The special theory of relativity as applied to the Born–Oppenheimer–Huang approach

In two recent publications (Int. J. Quant. Chem. 114, 1645 (2014) and Mole. Phys. 114, 227 (2016)) it was shown that the Born–Hwang (BH) treatment of a molecular system perturbed by an external field yields a set of decoupled vectorial wave equations, just like in electro-magnetism. This finding led us to declare on the existence of a new type of Fields, which were termed Molecular Fields. The fact that such fields exist implies that at the vicinity of conical intersections exist a mechanism that transforms a passing-by electric beam into a field which differs from the original electric field. This situation is reminiscent of what is encountered in astronomy where Black Holes formed by massive stars may affect the nature of a near-by beam of light. Thus, if the non-adiabatic-coupling-terms (NACT) with their singular points may affect the nature of such a beam (see the above two publications), then it would be interesting to know to what extend NACTs (and consequently also the BH equation) will be affected by the special theory of relativity as introduced by Dirac. Indeed, while applying the Dirac approach we derived the relativistic affected NACTs as well as the corresponding BH equation.     

Transmission through biased graphene strip

We solve the 2D Dirac equation describing graphene in the presence of a linear vector potential. The discretization of the transverse momentum due to the infinite mass boundary condition reduced our 2D Dirac equation to an effective massive 1D Dirac equation with an effective mass equal to the quantized transverse momentum. We use both a numerical Poincaré map approach, based on space discretization of the original Dirac equation, and a direct analytical method. These two approaches have been used to study tunneling phenomena through a biased graphene strip. The numerical results generated by the Poincaré map are in complete agreement with the analytical results.     

Elastic scattering of low-energy electrons by a uranium atom: Reliability of theoretical predictions of based on a model description

We used the method of phase functions to solve the radial relativistic Dirac equation and nonrelativistic Schroedinger equation. With these solutions, we investigated the elastic scattering of slow electrons by a uranium atom, and obtained numerical values for the total cross section and elastic scattering phases. In order to check the correctness of the results found from the method of phase functions, in all cases we also solved the Dirac and Schroedinger equations by direct numerical integration. Several types of polarization and exchange potentials were used to simulate the scattering process. We conclude that the strong dependence of the cross section for elastic scattering of an electron by uranium on the shape of the effective potential of the latter at small kinetic energies (E _k<5 eV) makes it impossible to predict the presence or absence of a Ramsauer effect reliably. Zh. éksp. Teor. Fiz. 111, 1214–1228 (April 1997)     

Berry phase and traversal time in asymmetric graphene structures

The Berry phase and the group-velocity-based traversal time have been calculated for an asymmetric non-contacted or contacted graphene structure, and significant differences have been observed compared to semiconductor heterostructures. These differences are related to the specific, Dirac-like evolution law of charge carriers in graphene, which introduces a new type of asymmetry. When contacted with electrodes, the symmetry of the Dirac equation is broken by the Schrödinger-type electrons in contacts, so that the Berry phase and traversal time behavior in contacted and non-contacted graphene differ significantly.     

Spin polarization of electrons elastically scattered from europium and bismuth atoms

We present calculations of differential, integrated elastic, total, momentum transfer cross-sections and spin-polarization parameters S, T and U for scattering of electrons from Eu and Bi atoms in the energy range 2.0 to 500.0 eV using semi-relativistic approach. The target-projectile interaction is represented both by real and complex parameter-free optical potentials in the solution of Dirac equation for the scattered electrons. The results for the differential cross-sections and spin-polarization parameters have been compared with the available calculations and experimental results. Received 17 February 2000 and Received in final form 15 June 2000     

Dirac quasinormal modes of Born-Infeld black hole spacetimes

Quasinormal modes (QNMs) for massless and massive Dirac perturbations of Born-Infeld black holes (BHs) in higher dimensions are investigated. Solving the corresponding master equation in accordance with hypergeometric functions and the QNMs are evaluated. We discuss the relationships between QNM frequencies and spacetime dimensions. Meanwhile, we also discuss the stability of the Born-Infeld BH by calculating the temporal evolution of the perturbation field. Both the perturbation frequencies and the decay rate increase with increasing dimension of spacetime n. This shows that the Born-Infeld BHs become more and more unstable at higher dimensions. Furthermore, the traditional finite difference method is improved, so that it can be used to calculate the massive Dirac field. We also elucidate the dynamic evolution of Born-Infeld BHs in a massive Dirac field. Because the number of extra dimensions is related to the string scale, there is a relationship between the spacetime dimension n and the properties of Born-Infeld BHs that might be advantageous for the development of extra-dimensional brane worlds and string theory.     

Spin-1/2 equation with indefinite metric

We study a Schrödinger equation involving a Hamiltonian that is a second-order differential operator, describes free spin-1/2 particles with both energy signs and a definite mass, and depends on a parameterG. One obtains the usual Dirac Hamiltonian by settingG=±i, but for real values ofG the one-particle theory developed here possesses an indefinite metric, so negative energy states have negative normalization. Although the new equation is not manifestly covariant, it is demonstrated that it can be made invariant under proper orthochronous Poincaré transformations; it is also invariant under the CPT transformation and charge conjugation, but not, as we interpret it, under space inversion.Supported in part by the U.S. Energy Research and Development Administration.     

Fermion tunnels of higher-dimensional anti-de Sitter Schwarzschild black hole and its corrected entropy

Applying the method beyond semiclassical approximation, fermion tunneling from higher-dimensional anti-de Sitter Schwarzschild black hole is researched. In our work, the “tortoise” coordinate transformation is introduced to simplify Dirac equation, so that the equation proves that only the (r−t) sector is important to our research. Because we only need to study the (r−t) sector, the Dirac equation is decomposed into several pairs of equations spontaneously, and we then prove the components of wave functions are proportional to each other in every pair of equations. Therefore, the suitable action forms of the wave functions are obtained, and finally the correctional Hawking temperature and entropy can be determined via the method beyond semiclassical approximation.     

Some extensions of the Einstein–Dirac equation

We considered an extension of the standard functional for the Einstein–Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler–Lagrange equations provide a new type of Einstein–Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein–Dirac system called the CL-Einstein–Dirac equation of type II (see Definition 3.1).     

Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

The impact of the strongly attractive electromagnetic field of heavy nuclei on electrons in quasi-elastic (e, e') scattering is often accounted for by the effective momentum approximation. This method is a plane wave Born approximation which takes the twofold effect of the attractive nucleus on initial- and final-state electrons into account, namely the modification of the electron momentum in the vicinity of the nucleus, and the focusing of electrons towards the nuclear region leading to an enhancement of the corresponding wave function amplitudes. The focusing effect due to the attractive Coulomb field of a homogeneously charged sphere on a classical ensemble of charged particles incident on the field is calculated in the highly relativistic limit and compared to results obtained from exact solutions of the Dirac equation. The result is relevant for the theoretical foundation of the effective momentum approximation and describes the high-energy behavior of the amplitude of continuum Dirac waves in the potential of a homogeneously charged sphere. Our findings indicate that the effective momentum approximation is a useful approximation for the calculation of Coulomb corrections in (e, e') scattering off heavy nuclei for sufficiently high electron energies and momentum transfer.     

From time operator to chronons

A time operator, which incorporates the idea of time as a dynamical variable, was first introduced in the context of a theory of irreversible evolution. The existence of a time operator has interesting implications in several areas of physics. Here we demonstrate a close link between the existence of the time operator for relativistic particles and the existence of an indivisible time interval or chronons for dynamical evolution. More explicitly, we consider a Klein-Gordon particle and require the existence of a time operator for its evolution. We also make a natural choice of the form of the time operator which expresses it in terms of the generators of the Poincaré group. These then imply that the physical time evolution group must be the discrete subgroup U_n (n integers) of the originally given evolution group U_t of the Klein-Gordon particle and the constant is given by =h/2mc². This means that the requirement of the existence of a time operator implies that the time evolution cannot be followed to time intervals smaller than and, as such, emerges as a chronon for the dynamical evolution. Expecting that the same results hold for a Dirac particle also, we conclude that the so-called Zitterbewegungdoes not occur in reality. Thus, possible confirmation of the existence of chronons would result if no observableconsequence of Zitterbewegungis actually realized in nature. This calls for a search of observable consequences of the Zitterbewegungand a re-examination of their agreement (if any) with experiments. A possible consequence of Zitterbewegung,the so-called Darwin term present in the Dirac Hamiltonian in an electric field, is briefly considered.     

Postulates for time evolution in quantum mechanics

A detailed list of postulates is formulated in an algebraic setting. These postulates are sufficient to entail the standard time evolution governed by the Schrödinger or Dirac equation. They are also necessary in a strong sense: Dropping any one of the postulates allows for other types of time evolution, as is demonstrated with examples. Some philosophical remarks hint on possible further investigations.     

The nonlinear Dirac equation in Bose-Einstein condensates: Foundation and symmetries

We show that Bose-Einstein condensates in a honeycomb optical lattice can be described by a nonlinear Dirac equation in the long wavelength, mean field limit. Unlike nonlinear Dirac equations posited by particle theorists, which are designed to preserve the principle of relativity, i.e., Poincaré covariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks this symmetry. We present a rigorous derivation of the nonlinear Dirac equation from first principles. We provide a thorough discussion of all symmetries broken and maintained.     

Dirac spinors and flavor oscillations

In the standard treatment of particle oscillations the mass eigenstates are implicitly assumed to be scalars and, consequently, the spinorial form of the neutrino wave functions is not included in the calculations. To analyze this additional effect, we discuss the oscillation probability formula obtained by using the Dirac equation as evolution equation for the neutrino mass eigenstates. The initial localization of the spinor state also implies an interference between positive and negative energy components of mass eigenstate wave packets which modifies the standard oscillation probability.Received: 8 April 2004, Revised: 12 July 2004, Published online: 20 October 2004     

Nonrelativistic Wave Equations with Gauge Fields

We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Lévy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.     

Solutions of the Dirac Equation for the Davidson Potential

In the present paper we solve the Dirac equation with Davidson potential by Nikiforov-Uvarov method. The Dirac Hamiltonian contains a scalar S and a vector V Davidson potentials. With equal scalar and vector potential, analytical solutions for bound states of the corresponding Dirac equations are found.     

Two-Dimensional Linear Dependencies on the Coordinate Time-Dependent Interaction in Relativistic Non-Commutative Phase Space

In this paper, the Non-Commutative phase space and Dirac equation, time-dependent Dirac oscillator are introduced. After presenting the desire general form of a two-dimensional linear dependency on the coordinate time-dependent potential, the Dirac equation is written in terms of Non-Commutative phase space parameters and solved in a general form by using Lewis-Riesenfield invariant method and the time-dependent invariant of Dirac equation with two-dimensional linear dependency on the coordinate time-dependent potential in Non-Commutative phase space has been constructed, then such latter operations are done for time-dependent Dirac oscillator. In order to solve the differential equation of wave function time evolution for Dirac equation and time-dependent Dirac oscillator which are partial differential equation some appropriate ordinary physical problems have been studied and at the end the interesting result has been achieved.     

Approximate scattering solutions of the dirac equation for electron-nucleus processes in light nuclei

Scattering solutions of the second-order Dirac equation for the case of the Coulomb potential and which are correct to first order in the coupling constantZe ²/hc are investigated and found to describe pure Coulomb scattering equally well as the Sommerfeld-Maue wave functions. Errors introduced by the use of these solutions are studied in a numerical calculation of cross sections for nuclear electric-quadrupole excitation by high-energy electrons. The use of these wave functions is suggested for simplified calculations of lowest-order Coulomb corrections to Born approximation results for various electron-nucleus processes.     

Dirac Equation at Finite Temperature

In this paper, we propose finite temperature Dirac equation, which can describe the quantum systems in an arbitrary temperature for a relativistic particle of spin-1/2. When the temperature T=0, it become Dirac equation. With the equation, we can study the relativistic quantum systems in an arbitrary temperature.     

Wave equation for a magnetic monopole

We show that there is room, in the Dirac equation, for a massless monopole. The basic idea is that the Dirac equation admits a second electromagnetic minimal coupling associated to the chiral gauge , which is only valid for a massless particle, but satisfies all the symmetry laws of a monopole. In the problem of the diffusion on a central electric field, we find the Poincaré integral and the Dirac relationeg/=n/2. The latter is deduced as a consequence of the fact (which is shown in this paper) thateg/c is the projection of the total angular momentum on the symmetry axis of the system formed by the monopole and the electric charge. Another important property is that a monopole and an antimonopole have opposite helicities (as for the neutrino), but do not have opposite charges: this precludes a vacuum magnetic polarization which would be analogous to the electric one, but allows us to imagine an aether made up of monopole-antimonopole pairs. The theory is then generalized on the basis of a nonlinear equation which is the most general invariant equation under the chiral gauge law. This equation admits solutions corresponding to massive monopoles, among which there are bradyons (i.e., ordinary massive particles) and tachyons. This equation is shown to be closely related to previous works initiated by Hermann Weyl, on Dirac's theory in the framework of general relativity. In conclusion, it is suggested that massless monopoles are perhaps excited states of the neutrino and that they may be produced in some weak interactions. Consequences on the solar activity are considered.