India)
2014-10-15
We introduce a method for constructing Darboux (or supersymmetric) pairs of pseudoscalar and scalar Dirac potentials that are associated with exceptional orthogonal polynomials. Properties of the transformed potentials and regularity conditions are discussed. As an application, we consider a pseudoscalar Dirac potential related to the Schrödinger model for the rationally extended radial oscillator. The pseudoscalar partner potentials are constructed under the first- and second-order Darboux transformations.
The massive Dirac field on a rotating black hole spacetime: Angular solutions
Dolan, Sam
2009-01-01
The massive Dirac equation on a Kerr-Newman background may be solved by the method of separation of variables. The radial and angular equations are coupled via an angular eigenvalue, which is determined from the Chandrasekhar-Page (CP) equation. Obtaining accurate angular eigenvalues is a vital first step in studying scattering, absorption and emission of the fermionic field. Here we introduce a new method for finding the angular solutions of the CP equation (so-called mass-dependent spin-half spheroidal harmonics). First, we introduce a novel representation for the spin-half spherical harmonics. Next, we decompose the spheroidal harmonics in the basis of spherical harmonics. The approach yields a three-term recurrence relation which may be solved numerically with continued fraction methods, or perturbatively to obtain a series expansion for the eigenvalues. In the case $\\mu = \\pm \\omega$ (where $\\omega$ and $\\mu$ are the frequency and mass of the fermion) we obtain eigenvalues and eigenfunctions in closed fo...
Production of Dirac particle in twisted Minkowsky space-time
Samary, Dine Ousmane; Kanfon, Antonin
2015-01-01
In this paper we study the Dirac equation interacting with external gravitation field. This curve background, which correspond to the deformation of Minkowsky space-time is described with the tetrad of the form $e_b^\\mu(x)=\\varepsilon(\\delta_b^\\mu+\\omega_{ba}^\\mu x^a)$, where $\\varepsilon=1$ for $\\mu=0$ and $\\varepsilon=i$ for $\\mu=1,2,3.$ Using separation of variables the corresponding Dirac equation is solved. The probability density of the vacuum-vacuum pair creation is given. In particular case of vanishing electromagnetic fields, we point out that, this external gravitation field modify weakly the well know probability of pair production of the Dirac particle given in ordinary space-time.
LHCb: DIRAC Secure Distributed Platform
Casajus, A
2009-01-01
DIRAC, the LHCb community grid solution, provides access to a vast amount of computing and storage resources to a large number of users. In DIRAC users are organized in groups with different needs and permissions. In order to ensure that only allowed users can access the resources and to enforce that there are no abuses, security is mandatory. All DIRAC services and clients use secure connections that are authenticated using certificates and grid proxies. Once a client has been authenticated, authorization rules are applied to the requested action based on the presented credentials. These authorization rules and the list of users and groups are centrally managed in the DIRAC Configuration Service. Users submit jobs to DIRAC using their local credentials. From then on, DIRAC has to interact with different Grid services on behalf of this user. DIRAC has a proxy management service where users upload short-lived proxies to be used when DIRAC needs to act on behalf of them. Long duration proxies are uploaded by us...
Consequences of a Relativistic Pseudospin Symmetry for Radial Nodes and Intruder Levels in Nuclei
Leviatan, A
2001-01-01
The identification of pseudospin symmetry as a relativistic symmetry of the Dirac Hamiltonian is used to explain the structure of radial nodes occurring in pseudospin doublets and to illuminate the special status of nodeless intruder states in nuclei.
CREUTZ, M.
2006-01-26
It is popular to discuss low energy physics in lattice gauge theory ill terms of the small eigenvalues of the lattice Dirac operator. I play with some ensuing pitfalls in the interpretation of these eigenvalue spectra. In short, thinking about the eigenvalues of the Dirac operator in the presence of gauge fields can give some insight, for example the elegant Banks-Casher picture for chiral symmetry breaking. Nevertheless, care is necessary because the problem is highly non-linear. This manifests itself in the non-intuitive example of how adding flavors enhances rather than suppresses low eigenvalues. Issues involving zero mode suppression represent one facet of a set of connected unresolved issues. Are there non-perturbative ambiguities in quantities such as the topological susceptibility? How essential are rough gauge fields, i.e. gauge fields on which the winding number is ambiguous? How do these issues interplay with the quark masses? I hope the puzzles presented here will stimulate more thought along these lines.
Dirac gauginos, gauge mediation and unification
Benakli, K., E-mail: kbenakli@lpthe.jussieu.f [Laboratoire de Physique Theorique et Hautes Energies, CNRS, UPMC Univ. Paris 06 Boite 126, 4 Place Jussieu, 75252 Paris cedex 05 (France); Goodsell, M.D., E-mail: mark.goodsell@desy.d [Deutsches Elektronen-Synchrotron, DESY, Notkestrasse 85, 22607 Hamburg (Germany)
2010-11-21
We investigate the building of models with Dirac gauginos and perturbative gauge coupling unification. Here, in contrast to the MSSM, additional fields are required for unification, and these can naturally play the role of the messengers of supersymmetry breaking. We present a framework within which such models can be constructed, including the constraints that the messenger sector must satisfy; and the renormalisation group equations for the soft parameters, which differ from those of the MSSM. For illustration, we provide the spectrum at the electroweak scale for explicit models whose gauge couplings unify at the scale predicted by heterotic strings.
Dirac gauginos, gauge mediation and unification
Benakli, K. [UPMC Univ. Paris 06 (France). Laboratoire de Physique Theorique et Hautes Energies, CNRS; Goodsell, M.D. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2010-03-15
We investigate the building of models with Dirac gauginos and perturbative gauge coupling unification. Here, in contrast to the MSSM, additional fields are required for unification, and these can naturally play the role of the messengers of supersymmetry breaking. We present a framework within which such models can be constructed, including the constraints that the messenger sector must satisfy; and the renormalisation group equations for the soft parameters, which differ from those of the MSSM. For illustration, we provide the spectrum at the electroweak scale for explicit models whose gauge couplings unify at the scale predicted by heterotic strings. (orig.)
Dirac's Constrained Hamiltonian Dynamics from an Unconstrained Dynamics
Rothe, Heinz J.
2003-01-01
We derive the Hamilton equations of motion for a constrained system in the form given by Dirac, by a limiting procedure, starting from the Lagrangean for an unconstrained system. We thereby ellucidate the role played by the primary constraints and their persistance in time.
From "Dirac combs" to Fourier-positivity
Giraud, Bertrand G
2015-01-01
Motivated by various problems in physics and applied mathematics, we look for constraints and properties of real Fourier-positive functions, i.e. with positive Fourier transforms. Properties of the "Dirac comb" distribution and of its tensor products in higher dimensions lead to Poisson resummation, allowing for a useful approximation formula of a Fourier transform in terms of a limited number of terms. A connection with the Bochner theorem on positive definiteness of Fourier-positive functions is discussed. As a practical application, we find simple and rapid analytic algorithms for checking Fourier-positivity in 1- and (radial) 2-dimensions among a large variety of real positive functions. This may provide a step towards a classification of positive positive-definite functions.
Blanchet, Steve
2007-01-01
I present here a concise summary of the preprint arXiv:0707.3024, written in collaboration with A. Anisimov and P. Di Bari. There we discuss leptogenesis when {\\em CP} violation stems exlusively from the Dirac phase in the PMNS mixing matrix. Under this assumption it turns out that the situation is very constrained when a hierarchical heavy right-handed (RH) neutrino spectrum is considered: the allowed regions are small and the final asymmetry depends on the initial conditions. On the other hand, for a quasi-degenerate spectrum of RH neutrinos, the {\\em CP} asymmetry can be enhanced and the situation becomes much more favorable, with no dependence on the initial conditions. Interestingly, in the extreme case of resonant leptogenesis, in order to match the observed baryon asymmetry of the Universe, we obtain a lower bound on \\sin \\q_{13} which depends on the lightest active neutrino mass m_1.
Spectrum of the Hermitian Wilson-Dirac Operator for a Uniform Magnetic Field in Two Dimensions
Kurokawa, H
2003-01-01
It is shown that the eigenvalue problem for the hermitian Wilson-Dirac operator of for a uniform magnetic field in two dimensions can be reduced to one-dimensional problem described by a relativistic analog of the Harper equation. An explicit formula for the secular equations is given in term of a set of polynomials. The spectrum exhibits a fractal structure in the infinite volume limit. An exact result concerning the index theorem for the overlap Dirac operator is obtained.
Saleh, Mahamat; Bouetou, Bouetou Thomas; Kofane, Timoleon Crepin
2016-04-01
In this work, quasinormal modes (QNMs) of the Schwarzschild black hole are investigated by taking into account the quantum fluctuations. Gravitational and Dirac perturbations were considered for this case. The Regge-Wheeler gauge and the Dirac equation were used to derive the perturbation equations of the gravitational and Dirac fields respectively and the third order Wentzel-Kramers-Brillouin (WKB) approximation method is used for the computing of the quasinormal frequencies. The results show that due to the quantum fluctuations in the background of the Schwarzschild black hole, the QNMs of the black hole damp more slowly when increasing the quantum correction factor (a), and oscillate more slowly.
On the Origin of the Charge-Asymmetric Matter. II. Localized Dirac Waveforms
Makhlin, Alexander
2016-01-01
This paper continues the author's work \\cite{PartI}, where a new framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation discovered. Here, the nonlinear Dirac equation is solved and the localized configurations are found analytically. Of the two possible types of the potentially stationary localized configurations of the Dirac field, only one is stable with respect to the action of an external field and it corresponds to a positive charge. A connection with the global charge asymmetry of matter in the Universe and with the recently observed excess of the cosmic positrons is discussed.
Saleh, Mahamat; Crépin, Kofané Timoléon
2016-01-01
In this work, quasinormal modes (QNMs) of the Schwarzschild black hole are investigated by taking into account the quantum fluctuations. Gravitational and Dirac perturbations were considered for this case. The Regge-Wheeler gauge and the Dirac equation were used to derive the perturbation equations of the gravitational and Dirac fields respectively and the third order Wentzel-Kramers-Brillouin (WKB) approximation method is used for the computing of the quasinormal frequencies. The results show that due to the quantum fluctuations in the background of the Schwarzschild black hole, the QNMs of the black hole damp more slowly when increasing the quantum correction factor (a), and oscillate more slowly.
Karbstein, Felix
2007-01-01
We use 1+1 dimensional large N Gross-Neveu models as a laboratory to derive microscopically effective Lagrangians for positive energy fermions only. When applied to baryons, the Euler-Lagrange equation for these effective theories assumes the form of a non-linear Dirac equation. Its solution reproduces the full semi-classical results including the Dirac sea to any desired accuracy. Dynamical effects from the Dirac sea are encoded in higher order derivative terms and multi-fermion interactions with perturbatively calculable, finite coefficients. Characteristic differences between models with discrete and continuous chiral symmetry are observed and clarified.
Babourova, O V; Kudlaev, P E; Romanova, E V
2016-01-01
The Poincare and Poincare-Weyl gauge theories of gravitation with Lagrangians quadratic on curvature and torsion in post-Riemannian spaces with the Dirac scalar field is discussed in a historical aspect. The various hypothesizes concerning the models of a dark matter with the help of a scalar field are considered. The new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl space-time with the Dirac scalar field representing the dark matter model. A static spherically symmetric solution of the field equations in vacuum for a central compact mass is obtained as the metrics conformal to the Yilmaz-Rosen metrics. On the base of this solution one considers a radial movement of an interplanetary spacecraft starting from the Earth. Using the Newton approximation one obtains that the asymptotic line-of-sight velocity in this case depends from the parameters of the solution, and therefore one can obtain on basis of the observable data the values of these parameters.
Dirac particles tunneling from black holes with topological defects
Jusufi, Kimet
2015-01-01
We study Hawking radiation of Dirac particles with spin-$1/2$ as a tunneling process from Schwarzschild-de Sitter and Reissner-Nordstr\\"{o}m-de Sitter black holes in background spacetimes with a spinning cosmic string and a global monopole. Solving Dirac's equation by employing the Hamilton-Jacobi method and WKB approximation we find the corresponding tunneling probabilities and the Hawking temperature. Furthermore, we show that the Hawking temperature of black holes remains unchanged in presence of topological defects in both cases.
Noncommutative Dirac quantization condition using the Seiberg-Witten map
Maceda, Marco; Martínez-Carbajal, Daniel
2016-11-01
The Dirac quantization condition (DQC) for magnetic monopoles in noncommutative space-time is analyzed. For this a noncommutative generalization of the method introduced by Wu and Yang is considered; the effects of noncommutativity are analyzed using the Seiberg-Witten map and the corresponding deformed Maxwell's equations are discussed. By using a perturbation expansion in the noncommutativity parameter θ , we show first that the DQC remains unmodified up to the first and second order. This result is then generalized to all orders in the expansion parameter for a class of noncommutative electric currents induced by the Seiberg-Witten map; these currents reduce to the Dirac delta function in the commutative limit.
Weak cosmic censorship, dyonic Kerr-Newman black holes and Dirac fields
Zsolt Tóth, Gábor
2016-06-01
It was investigated recently, with the aim of testing the weak cosmic censorship conjecture, whether an extremal Kerr black hole can be converted into a naked singularity by interaction with a massless classical Dirac test field, and it was found that this is possible. We generalize this result to electrically and magnetically charged rotating extremal black holes (i.e. extremal dyonic Kerr-Newman black holes) and massive Dirac test fields, allowing magnetically or electrically uncharged or nonrotating black holes and the massless Dirac field as special cases. We show that the possibility of the conversion is a direct consequence of the fact that the Einstein-Hilbert energy-momentum tensor of the classical Dirac field does not satisfy the null energy condition, and is therefore not in contradiction with the weak cosmic censorship conjecture. We give a derivation of the absence of superradiance of the Dirac field without making use of the complete separability of the Dirac equation in the dyonic Kerr-Newman background, and we determine the range of superradiant frequencies of the scalar field. The range of frequencies of the Dirac field that can be used to convert a black hole into a naked singularity partially coincides with the superradiant range of the scalar field. We apply horizon-penetrating coordinates, as our arguments involve calculating quantities at the event horizon. We describe the separation of variables for the Dirac equation in these coordinates, although we mostly avoid using it.
Aloisi, A M
2016-01-01
In 1931, Dirac advanced a startling prediction about the existence of a new elementary particle, characterized by a magnetic charge of a single polarity: the magnetic monopole. This prediction, that was not based on experimental reasons but on mathematical consistency considerations and the generalization of the formalism of quantum mechanics, illustrates emblematically the Dirac conception of the relationship between physics and mathematics. ----- Nel 1931 Dirac avanz\\`o una sorprendente previsione circa l'esistenza di una nuova particella elementare, caratterizzata da una carica magnetica di un'unica polarit\\`a: il monopolo magnetico. Questa previsione, che non era fondata su ragioni sperimentali ma su considerazioni di consistenza matematica e sulla generalizzazione del formalismo della meccanica quantistica, illustra emblematicamente la concezione di Dirac del rapporto tra fisica e matematica.
Peter Zhidkov
2009-01-01
We consider the following eigenvalue problem: − Δ + ( ) = , = ( ) , ∈ = { ∈ ℝ 3 ∶ | | 0 , | | | = 1 = 0 , where is an arbitrary fixed parameter and is an odd smooth function. First, we prove that for each integer ≥ 0 there exists a radially symmetric eigenfunction which possesses precisely zeros being regarded as a function of = | | ∈ [ 0 , 1 ) . For > 0 sufficiently small, such an eigenfunction is unique for each . Then, w...
Electronic structure of a graphene superlattice with massive Dirac fermions
Lima, Jonas R. F., E-mail: jonas.iasd@gmail.com [Instituto de Ciencia de Materiales de Madrid (CSIC) - Cantoblanco, Madrid 28049 (Spain)
2015-02-28
We study the electronic and transport properties of a graphene-based superlattice theoretically by using an effective Dirac equation. The superlattice consists of a periodic potential applied on a single-layer graphene deposited on a substrate that opens an energy gap of 2Δ in its electronic structure. We find that extra Dirac points appear in the electronic band structure under certain conditions, so it is possible to close the gap between the conduction and valence minibands. We show that the energy gap E{sub g} can be tuned in the range 0 ≤ E{sub g} ≤ 2Δ by changing the periodic potential. We analyze the low energy electronic structure around the contact points and find that the effective Fermi velocity in very anisotropic and depends on the energy gap. We show that the extra Dirac points obtained here behave differently compared to previously studied systems.
CERN Bulletin
2010-01-01
When a group of physicists entered the Main Auditorium, during the evening of 29 June, they felt they had opened a time portal. Paul Dirac in front of a blackboard showing his formula. ©Sandra Hoogeboom An attentive audience, dressed in early 1900 costumes, were watching a lecture by the elusive Paul Dirac, presenting for the first time his famous formula on the blackboard. Paul Adrien Maurice Dirac (1902-1984) was a British mathematical physicist at Cambridge, and one of the "fathers" of quantum mechanics. When he first wrote it, in 1928, Dirac was not sure what his formula really meant. As demonstrated by Andersson four year later, what Dirac had written on the blackboard was the first definition of a positron, hence he is credited with having anticipated the existence of antimatter. The actor John Kohl performs as Paul Dirac. ©Sandra Hoogeboom What the group of puzzled physicists were really observing when they entered the CERN Auditorium was the shoo...
LHCb: LHCbDirac is a DIRAC extension to support LHCb specific workflows
Stagni, Federico
2012-01-01
We present LHCbDIRAC, an extension of the DIRAC community Grid solution to handle the LHCb specificities. The DIRAC software has been developed for many years within LHCb only. Nowadays it is a generic software, used by many scientific communities worldwide. Each community wanting to take advantage of DIRAC has to develop an extension, containing all the necessary code for handling their specific cases. LHCbDIRAC is an actively developed extension, implementing the LHCb computing model and workflows. LHCbDIRAC extends DIRAC to handle all the distributed computing activities of LHCb. Such activities include real data processing (reconstruction, stripping and streaming), Monte-Carlo simulation and data replication. Other activities are groups and user analysis, data management, resources management and monitoring, data provenance, accounting for user and production jobs. LHCbDIRAC also provides extensions of the DIRAC interfaces, including a secure web client, python APIs and CLIs. While DIRAC and LHCbDIRAC f...
A convergent 2D finite-difference scheme for the Dirac-Poisson system and the simulation of graphene
Brinkman, Daniel
2014-01-01
We present a convergent finite-difference scheme of second order in both space and time for the 2D electromagnetic Dirac equation. We apply this method in the self-consistent Dirac-Poisson system to the simulation of graphene. The model is justified for low energies, where the particles have wave vectors sufficiently close to the Dirac points. In particular, we demonstrate that our method can be used to calculate solutions of the Dirac-Poisson system where potentials act as beam splitters or Veselago lenses. © 2013 Elsevier Inc.
Formulas for Radial Transport in Protoplanetary Disks
Desch, Steven J.; Estrada, Paul R.; Kalyaan, Anusha; Cuzzi, Jeffrey N.
2017-05-01
The quantification of the radial transport of gaseous species and solid particles is important to many applications in protoplanetary disk evolution. An especially important example is determining the location of the water snow lines in a disk, which requires computing the rates of outward radial diffusion of water vapor and the inward radial drift of icy particles; however, the application is generalized to evaporation fronts of all volatiles. We review the relevant formulas using a uniform formalism. This uniform treatment is necessary because the literature currently contains at least six mutually exclusive treatments of radial diffusion of gas, only one of which is correct. We derive the radial diffusion equations from first principles using Fick's law. For completeness, we also present the equations for radial transport of particles. These equations may be applied to studies of diffusion of gases and particles in protoplanetary and other accretion disks.
Excitonic pairing and insulating transition in two-dimensional semi-Dirac semimetals
Wang, Jing-Rong; Liu, Guo-Zhu; Zhang, Chang-Jin
2017-02-01
A sufficiently strong long-range Coulomb interaction can induce excitonic pairing in gapless Dirac semimetals, which generates a finite gap and drives the semimetal-insulator quantum phase transition. This phenomenon is in close analogy to dynamical chiral symmetry breaking in high-energy physics. In most realistic Dirac semimetals, including suspended graphene, the Coulomb interaction is too weak to open an excitonic gap. The Coulomb interaction plays a more important role at low energies in a two-dimensional semi-Dirac semimetal, in which the fermion spectrum is linear in one component of momenta and quadratic in the other, than a Dirac semimetal, and indeed leads to breakdown of Fermi liquid theory. We study dynamical excitonic gap generation in a two-dimensional semi-Dirac semimetal by solving the Dyson-Schwinger equation, and show that a moderately strong Coulomb interaction suffices to induce excitonic pairing. Additional short-range four-fermion coupling tends to promote excitonic pairing. Among the available semi-Dirac semimetals, we find that the TiO2/VO2 nanostructure provides a promising candidate for the realization of an excitonic insulator. We also apply the renormalization group method to analyze the strong coupling between the massless semi-Dirac fermions and the quantum critical fluctuation of the excitonic order parameter at the semimetal-insulator quantum critical point, and reveal non-Fermi liquid behaviors of semi-Dirac fermions.
DIRAC: a community grid solution
Tsaregorodtsev, A [Centre de Physique des Particules de Marseille, 163 Av de Luminy Case 902 13288 Marseille (France); Bargiotti, M; Castellani, G; Charpentier, P; Closier, J; Paterson, S; Santinelli, R [CERN CH-1211 Geneve 23 (Switzerland); Brook, N [H. H. Wills Physics Laboratory, Royal Fort, Tyndal Avenue, Bristol BS8 1TL (United Kingdom); Ramo, A C; Diaz, R G [University of Barcelona, Diagonal 647, ES-08028 Barcelona (Spain); Cioffi, C [University of Oxford, 1, Keble Road, Oxford OX1 3NP (United Kingdom); Kuznetsov, G; Nandakumar, R [Rutherford Appleton Laboratory, Chilton, Didcot Oxon. OX11 0QX (United Kingdom); Li, Y Y [University of Cambridge, Wilberforce Road, Cambridge CB3 OWA (United Kingdom); Miguelez, M S [University of Santiago de Compostela, Campus Universitario Sur, ES-15706 Santiago de Compostela (Spain); Jimenez, S G [University Rovira i Virgili, Campus Sescelades, Avinguda dels Paisos Catalans, 26 Tarragona (Spain); Smith, A C, E-mail: atsareg@in2p3.fr
2008-07-15
The DIRAC system was developed in order to provide a complete solution for using the distributed computing resources of the LHCb experiment at CERN for data production and analysis. It allows a concurrent use of over 10K CPUs and 10M file replicas distributed over many tens of sites. The sites can be part of a Computing Grid such as WLCG or standalone computing clusters all integrated in a single management structure. DIRAC is a generic system with the LHCb specific functionality incorporated through a number of plug-in modules. It can be easily adapted to the needs of other communities. Special attention is paid to the resilience of the DIRAC components to allow an efficient use of non-reliable resources. The DIRAC production management components provide a framework for building highly automated data production systems including data distribution and data driven workload scheduling. In this paper we give an overview of the DIRAC system architecture and design choices. We show how different components are put together to compose an integrated data processing system including all the aspects of the LHCb experiment - from the MC production and raw data reconstruction to the final user analysis.
DIRAC: a community grid solution
Tsaregorodtsev, A; Brook, N; Ramo, A C; Castellani, G; Charpentier, P; Cioffi, C; Closier, J; Díaz, R G; Kuznetsov, G; Li, Y Y; Nandakumar, R; Paterson, S; Santinelli, R; Smith, A C; Miguelez, M S; Jimenez, S G
2008-01-01
The DIRAC system was developed in order to provide a complete solution for using the distributed computing resources of the LHCb experiment at CERN for data production and analysis. It allows a concurrent use of over 10K CPUs and 10M file replicas distributed over many tens of sites. The sites can be part of a Computing Grid such as WLCG or standalone computing clusters all integrated in a single management structure. DIRAC is a generic system with the LHCb specific functionality incorporated through a number of plug-in modules. It can be easily adapted to the needs of other communities. Special attention is paid to the resilience of the DIRAC components to allow an efficient use of non-reliable resources. The DIRAC production management components provide a framework for building highly automated data production systems including data distribution and data driven workload scheduling. In this paper we give an overview of the DIRAC system architecture and design choices. We show how different components are pu...
Non-uniqueness of the Dirac theory in a curved spacetime
Arminjon, Mayeul
2010-01-01
We summarize a recent work on the subject title. The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. In this paper, we focus on the standard version of the gravitational Dirac equation, but the non-uniqueness applies also to our alternative versions. We find that the changes which lead to an equivalent operator H, or respectively to an equivalent operator E, are determined by initial data, or respectively have to make some point-dependent antihermitian matrix vanish. Thus, the vast majority of the possible coefficient changes lead neither to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. We show that even the Dirac energy spectrum is not unique.
Non-uniqueness of the Dirac theory in a curved spacetime
Arminjon, Mayeul; Reifler, Frank
2010-04-01
We summarize a recent work on the subject title. The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. In this paper, we focus on the standard version of the gravitational Dirac equation, but the non-uniqueness applies also to our alternative versions. We find that the changes which lead to an equivalent operator H, or respectively to an equivalent operator E, are determined by initial data, or respectively have to make some point-dependent antihermitian matrix vanish. Thus, the vast majority of the possible coefficient changes lead neither to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. We show that even the Dirac energy spectrum is not unique.
Das, Tapas
2015-01-01
The second order $N$-dimensional Schr\\"odinger equation with pseudoharmonic potential is reduced to a first order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Our results generalize all other previous works that done for various potential combinations in the case of lower dimensions.The Ladder operators are also constructed for the pseudoharmonic potential in $N$-dimensions.Lie algebra associated with these operators are studied and found that they satisfy the commutation relations for the SU(1,1) group. Matrix elements of different operators such as $z$, $z\\frac{d}{dz}$ are derived and finally the Casimir operator is discussed briefly.
Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension
Bisio, Alessandro; D’Ariano, Giacomo Mauro; Tosini, Alessandro, E-mail: alessandro.tosini@unipv.it
2015-03-15
We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be efficiently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing. In this framework, the discretization is a central ingredient and not only a tool for performing non-perturbative calculation as in lattice gauge theory. The automaton model, endowed with a precise notion of local observables and a full probabilistic interpretation, could lead to a coherent unification of a hypothetical discrete Planck scale with the usual Fermi scale of high-energy physics. - Highlights: • The free Dirac field in one space dimension as a quantum cellular automaton. • Large scale limit of the automaton and the emergence of the Dirac equation. • Dispersive differential equation for the evolution of smooth states on the automaton. • Optimal discrimination between the
Finster, Felix
2016-01-01
The massive Dirac equation is considered in the non-extreme Kerr geometry in horizon-penetrating Eddington-Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac waves outside and across the event horizon, up to the Cauchy horizon. For the proof, we write the Dirac equation in Hamiltonian form. One of the main difficulties is that the time evolution is not unitary, because the wave may "hit" the singularity. This problem is resolved by imposing suitable Dirichlet-type boundary conditions inside the Cauchy horizon, having no effect on the outside dynamics. Another main difficulty is that the Dirac Hamiltonian fails to be elliptic at the horizons. Combining the theory of symmetric hyperbolic systems with elliptic methods near the boundary, we construct a self-adjoint extension of the resulting Hamiltonian. We finally apply S...
Hydrodynamics of the Chiral Dirac Spectrum
Liu, Yizhuang; Zahed, Ismail
2016-01-01
We derive a hydrodynamical description of the eigenvalues of the chiral Dirac spectrum in the vacuum and in the large $N$ (volume) limit. The linearized hydrodynamics supports sound waves. The stochastic relaxation of the eigenvalues is captured by a hydrodynamical instanton configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of localized eigenvalues and unbroken chiral symmetry to a phase of de-localized eigenvalues and broken chiral symmetry occurs over a time set by the speed of sound. We show that the time is $\\Delta \\tau=\\pi\\rho(0)/2\\beta N$ with $\\rho(0)$ the spectral density at zero virtuality and $\\beta=1,2,4$ for the three Dyson ensembles that characterize QCD with different quark representations in the ergodic regime.
On regularizations of the Dirac delta distribution
Hosseini, Bamdad; Nigam, Nilima; Stockie, John M.
2016-01-01
In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the convergence of a sequence of distributions SH to a singular term S as a parameter H (associated with the support size of SH) shrinks to zero. We characterize this convergence in both the weak-* topology of distributions and a weighted Sobolev norm. These notions motivate a framework for constructing regularizations of the delta distribution that includes a large class of existing methods in the literature. This framework allows different regularizations to be compared. The convergence of solutions of PDEs with these regularized source terms is then studied in various topologies such as pointwise convergence on a deleted neighborhood and weighted Sobolev norms. We also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems.
Quantum Einstein-Dirac Bianchi Universes
Damour, Thibault
2011-01-01
We study the mini--superspace quantization of spatially homogeneous (Bianchi) cosmological universes sourced by a Dirac spinor field. The quantization of the homogeneous spinor leads to a finite-dimensional fermionic Hilbert space and thereby to a multi-component Wheeler-DeWitt equation whose main features are: (i) the presence of spin-dependent Morse-type potentials, and (ii) the appearance of a q-number squared-mass term, which is of order ${\\cal O}(\\hbar^2)$, and which is affected by ordering ambiguities. We give the exact quantum solution of the Bianchi type-II system (which contains both scattering states and bound states), and discuss the main qualitative features of the quantum dynamics of the (classically chaotic) Bianchi type-IX system. We compare the exact quantum dynamics of fermionic cosmological billiards to previous works that described the spinor field as being either classical or Grassmann-valued.
Dirac-graphene quasiparticles in strong slow-light pulse
Golovinski, P. A.; Astapenko, V. A.; Yakovets, A. V.
2017-02-01
An analytical Volkov's solution of the massless Dirac equation for graphene in the field of slow-light pulse with arbitrary time dependence is obtained. Exact solutions are presented for special cases of monochromatic field and a single-cycle pulse. Following the Fock-Schwinger proper time method, the Green's function for quasiparticles is derived with the account of the influence an external classical electromagnetic wave field.
Construction of the Pauli-Villars-Regulated Dirac Vacuum in Electromagnetic Fields
Gravejat, Philippe; Hainzl, Christian; Lewin, Mathieu; Séré, Éric
2013-05-01
Using the Pauli-Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac's vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics.
Superconductivity in doped Dirac semimetals
Hashimoto, Tatsuki; Kobayashi, Shingo; Tanaka, Yukio; Sato, Masatoshi
2016-07-01
We theoretically study intrinsic superconductivity in doped Dirac semimetals. Dirac semimetals host bulk Dirac points, which are formed by doubly degenerate bands, so the Hamiltonian is described by a 4 ×4 matrix and six types of k -independent pair potentials are allowed by the Fermi-Dirac statistics. We show that the unique spin-orbit coupling leads to characteristic superconducting gap structures and d vectors on the Fermi surface and the electron-electron interaction between intra and interorbitals gives a novel phase diagram of superconductivity. It is found that when the interorbital attraction is dominant, an unconventional superconducting state with point nodes appears. To verify the experimental signature of possible superconducting states, we calculate the temperature dependence of bulk physical properties such as electronic specific heat and spin susceptibility and surface state. In the unconventional superconducting phase, either dispersive or flat Andreev bound states appear between point nodes, which leads to double peaks or a single peak in the surface density of states, respectively. As a result, possible superconducting states can be distinguished by combining bulk and surface measurements.
Universal Behavior in Dirac Spectra
Verbaarschot, J J M
1997-01-01
In these lectures we review recent results on universal fluctuations of QCD Dirac spectra and applications of Random Matrix Theory (RMT) to QCD. We review general properties of Dirac spectra and discuss the relation between chiral symmetry breaking and correlations of Dirac eigenvalues. In particular, we will focus on the microscopic spectral density density, i.e. the spectral density near zero virtuality on the scale of a typical level spacing. The relation with Leutwyler-Smilga sum-rules will be discussed. The success of applications of RMT to spectra of 'complex' systems leads us to the introduction of a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. Our central conjecture is that it decribes correlations of QCD Dirac spectra. We will review recent universality proofs supporting this conjecture. Lattice QCD results for the microscopic spectral density and for correlations in the bulk of the spectrum are shown to be in perfect agreement with chRMT. We close wit...
Patrice Loïez
2002-01-01
Photo 01: The DIRAC upstream vacuum channel placed between the target and the upstream detector region. Both the non-intracting primary proton beam and the seconday particle channel travel inside the shown vacuum channel. Photo 02: The DIRAC upstream detector region consisting of 4 planes of GEM/MSGC; 3 planes of Scintillating Fibres; 4 planes of Ionisation hodospope. The photo shows the cabling of GEM/MSGC (right end) and Scintillating Fibres (left end) detectors. Photo 03: Detailed view of the 4 GEM/MSGC planes. The secondary particle channel and the detectors are tilted by 5.7 degrees with respect to the primary proton beam channel visible on the bottom. Photo 04: View of the downstream part of the double arm DIRAC spectrometer, facing the direction of incoming particles. The Drift Chamber system, the scintillation hodoscopes and the threshold Cherenkov counters are shown in the picture. Photo 05: The DIRAC vacuum region between upstream detectors and the dipole magnet. The shielding around the primary pro...
dftatom: A robust and general Schrödinger and Dirac solver for atomic structure calculations
Čertík, Ondřej; Pask, John E.; Vackář, Jiří
2013-07-01
A robust and general solver for the radial Schrödinger, Dirac, and Kohn-Sham equations is presented. The formulation admits general potentials and meshes: uniform, exponential, or other defined by nodal distribution and derivative functions. For a given mesh type, convergence can be controlled systematically by increasing the number of grid points. Radial integrations are carried out using a combination of asymptotic forms, Runge-Kutta, and implicit Adams methods. Eigenfunctions are determined by a combination of bisection and perturbation methods for robustness and speed. An outward Poisson integration is employed to increase accuracy in the core region, allowing absolute accuracies of 10-8 Hartree to be attained for total energies of heavy atoms such as uranium. Detailed convergence studies are presented and computational parameters are provided to achieve accuracies commonly required in practice. Comparisons to analytic and current-benchmark density-functional results for atomic number Z=1-92 are presented, verifying and providing a refinement to current benchmarks. An efficient, modular Fortran 95 implementation, dftatom, is provided as open source, including examples, tests, and wrappers for interface to other languages; wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program summaryProgram title:dftatom Catalogue identifier: AEPA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPA_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: MIT license No. of lines in distributed program, including test data, etc.: 14122 No. of bytes in distributed program, including test data, etc.: 157453 Distribution format: tar.gz Programming language: Fortran 95 with interfaces to Python and C. Computer: Any computer with a Fortran 95 compiler. Operating system: Any OS with a Fortran 95 compiler. RAM: 500 MB
A mathematical introduction to Dirac's formalism
van Eijndhoven, SJL
1986-01-01
This monograph contains a functional analytic introduction to Dirac''s formalism. The first part presents some new mathematical notions in the setting of triples of Hilbert spaces, mentioning the concept of Dirac basis. The second part introduces a conceptually new theory of generalized functions, integrating the notions of the first part.The last part of the book is devoted to a mathematical interpretation of the main features of Dirac''s formalism. It involves a pairing between distributional bras and kets, continuum expansions and continuum matrices.
A trace formula for Dirac Green's functions related by Darboux transformations
Pozdeeva, Ekaterina [Department of Quantum Field Theory, Tomsk State University, 36 Lenin Avenue, Tomsk 634050 (Russian Federation); Schulze-Halberg, Axel [Department of Mathematics, University of Colima, Bernal Diaz del Castillo 340, Colima, Col. 28045 (Mexico)], E-mail: ekatpozdeeva@mail.ru, E-mail: xbat@ucol.mx
2008-07-04
We construct Green's functions and a trace formula for pairs of stationary Dirac equations under Sturm-Liouville boundary conditions, where the equations are related to each other by a Darboux transformation. Our findings generalize former results (Pozdeeva E 2008 J. Phys. A: Math. Theor at press)
DIRAC pilot framework and the DIRAC Workload Management System
Casajus, Adrian; Graciani, Ricardo; Paterson, Stuart; Tsaregorodtsev, Andrei; LHCb DIRAC Team
2010-04-01
DIRAC, the LHCb community Grid solution, has pioneered the use of pilot jobs in the Grid. Pilot Jobs provide a homogeneous interface to an heterogeneous set of computing resources. At the same time, Pilot Jobs allow to delay the scheduling decision to the last moment, thus taking into account the precise running conditions at the resource and last moment requests to the system. The DIRAC Workload Management System provides one single scheduling mechanism for jobs with very different profiles. To achieve an overall optimisation, it organizes pending jobs in task queues, both for individual users and production activities. Task queues are created with jobs having similar requirements. Following the VO policy a priority is assigned to each task queue. Pilot submission and subsequent job matching are based on these priorities following a statistical approach.
Quarkonium and hydrogen spectra with spin-dependent relativistic wave equation
V H Zaveri
2010-10-01
The non-linear non-perturbative relativistic atomic theory introduces spin in the dynamics of particle motion. The resulting energy levels of hydrogen atom are exactly the same as that of Dirac theory. The theory accounts for the energy due to spin-orbit interaction and for the additional potential energy due to spin and spin-orbit coupling. Spin angular momentum operator is integrated into the equation of motion. This requires modification to classical Laplacian operator. Consequently, the Dirac matrices and the k operator of Dirac’s theory are dispensed with. The theory points out that the curvature of the orbit draws on certain amount of kinetic and potential energies affecting the momentum of electron and the spin-orbit interaction energy constitutes a part of this energy. The theory is developed for spin-1/2 bound state single electron in Coulomb potential and then extended further to quarkonium physics by introducing the linear confining potential. The unique feature of this quarkonium model is that the radial distance can be exactly determined and does not have a statistical interpretation. The established radial distance is then used to determine the wave function. The observed energy levels are used as the input parameters and the radial distance and the string tension are predicted. This ensures 100% conformance to all observed energy levels for the heavy quarkonium.
Gherase, Mihai R
2012-01-01
Diffusive spin exchange is one of the most important relaxation mechanisms in the Nuclear Magnetic Resonance (NMR) applications to medicine and biology. Two models based on the Bloch-McConnell (B-M) and the Bloch-Torrey (B-T) equations are commonly used for modelling the physical processes which determine the NMR lineshapes. Qualitative arguments for each of the two methods can be found in various studies in the literature. However, there is a lack of systematic quantitative investigations of the diffusive exchange spectra calculated with the two methods for the same physical system or model. In this work exact frequency-domain transverse magnetization solutions of the B-M and the B-T equations with boundary conditions for a two-compartment radial diffusive exchange model are presented. Theoretical spectra and the two corresponding metrics were computed by varying three different parameters: diffusive permeability of the separating membrane between the two compartments (P), the radius of the inner spherical c...
Equations of motion for charged particles in strong laser fields
Ruhl, Hartmut
2016-01-01
Starting from the Dirac equation coupled to a classical radiation field a set of equations of motion for charged quasi-particles in the classical limit for slowly varying radiation and matter fields is derived. The radiation reaction term derived in the paper is the Abraham-Lorentz-Dirac term.
On Dirac theory in the space with deformed Heisenberg algebra. Exact solutions
Vakarchuk, I O
2005-01-01
The Dirac equation has been studied in which the Dirac matrices $\\hat{\\boldmath$\\alpha$}, \\hat\\beta$ have space factors, respectively $f$ and $f_1$, dependent on the particle's space coordinates. The $f$ function deforms Heisenberg algebra for the coordinates and momenta operators, the function $f_1$ being treated as a dependence of the particle mass on its position. The properties of these functions in the transition to the Schr\\"odinger equation are discussed. The exact solution of the Dirac equation for the particle motion in the Coulomnb field with a linear dependence of the $f$ function on the distance $r$ to the force centre and the inverse dependence on $r$ for the $f_1$ function has been found.
On Dirac theory in the space with deformed Heisenberg algebra: exact solutions
Vakarchuk, I. O.
2005-08-01
The Dirac equation has been studied in which the Dirac matrices \\hat{{\\bm \\alpha}}, \\skew4\\hat{\\beta} have space factors, respectively f and f1, dependent on the particle's space coordinates. The function f deforms Heisenberg algebra for the coordinates and momenta operators, the function f1 being treated as a dependence of the particle mass on its position. The properties of these functions in the transition to the Schrödinger equation are discussed. The exact solution of the Dirac equation for the particle motion in the Coulomb field with a linear dependence of the function f on the distance r to the force centre and the inverse dependence on r for the function f1 has been found.
Distributions of Dirac Operator Eigenvalues
Akemann, G
2004-01-01
The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory.
Dirac neutrinos from flavor symmetry
Aranda, Alfredo; Morisi, S; Peinado, E; Valle, J W F
2013-01-01
We present a model where Majorana neutrino mass terms are forbidden by the flavor symmetry group Delta(27). Neutrinos are Dirac fermions and their masses arise in the same way as that of the charged fermions, due to very small Yukawa couplings. The model fits current neutrino oscillation data and correlates the octant of the atmospheric angle with the magnitude of the lightest neutrino mass, with maximal mixing excluded for any neutrino mass
Parabolic metamaterials and Dirac bridges
Colquitt, D. J.; Movchan, N. V.; Movchan, A. B.
2016-10-01
A new class of multi-scale structures, referred to as `parabolic metamaterials' is introduced and studied in this paper. For an elastic two-dimensional triangular lattice, we identify dynamic regimes, which corresponds to so-called `Dirac Bridges' on the dispersion surfaces. Such regimes lead to a highly localised and focussed unidirectional beam when the lattice is excited. We also show that the flexural rigidities of elastic ligaments are essential in establishing the `parabolic metamaterial' regimes.
Renormalization of Dirac's Polarized Vacuum
Lewin, Mathieu
2010-01-01
We review recent results on a mean-field model for relativistic electrons in atoms and molecules, which allows to describe at the same time the self-consistent behavior of the polarized Dirac sea. We quickly derive this model from Quantum Electrodynamics and state the existence of solutions, imposing an ultraviolet cut-off $\\Lambda$. We then discuss the limit $\\Lambda\\to\\infty$ in detail, by resorting to charge renormalization.
陈元千
2011-01-01
Bear first presented a physical concept and discriminant of the starting pressure gradient in 1972 when he studied the applied lower limit of the Darcy law. And then Professor Ge Jiali introduced the starting pressure gradient to China in 1982. The so-called starting pressure gradient refers to a pressure gradient that makes a fluid in fluid-saturated cores begin to flow. It should be pointed out that the pressure gradient of linear flow is directly proportional to the flow rate, while the starting pressure gradient is a constant. The pressure gradient of plane radial flow is directly proportional to the flow rate but inversely to the radial radius. Moreover, the starting pressure7 gradient at a position of different radial radius is variable. It is controversial for the correctness to have directly applied the Bear's starting pressure gradient and discriminant of linear flow to the plane radial flow equation by some researchers. Theoretically, the paper analyzed both the pressure gradient and starting pressure gradient of linear flow and plane radial flow and proposed the conception of starting flow rate. At the same time, a more applicable method to evaluate the starting drawdown pressure and starting bottomhole flowing pressure of low permeability tight reservoirs was proposed.%启动压力梯度的物理概念及判别式是Bear于1972年在利用岩心测试资料研究达西定律的应用下限时提出来的,葛家理教授首次介绍到我国.所谓启动压力梯度,是指流体在饱和的岩心开始发生流动时的压力梯度.应当指出,线性流的压力梯度与流量成正比,启动压力梯度为常数；平面径向流的压力梯度与流量成正比,与径向半径成反比,而且,不同径向半径位置的启动压力梯度是不同的.有关学者将线性流启动的压力梯度及判别式直接用于平面径向流方程的正确性值得质疑.笔者对线性流和平面径向流的压力梯度和启动压力梯度问题进行了
Generalization of the Dirac’s Equation and Sea
Javadi, Hossein; Forouzbakhsh, Farshid; Daei Kasmaei, Hamed
2016-01-01
. In quantum mechanics, Schrodinger equation is similar to Newton's second law in classic mechanics. Quantum mechanics is also extension of Newtonian mechanics to atomic and subatomic scales and relativistic mechanics is extension of Newtonian mechanics to high velocities near to velocity of light too....... Schrodinger equation is not a relativistic equation, because it is not invariant under Lorentz transformations. Dirac expanded The Schrodinger equation by presenting Dirac Sea and founded relativistic quantum mechanics. In this paper by reconsidering the Dirac Sea and his equation, the structure of photon...
Generalization of the Majorana equation for real spinors
Teruel, Ginés R Pérez
2016-01-01
We show that the Dirac equation for real spinors can be naturally decomposed into a system of two first-order relativistic wave equations. The decomposition separates in a transparent way the real and imaginary parts of the Dirac equation by means of two algebraic differential operators, allowing to describe real spinors in any representation of the Dirac matrices maintaining the reality condition $\\tilde{\\Psi}=\\tilde{\\Psi}^{*}$ unaltered. In addition, it is shown that the Majorana wave equation is a particular case of the relativistic system of equations deduced in this paper. We also briefly discuss how the formalism can be extended to deal with complex (charged) spinors.
Critical exact solutions for self-gravitating Dirac fields
Cianci, Roberto; Fabbri, Luca; Vignolo, Stefano [Universita di Genova, DIME Sez. Metodi e Modelli Matematici, Genova (Italy)
2016-11-15
We consider the Einstein-Dirac field equations describing a self-gravitating massive neutrino, looking for axially symmetric exact solutions; in the search of general solutions, we find some that are specific and which have critical features, such as the fact that the space-time curvature turns out to be flat and the spinor field gives rise to a vanishing bi-linear scalar ψψ =0 with non-vanishing bi-linear pseudo-scalar iψγ{sup 5}ψ ≠ 0: because in quantum-field theory general computational methods are built on plane-wave solutions, for which the bi-linear pseudo-scalar vanishes while the bi-linear scalar does not vanish, then the solutions we found cannot be treated with the usual machinery of quantum-field theory. This means that for the Einstein-Dirac system there exist admissible solutions which nevertheless cannot be quantized with the common prescriptions; we regard this situation as yet another issue of tension between Einstein gravity and quantum principles. Possible ways to quench this tension can be seen either in enlarging the validity of quantum-field theory or by restricting the space of the solutions of the Einstein-Dirac system of field equations. (orig.)
Massless Dirac fields and Barbero-Immirzi parameter in Cosmology
Berredo-Peixoto, Guilherme de; Shapiro, Ilya Lvovich; Souza, Cleber Abrahao de [Universidade Federal de Juiz de Fora (ICE/UFJF), MG (Brazil). Instituto de Ciencias Exatas. Dept. de Fisica
2011-07-01
We consider cosmological solution for Einstein gravity with massless fermions with a four-fermion coupling, which emerges from the Holst action and is related to the Barbero-Immirzi (BI) parameter. The gravitational action of this sort is a popular object of investigation in a non-perturbative formalism of quantum gravity. After exploring the consistency conditions for Dirac field within the standard Friedman-Robertson-Walker (FRW) metric, one can rule out some classes of simplest solutions, related to conformal transformation of the field. It can be shown that the Dirac spinor components should be distinct complex functions of time. Finally, the theory with BI parameter and minimally coupling massless Dirac field is equivalent to a perfect fluid with the equation of state p = wρ, with w = 1/7. It is remarkable that the equation of state of the self-interacting spinor matter does not depend on the BI parameter. As a result, the theory does not allow smooth transition to the usual GR without Holst term. (author)
DIRAC: Secure web user interface
Casajus Ramo, A [University of Barcelona, Diagonal 647, ES-08028 Barcelona (Spain); Sapunov, M, E-mail: sapunov@in2p3.f [Centre de Physique des Particules de Marseille, 163 Av de Luminy Case 902 13288 Marseille (France)
2010-04-01
Traditionally the interaction between users and the Grid is done with command line tools. However, these tools are difficult to use by non-expert users providing minimal help and generating outputs not always easy to understand especially in case of errors. Graphical User Interfaces are typically limited to providing access to the monitoring or accounting information and concentrate on some particular aspects failing to cover the full spectrum of grid control tasks. To make the Grid more user friendly more complete graphical interfaces are needed. Within the DIRAC project we have attempted to construct a Web based User Interface that provides means not only for monitoring the system behavior but also allows to steer the main user activities on the grid. Using DIRAC's web interface a user can easily track jobs and data. It provides access to job information and allows performing actions on jobs such as killing or deleting. Data managers can define and monitor file transfer activity as well as check requests set by jobs. Production managers can define and follow large data productions and react if necessary by stopping or starting them. The Web Portal is build following all the grid security standards and using modern Web 2.0 technologies which allow to achieve the user experience similar to the desktop applications. Details of the DIRAC Web Portal architecture and User Interface will be presented and discussed.
Baryon spectrum analysis using Dirac's covariant constraint dynamics
Whitney, Joshua F.; Crater, Horace W.
2014-01-01
We present a relativistic quark model for the baryons that combines three related relativistic formalisms. The three-body constraint formalism of Sazdjian is used to recast three relativistic two-body equations for the three pairs of interacting quarks into a single relativistically covariant three-body equation for the bound state energies, having a Schrodinger-like structure. The two-body equations are the two-body Dirac equations of constraint dynamics derived by Crater and Van Alstine for combined world vector and scalar interactions providing the necessary spin dependent and spin independent interaction terms. The minimal quasipotential formalism of Todorov is used to provide an invariant framework for the vector and scalar dynamics used in the two-body Dirac equations into which is inserted a local simplified version of the Richardson potential. The spectral results are analyzed and compared to experiment using a best fit method and several different algorithms, including a gradient approach, and a Monte Carlo method.
A.I.Arbab
2013-01-01
A unified complex model of Maxwell's equations is presented.The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities.A new vacuum solution is obtained,and a new transformation under which Maxwell's equations are invariant is proposed.This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential.An electric dipole moment is found to be related to the magnetic charges,and Dirac's quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges.We generalize Maxwell's equations to include longitudinal waves.A formal analogy between this formulation and Dirac's equation is also discussed.
Noncommutative Dirac quantization condition using the Seiberg-Witten map
Maceda, Marco
2016-01-01
We investigate the validity of the Dirac quantization condition (DQC) for magnetic monopoles in noncommutative space-time. We use an approach based on an extension of the method introduced by Wu and Yang; the effects of noncommutativity are analyzed using the Seiberg-Witten map and the corresponding deformed Maxwell's equations are discussed. By means of a perturbation expansion in the noncommutativity parameter $\\theta$, we show first that the DQC remains unmodified up to the first and second order. This result is then generalized to all orders in the expansion parameter for a class of noncommutative electric currents induced by the Seiberg-Witten map; these currents reduce to the Dirac delta function in the commutative limit.
Double Dirac cones in phononic crystals
Li, Yan
2014-07-07
A double Dirac cone is realized at the center of the Brillouin zone of a two-dimensional phononic crystal (PC) consisting of a triangular array of core-shell-structure cylinders in water. The double Dirac cone is induced by the accidental degeneracy of two double-degenerate Bloch states. Using a perturbation method, we demonstrate that the double Dirac cone is composed of two identical and overlapping Dirac cones whose linear slopes can also be accurately predicted from the method. Because the double Dirac cone occurs at a relatively low frequency, a slab of the PC can be mapped onto a slab of zero refractive index material by using a standard retrieval method. Total transmission without phase change and energy tunneling at the double Dirac point frequency are unambiguously demonstrated by two examples. Potential applications can be expected in diverse fields such as acoustic wave manipulations and energy flow control.
Data Management System of the DIRAC Project
Haen, Christophe; Tsaregorodtsev, Andrei
2015-01-01
The DIRAC Interware provides a development framework and a complete set of components for building distributed computing systems. The DIRAC Data Management System (DMS) offers all the necessary tools to ensure data handling operations for small and large user communities. It supports transparent access to storage resources based on multiple technologies, and is easily expandable. The information on data files and replicas is kept in a File Catalog of which DIRAC offers a powerful and versatile implementation (DFC). Data movement can be performed using third party services including FTS3. Bulk data operations are resilient with respect to failures due to the use of the Request Management System (RMS) that keeps track of ongoing tasks. In this contribution we will present an overview of the DIRAC DMS capabilities and its connection with other DIRAC subsystems such as the Transformation System. The DIRAC DMS is in use by several user communities now. The contribution will present the experience of the LHCb exper...
Savvidy, G K
1998-01-01
We discuss the basic properties of the gonihedric string and the problem of its formulation in continuum. We propose a generalization of the Dirac equation and of the corresponding gamma matrices in order to describe the gonihedric string. The wave function and the Dirac matrices are infinite-dimensional. The spectrum of the theory consists of particles and antiparticles of increasing half-integer spin lying on quasilinear trajectories of different slope. Explicit formulas for the mass spectrum allow to compute the string tension and thus demonstrate the string character of the theory.
A relativistic correlationless kinetic equation with radiation reaction fully incorporated
Lai, H. M.
1984-06-01
The Landau-Lifshitz expression for the Lorentz-Dirac equation is used to derive a relativistic correlationless kinetic equation for a system of electrons with radiation reaction fully incorporated. Various situations and possible applications are discussed.
Jusufi, Kimet; Apostolovska, Gordana
2016-12-01
In this paper we study the quantum tunneling of Dirac magnetic monopoles from the global monopole black hole under quantum gravity effects. We start from the modified Maxwell's equations and the Generalized Uncertainty Relation (GUP), to recover the GUP corrected temperature for the global monopole black hole by solving the modified Dirac equation via Hamilton-Jacobi method. Furthermore, we also include the quantum corrections beyond the semiclassical approximation, in particular, first we find the logarithmic corrections of GUP corrected entropy and finally we calculate the GUP corrected specific heat capacity. It is argued that the GUP effects may prevent a black hole from complete evaporation and leave remnants.
The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space
B. Mirza; M. Mohadesi
2004-01-01
We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics ofa particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.
The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space
B.Mirza; M.Mohadesi
2004-01-01
We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.
Dirac structures on generalized Riemannian manifolds
Vaisman, Izu
2011-01-01
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
HUANG Zeng-Guang; FANG Wei; LU Hui-Qing
2011-01-01
@@ We discuss Bianchi type-Ⅶ0 cosmology with a Dirac field in the Einstein-Cartan(E-C) theory and obtain the equations of the Dirac and gravitational fields in the E-C theory.A Bianchi type-Ⅶ0 inflationary solution is found.When(3/16)S2-σ2＞0, the Universe may avoid singularity.
Bakke, K., E-mail: kbakke@fisica.ufpb.br; Furtado, C., E-mail: furtado@fisica.ufpb.br
2013-09-15
In this paper, we study the influence of the Aharonov–Casher effect [Y. Aharonov, A. Casher, Phys. Rev. Lett. 53 (1984) 319] on the Dirac oscillator in three different scenarios of general relativity: the Minkowski spacetime, the cosmic string spacetime and the cosmic dislocation spacetime. In this way, we solve the Dirac equation and obtain the energy levels for bound states and the Dirac spinors for positive-energy solutions. We show that the relativistic energy levels depend on the Aharonov–Casher geometric phase. We also discuss the influence of curvature and torsion on the relativistic energy levels and the Dirac spinors due to the topology of the cosmic string and cosmic dislocation spacetimes. -- Highlights: •Influence of the Aharonov–Casher effect on the Dirac oscillator. •Exact solutions of the Dirac equation in topological defect spacetimes. •Dependence of the relativistic energy levels on the Aharonov–Casher geometric phase. •Influence of curvature and torsion on the relativistic energy levels and the Dirac spinors.
LI Chang-Hui; DING Hao-Gang; DAI Jian; SONG Xing-Chang
2001-01-01
Several models in noncommutative geometry (NCG) with mild changes to the standard model are introduced to discuss the neutrino mass problem. We use two constraints, Poincaré duality and gauge anomaly free, to discuss the possibility of containing right-handed neutrinos in them. Our work shows that no model in this paper, with each generation containing a right-handed neutrino, can satisfy these two constraints at the same time. So, to consist with neutrino oscillation experiment results, maybe fundamental changes to the present version of NCG are usually needed to include Dirac massive neutrinos.
Dirac tensor with heavy photon
Bytev, V.V.; Kuraev, E.A. [Joint Institute of Nuclear Research, Moscow (Russian Federation). Bogoliubov Lab. of Theoretical Physics; Scherbakova, E.S. [Hamburg Univ. (Germany). 1. Inst. fuer Theoretische Physik
2012-01-15
For the large-angles hard photon emission by initial leptons in process of high energy annihilation of e{sup +}e{sup -} {yields} to hadrons the Dirac tensor is obtained, taking into account the lowest order radiative corrections. The case of large-angles emission of two hard photons by initial leptons is considered. This result is being completed by the kinematics case of collinear hard photons emission as well as soft virtual and real photons and can be used for construction of Monte-Carlo generators. (orig.)
Superalgebraic representation of Dirac matrices
Monakhov, V. V.
2016-01-01
We consider a Clifford extension of the Grassmann algebra in which operators are constructed from products of Grassmann variables and derivatives with respect to them. We show that this algebra contains a subalgebra isomorphic to a matrix algebra and that it additionally contains operators of a generalized matrix algebra that mix states with different numbers of Grassmann variables. We show that these operators are extensions of spin-tensors to the case of superspace. We construct a representation of Dirac matrices in the form of operators of a generalized matrix algebra.
Phenomenology of Pseudo Dirac Neutrinos
Joshipura, A S; Joshipura, Anjan S.; Rindani, Saurabh D.
2000-01-01
We formulate general conditions on $3\\times 3$ neutrino mass matrices under which a degenerate pair of neutrinos at a high scale would split at low scale by radiative corrections involving only the standard model fields. This generalizes the original observations of Wolfenstein on pseudo Dirac neutrinos to three generations. A specific model involving partially broken discrete symmetry and solving the solar and atmospheric anomalies is proposed. The symmetry pattern of the model naturally generates two large angles one of which can account for the large angle MSW solution to the solar neutrino problem.
Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges
Nam, Phan Thành, E-mail: pnam@ist.ac.at [Institute of Science and Technology Austria (Austria); Den Bosch, Hanne Van, E-mail: hannevdbosch@fis.puc.cl [Pontificia Universidad Católica de Chile, Instituto de Física (Chile)
2017-06-15
We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.
a New Formalism for Dirac-Like Theories with Curved Space-Time
Halliday, David Wayne
This paper develops a formalism for Dirac-like equations (linear complex differential equations, linear in all derivatives), allowing for general coordinate and "spin-space" (internal space) transformations. A correspondence principle is also developed by requiring solutions to the Dirac-like equations to be solutions to a Klein-Gordon equation, that is likewise generally invariant. Through this treatment, previous generalizations of the Dirac equation are incorporated, and various aspects of these methods are analyzed. Furthermore, the Yang-Mills-like gauge fields allowed, or required, by the formalism are expressed, and found to be associated with much larger symmetries than most would desire, suggesting either there has been much greater symmetry breaking than expected, or else few of the particles we accept as fundamental really are. It is also found that unless the space-time is "parallelizable" (so there exist fields that are everywhere parallel transported into themselves, which is not generally the case), or some of the wave function components (and separately some of the Yang-Mills fields) are interdependent, we cannot have the Dirac gamma operators commuting with the momentum operators, while simultaneously having a spin-space metric that is compatible with the Yang-Mills fields.
Weak cosmic censorship, dyonic Kerr-Newman black holes and Dirac fields
Toth, Gabor Zsolt
2015-01-01
It was investigated recently, with the aim of testing the weak cosmic censorship conjecture, whether an extremal Kerr black hole can be converted into a naked singularity by interaction with a massless classical Dirac test field, and it was found that this is possible. We generalize this result to electrically and magnetically charged rotating extremal black holes (i.e. extremal dyonic Kerr-Newman black holes) and massive Dirac test fields, allowing magnetically or electrically uncharged or nonrotating black holes and the massless Dirac field as special cases. We show that the possibility of the conversion is a direct consequence of the fact that the Einstein-Hilbert energy-momentum tensor of the classical Dirac field does not satisfy the null energy condition, and is therefore not in contradiction with the weak cosmic censorship conjecture. We give a derivation of the absence of superradiance of the Dirac field without making use of the complete separability of the Dirac equation in dyonic Kerr-Newman backgr...
Merdaci, Abdeldjalil; Jellal, Ahmed; Chetouani, Lyazid
2017-09-01
It is shown that the propagator of the neutral Pauli-Dirac particle with an anomalous magnetic moment μ in an external linear magnetic field B(x) = B +B‧ x is the causal Green function Sc(xb ,xa) of the Pauli-Dirac equation. The corresponding Green function is calculated via path integral method in global projection, giving rise to the exact eigenspinor expressions. The effective action is used to explicitly determine the production rate in vacuum of neutral Dirac particle in terms of B‧ and μ, which is B independent.
Bhatnagar, Shashank; Mengesha, Yikdem
2013-01-01
In this work we have employed Bethe-Salpeter equation (BSE) under covariant instantaneous ansatz (CIA) to study electromagnetic decays of ground state equal mass vector mesons: $\\rho$, $\\omega$, $\\phi$, $\\psi$ and $Y$ through the process $V\\rightarrow\\gamma*\\rightarrow e^+ + e^-$. We employ the generalized structure of hadron-quark vertex function $\\Gamma$ which incorporates various Dirac structures from their complete set order-by-order in powers of inverse of meson mass. The electromagnetic decay constants for the above mesons are calculated using the leading order (LO) and the next-to-leading order (NLO) Dirac structures. The relevance of various Dirac structures in this calculation is studied.
HILBERT-DIRAC OPERATORS IN CLIFFORD ANALYSIS
F.BRACKX; H.DE SCHEPPER
2005-01-01
Around the central theme of "square root" of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.
Representation-independent manipulations with Dirac spinors
Pal, P B
2007-01-01
Dirac matrices, also known as gamma matrices, are defined only up to a similarity transformation. Usually, some explicit representation of these matrices is assumed in order to deal with them. In this article, we show how it is possible to proceed without any such assumption. Various important identities involving Dirac matrices and spinors have been derived without assuming any representation at any stage.
LHCbDIRAC as Apache Mesos microservices
Couturier, Ben
2016-01-01
The LHCb experiment relies on LHCbDIRAC, an extension of DIRAC, to drive its offline computing. This middleware provides a development framework and a complete set of components for building distributed computing systems. These components are currently installed and ran on virtual machines (VM) or bare metal hardware. Due to the increased load of work, high availability is becoming more and more important for the LHCbDIRAC services, and the current installation model is showing its limitations. Apache Mesos is a cluster manager which aims at abstracting heterogeneous physical resources on which various tasks can be distributed thanks to so called "framework". The Marathon framework is suitable for long running tasks such as the DIRAC services, while the Chronos framework meets the needs of cron-like tasks like the DIRAC agents. A combination of the service discovery tool Consul together with HAProxy allows to expose the running containers to the outside world while hiding their dynamic placements. Such an arc...
Neutrinos Are Nearly Dirac Fermions
Cahill, K E
1999-01-01
Neutrino masses and mixings are analyzed in terms of left-handed fields and a 6x6 complex symmetric mass matrix whose singular values are the neutrino masses. An angle theta_nu characterizes the kind of the neutrinos, with theta_nu = 0 for Dirac neutrinos and theta_nu = pi/2 for Majorana neutrinos. If theta_nu = 0, then baryon-minus-lepton number is conserved. When theta_nu is approximately zero, the six neutrino masses coalesce into three nearly degenerate pairs. Thus the smallness of the differences in neutrino masses exhibited in the solar and atmospheric neutrino experiments and the stringent limits on neutrinoless double-beta decay are naturally explained if B-L is approximately conserved and neutrinos are nearly Dirac fermions. If one sets theta_nu = 0.0005, suppresses inter-generational mixing, and imposes a quark-like mass hierarchy, then one may fit the essential features of the solar, reactor, and atmospheric neutrino experiments with otherwise random mass matrices in the eV range. This B-L model le...
Finster, Felix
2015-01-01
We consider a boundary value problem for the Dirac equation in a four-dimensional, smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the space-time includes horizons, where the Hamiltonian fails to be elliptic.
Babourova, O V; Kudlaev, P E
2016-01-01
On the basis of the Poincare-Weyl gauge theory of gravitation, a new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl spacetime with the Dirac scalar field representing the dark matter model. A static approximate axially symmetric solution of the field equations in vacuum is obtained. On the base of this solution in the Newtonian approximation one considers the problem of rotation velocities in spiral components of galaxies.
Low regularity and local well-posedness for the 1+3 dimensional Dirac-Klein-Gordon system
Achenef Tesfahun
2007-11-01
Full Text Available We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known results for the same problem. Our proof relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.
Relativistic bound-state equations for fermions with instantaneous interactions
Suttorp, L.G.
1979-01-01
Three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary
Radial sine-Gordon kinks as sources of fast breathers
Caputo, Jean Guy; Sørensen, Mads Peter
2013-01-01
We consider radial sine-Gordon kinks in two, three, and higher dimensions. A full two-dimensional simulation showing that azimuthal perturbations remain small allows us to reduce the problem to the one-dimensional radial sine-Gordon equation. We solve this equation on an interval [r, r1] and abso...
A generally-relativistic gauge classification of the Dirac fields
Fabbri, Luca
2016-01-01
We consider generally-relativistic gauge transformations for the spinorial fields finding two mutually exclusive but together exhaustive classes in which fermions are placed adding supplementary information to the results obtained by Lounesto, and identifying quantities analogous to the momentum vector and the Pauli-Lubanski axial vector we discuss how our results are similar to those obtained by Wigner; by taking into account the most general Dirac equations we will investigate the consequences for the dynamics: and in particular we shall address the problem of getting the non-relativistic approximation in a consistent way. We are going to comment on extensions.
Dirac particle in a square well and in a box
Alhaidari, A D
2009-01-01
We obtain an exact solution of the 1D Dirac equation for a square well potential of depth greater then twice the particle's mass. The energy spectrum formula in the Klein zone is surprisingly very simple and independent of the depth of the well. This implies that the same solution is also valid for the potential box (infinitely deep well). In the nonrelativistic limit, the well-known energy spectrum of a particle in a box is obtained. We also provide in tabular form the elements of the complete solution space of the problem for all energies.
Critical exact solutions for self-gravitating Dirac fields
Cianci, Roberto; Vignolo, Stefano
2016-01-01
We consider the Einstein-Dirac field equations describing a self-gravitating massive neutrino, looking for axially-symmetric exact solutions; in the search of general solutions, we find some that are specific and which have critical features, such as the fact that the space-time curvature turns out to be flat and the spinor field gives rise to a vanishing bi-linear scalar $\\overline{\\psi}\\psi=0$ with non-vanishing bi-linear pseudo-scalar $i\\overline{\\psi}\\gamma^5\\psi\