Relativistic Wave Equations Research Articles

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

Positive-Energy Dirac Particles and Dark Matter

The relativistic positive-energy wave equation proposed by P. Dirac in 1971 is an old but largely forgotten subject. The purpose of this note is to speculate that particles described by this equation (called here Dirac particles) are natural candidates for the dark matter. The reasoning is based on a fact that the internal structure of such particles simply prohibits their interaction with electromagnetic fields (at least with the minimal coupling) which is exactly what is required for dark matter. Dirac particles have quite unusual properties. In particular, they are transformed by an infinite-dimensional representation of the homogeneous Lorentz group, which clearly distinguishes them from all known elementary particles described by finite-dimensional representations and hints to a physics beyond the Standard Model. To clarify the topic, a brief review of the main features of the above-mentioned Dirac equation is given.

Q−Relativistic wave equation of the form i∂^q·ψq + mψ_0 = Eψ

In this paper we introduce a q relativistic wave equation of the form i∂^q·ψq + mψ_0 = Eψ. We present the q-spinorial solutions using the method of separated variables in the q-relativistic wave equation. Some comments are mentioned at the end of the paper.

Exact massive and massless scalar quasibound states solutions of the Einstein–Maxwell-dilaton (EMD) black hole

In this letter, we will focus on the Klein–Gordon equation with static spherically symmetric black hole solution of the Einstein–Maxwell-dilaton (EMD) theory as its 3+1 background space-time. The Klein–Gordon equation represents quasibound states of both massive and massless scalar fields which are localized in the black hole potential well. By using the covariant Klein–Gordon equation, we investigate the behaviour of both massive and massless scalars in the EMD black hole space-time. We successfully exactly solved the relativistic wave equation and are going to present the novel exact results in this letter. The exact solutions, the wave functions and the energy levels, describe the decaying nature of the relativistic scalar field bound in the curved space-time. The massive scalar quasibound state has complex-valued energy levels where the real part is the massive scalar’s energy while the imaginary part represents the decay. For the massless scalar quasibound state, pure imaginary energy levels are discovered. In this letter, by using the obtained exact scalar particle’s wave functions, we also consider the Hawking radiation of the apparent horizon of the EMD black hole that is calculated via Damour–Ruffini method. In principle, the investigation of black hole quasibound states could provide possibility for laboratory testing of effects whose nature are absolutely related with quantum effects in gravity.

The electronic and electromagnetic Dirac equations

Maxwell’s equations and the Dirac equation are the first-order differential relativistic wave equation for electromagnetic waves and electronic waves respectively. Hence, there is a notable similarity between these two wave equations, which has been widely researched since the Dirac equation was proposed. In this paper, we show that the Maxwell equations can be written in an exact form of the Dirac equation by representing the four Dirac operators with 8×8 matrices. Unlike the ordinary 4×4 Dirac equation, both spin–1/2 and spin–1 operators can be derived from the 8×8 Dirac equation, manifesting that the 8×8 Dirac equation is able to describe both electrons and photons. As a result of the restrictions that the electromagnetic wave is a transverse wave, the photon is a spin–1 particle. The four–current in the Maxwell equations and the mass in the electronic Dirac equation also force the electromagnetic field to transform differently to the electronic field. We use this 8×8 representation to find that the Zitterbewegung of the photon is actually the oscillatory part of the Poynting vector, often neglected upon time averaging.

Exact scalar quasibound states solutions of f(R) theory's static spherically symmetric black hole

Following the recent developments in black holes' quasibound states exact solutions by Senjaya (2023b,c), in this letter, we present a novel exact solution of quasibound states of a scalar around a static spherically symmetric black hole solution of f(R) theory of gravity. We start with the relativistic scalar field wave equation, i.e. the Klein-Gordon equation, isolating the radial equation and finally successfully obtain the exact radial wave solutions in terms of General Heun functions. Having the exact solutions in hand, the energy levels expression is obtained from the radial wave polynomial conditions. In the last part of this letter, the Hawking radiation is investigated via the Damour-Ruffini method and the Hawking temperature is obtained from radiation distribution function.

Detectability of Single Spinless Stellar-Mass Black Holes through Gravitational Lensing of Gravitational Waves with Advanced LIGO.

We investigate the detectability of gravitational waves that have been lensed by a spinless stellar-mass black hole, with respect to the advanced LIGO. By solving the full relativistic linear wave equations in the spacetime of a Schwarzschild black hole, we find that the strong gravity can create unique signals in the lensed waveform, particularly during the merger and ring-down stages. The differences in terms of fitting factor between the lensed waveform and best-fitted unlensed general relativity template with spin precession and higher-order multipoles are greater than 5% for the lens black hole mass within 70M_{⊙}<M_{lens}<133.33M_{⊙} under advanced LIGO's sensitivity. This is up to 5 times more detectable than the previous analysis based on the weak field approximation for a point mass and covers most part of the black hole mass gap predicted by stellar evolution theory. Based on Bayesian inference, the lensing feature can be distinguished with a signal-to-noise ratio of 12.5 for M_{lens}=70M_{⊙} and 19.2 for M_{lens}=250M_{⊙}, which is attainable for advanced LIGO.

On auxiliary fields and Lagrangians for relativistic wave equations

We address the problem of the existence of a Lagrangian for a given system of linear partial differential equation with constant coefficients. As a subtask, this involves bringing the system into a pre-Lagrangian form, wherein the number of equations matches the number of unknowns. We introduce a class of overdetermined systems, called co-flat, and show that they always admit a pre-Lagrangian form, which can be explicitly constructed by means of auxiliary variables. Moreover, we argue that such systems enjoy pre-Lagrangian formulations without auxiliary variables at all. As an application of our method, we construct new pre-Lagrangian and Lagrangian formulations for free massive fields of arbitrary integer spin. In contrast to the well-known models of Singh and Hagen, our Lagrangians involve much fewer auxiliary fields.

Conformal symmetry in the Bhabha theory

The first-order formalism of relativistic wave equations, such as the Dirac (spin s = 1=2) and the Kemmer (s = 1) equations, are generalized to the Bhabha (one s) equation, where the generalized gamma matrices for space-time dimension (3+1) are labeled with not only s but also s′ ∈ {s, s − 1,… s − ⎣s⎦}. We show that for s′ = s, the generators (D, Pμ , Kμ , Sμν ) constructed from the generalized gamma matrices satisfy the conformal algebra, in which we find that the null-eigenstates |s ±〉 with respect to Pμ and Kμ as Pμ |s +〉 = 0 = Kμ |s −〉 have the two following properties: the dimension of the eigenspace for|s ±〉 is given by 2s + 1, and , where represents the spin magnitude. In this sense, we can regard |s ±〉 as physical states for a massive particle.

Interaction of a photon with the surface and the bulk potentials of dielectric medium and a relativistic wave equation describing reflection and refraction phenomena and material dispersion

The surface potential differs from that of bulk, due to different atomic structures and vibration modes of a solid. Interaction of a photon with the potentials leads to material dispersion. Experimentally the surface and the bulk potentials can be determined from the wavelength-dependent refractive index. They are found to be the correlated quantities at different temperatures. A relativistic wave equation is proposed in this work, which contains the potentials. It provides a consistent description of reflection and refraction phenomena and material dispersion. Using this approach we investigate the surface and the bulk potentials of silica glass. The potentials decrease with rising temperature. The bulk potential increases with the density of the glass, whereas the surface potential remains almost unchanged at room temperature.

Propagation of anisotropic gravitational and electromagnetic waves at very high energies

We analyze the dispersion relation for an anisotropic gravity-electromagnetic theory at very high energies. In particular for photons of very high energy. We start by introducing the anisotropic gravity-gauge vector field model. It is invariant under spacelike diffeomorphisms, time parametrization, and U(1) gauge transformations. It includes high-order spacelike derivatives as well as polynomial expressions of the Riemann and field strength tensor fields. It is based on the Hořava-Lifshitz anisotropic proposal. We show its consistency, and the stability of the Minkowski ground state. Finally, we determine the exact zone at which the physical degrees of freedom, i.e. the transverse-traceless tensorial degrees of freedom and the transverse vectorial degrees of freedom propagate according to a linear wave equation. This is so, in spite of the fact that there exists in the zone a non-trivial Newtonian background of the same order. The wave equation contains spatial derivatives up to the sixth order, in the lowest order it exactly matches the relativistic wave equation. We then analyze the dispersion relation at very high energies in the context of recent experimental data. The qualitative predictions of the proposed model, concerning the propagation of highly energetic photons, are different from the ones obtained from the modified dispersion relation of the LIV models.

Topological effects on generalized Duffin-Kemmer-Petiau oscillator under Aharonov–Bohm flux field and Coulomb potential

In this paper, we study the relativistic quantum motions of the oscillator field of the wave equation under the influence of the Aharonov–Bohm (AB) flux field with a Coulomb vector potential in the background of the topological defects produced by a cosmic string and global monopole space-time. We derive the radial equation of the generalized Duffin-Kemmer-Petiau (DKP) oscillator in a static cosmic string space-time and solve it through the Heun function equation. Afterwards, we derive the radial equation of the same generalized DKP oscillator in a point-like global monopole background and obtain the eigenvalue solutions using the same procedure. The generalized oscillator field is studied by substituting the radial momentum operator ∂ r → (∂ r + i M ω η 0 f(r)), where f(r) is an arbitrary function other than linear and introduces a vector potential of Coulomb-types through a minimal substitution via ∂ μ → (∂ μ − i q A μ ) in the relativistic wave equation. It is shown that the eigenvalue solutions of the oscillator field are influenced by the topological defects of the cosmic string and point-like global monopole space-times and get them modified. Furthermore, we see that the eigenvalue solutions depend on the geometric quantum phase, and hence, shifted them more in addition to the topological defects that show the gravitational analogue to the Aharonov–Bohm effect for the bound-states.

EFFECTS OF GRAVITY’S RAINBOW ON A RELATIVISTIC SPIN-1 OSCILLATOR

We consider a relativistic spin-1 particle with non-minimal coupling in the context of gravity’s rainbow in the three dimensional background spacetime spanned by static cosmic string. In this context, we acquire an exact solution of the associated spin-1 equation in the modified three dimensional static cosmic string-spanned background spacetime. This relativistic wave equation includes a reducible spinor and this allows us to acquire a non-perturbative expression including the modification functions in the energy domain. In the low energy limit, our results agree well with current literature and provide a basis to discuss the fundamental features of the relativistic spin-1 oscillator. Afterwards, we try to discuss the effects of gravity rainbow functions on the considered spin-1 oscillator in three different scenarios for the modification functions.

Geometrizing the Klein–Gordon and Dirac equations in doubly special relativity

In this work we discuss the deformed relativistic wave equations, namely the Klein–Gordon and Dirac equations in a doubly special relativity scenario. We employ what we call a geometric approach, based on the geometry of a curved momentum space, which should be seen as complementary to the more spread algebraic one. In this frame we are able to rederive well-known algebraic expressions, as well as to treat yet unresolved issues, to wit, the explicit relation between both equations, the discrete symmetries for Dirac particles, the fate of covariance, and the formal definition of a Hilbert space for the Klein–Gordon case.

The Bethe-Salpeter QED Wave Equation for Bound-State Computations of Atoms and Molecules.

Interactions in atomic and molecular systems are dominated by electromagnetic forces and the theoretical framework must be in the quantum regime. The physical theory for the combination of quantum mechanics and electromagnetism, quantum electrodynamics has been "established" by the mid-twentieth century, primarily as a scattering theory. To describe atoms and molecules, it is important to consider bound states. In the nonrelativistic quantum mechanics framework, bound states can be efficiently computed using robust and general methodologies with systematic approximations developed for solving wave equations. With the sight of the development of a computational quantum electrodynamics framework for atomic and molecular matter, the field theoretic Bethe-Salpeter wave equation expressed in space-time coordinates, its exact equal-time variant, and emergence of a relativistic wave equation, is reviewed. A computational framework, with initial applications and future challenges in relation with precision spectroscopy, is also highlighted.

Tachyon Interactions

A consistent theory of free tachyons has shown how tachyon neutrinos can explain major cosmological phenomena, dark energy and dark matter. Now, we investigate how tachyon neutrinos might interact with other particles: the weak interactions. Using the quantized field operators for electrons and tachyon neutrinos, the simplest interaction shows how the chirality selection rule, put in by force in the Standard Model, comes out naturally. Then, I wander into a re-study of what we do with negative frequencies of plane wave solutions of relativistic wave equations. The findings are simple and surprising, leading to a novel understanding of how to construct quantum field theories.

Qualitative results for a relativistic wave equation with multiplicative noise and damping terms

&lt;abstract&gt;&lt;p&gt;Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.&lt;/p&gt;&lt;/abstract&gt;

How Classical, Quantum-Mechanical, and Relativistic Wave and Field Equations Are Uniformly Generated by Velocity-Field Divergence Equations

How Classical, Quantum-Mechanical, and Relativistic Wave and Field Equations Are Uniformly Generated by Velocity-Field Divergence Equations

Analysis of local fractional Klein-Gordon equations arising in relativistic fractal quantum mechanics

In this paper, we present the implementation of a local fractional homotopy perturbation method pertaining to the local fractional natural transform (LFNT) operator for local fractional Klein-Gordon equations (LFKGEs) under distinct fractal initial conditions. The Klein-Gordon equation is a relativistic wave equation which estimates the nature and movement of zero spin particles at high velocities and energies resembling the lightspeed. This equation admits the laws of special relativity and describes zero spin particles with relativistic energy. This work also examines uniqueness and convergence analyses of the obtained solution for a general local fractional partial differential equation via applied method. The numerical simulations are presented for each LFKG equation on the Cantor set. The computational process ensures that the copulation of local fractional homotopy perturbation method and LFNT operator provides series solutions in closed form with fast convergence for local fractional Klein-Gordon equation in a very lucid way. Furthermore, the results have also been compared with solutions obtained by other methods.

Scattering of e± from CF3I Molecule

Theoretical investigation of the scattering of electrons and positrons from the plasma etching gas trifluoroiodomethane (CF3I) is presented in the present work. The investigation is carried out by taking into account the screening correction arising from a semiclassical analysis of atomic geometrical overlapping of the scattering cross-sections calculated in the independent atom approximation. The scattering system e±-CF3I is studied through the calculations of the observable quantities, namely, absolute differential, Sherman function, total elastic and inelastic, momentum transfer, viscosity, ionization and total cross sections over the energy range 1 eV–1 MeV. Energy dependency of the differential cross section and Sherman function are also picturized in this work. A comparative study is carried out between scattering observables for electron impact with those for positron impact to get a better understanding of the interaction and dynamics of the collision process. The corresponding scattering quantities of the constituent atoms are calculated employing a complex optical model potential by solving the Dirac relativistic wave equations in the framework of partial wave analysis. The comparison of our results with the available experimental and theoretical data shows a reasonable agreement.