Dirac Equation Research Articles

Dynamics of graphene as an open quantum system

We examine the imbalance thermodynamics of uncontaminated decoherence in graphene. In a uncontaminated decoherence process, no exchange of energy takes place directly between graphene and the surroundings. It is seen that due to decoherence heat dissolution takes place due to which energy of the environment cannot remain conserved. In this paper, we find out the energy eigenvalues and wave function in the graphene from the perturbed Hamiltonian to find out the density matrix and entropy of the system and their unique relation between the density matrix, momentum of the graphene and entropy where the graphene is taken as an open system. In graphene, the behavior of electrons is controlled by the massless Hamiltonian and ([Formula: see text]) Dirac equation.

New results on the two-body decay of neutrons shed new light on neutron stars

In attempts to resolve the neutron lifetime puzzle, there was suggested a hypothetical decay of neutrons into some unspecified dark matter (DM) particles. Later there were performed studies on how the hypothetical decay of neutrons would affect neutron stars. Recently it was shown that with the allowance for the second solution of Dirac equation for hydrogen atoms, the theoretical branching ratio (BR) for the two-body decay of neutrons (compared to their three-body decay) is amplified by a factor of 3300 from 0.000004. So, the BR becomes about 1.3% in the excellent agreement with the “experimental” BR = (1.15 ± 0.27)% required for reconciling the two distinct experimental values of the neutron lifetime: one from the beam experiments, another from the trap experiments. This meant that the two-body decay of neutrons in the beam experiments (that count only the protons) plays a much more sizable part in the overestimation of the lifetime of neutrons in these experiments than previously thought. Hydrogen atoms corresponding to the second solution of Dirac equations are called the second flavor of hydrogen atoms (SFHA) by the analogy with the flavors of quarks. The existence of the SFHA is evidenced by four different types of atomic/molecular experiments. The primary feature of the SFHA is that due to having only the s-states, they do not emit or absorb the electromagnetic radiation (except for the 21 cm line): they are practically dark. The SFHA became a candidate for a part of DM for the first time after the SFHA-based successful qualitative and quantitative explanation of the perplexing observation by Bowman et al. of the anomalous absorption in the redshifted 21 cm line from the early Universe. In the present paper we analyzed how this neutron decay into the SFHA affects neutron stars. We showed that old neutron stars could very slowly generate the new specific, described in detail baryonic DM in the form of the SFHA. Some old neutron stars would release it into their tiny atmospheres, while some other old neutron stars would release it into the interstellar medium. Besides, mergers of a neutron star with another neutron star or with a black hole, accompanied by the ejection of neutron-rich material, can also lead to the formation of SFHA as the ejecta cools down. This is another interesting aspect of the multi-messenger astronomy focused on studying these mergers through the gravitational waves they generate. These mechanisms of generating new baryonic DM in the universe should have the fundamental importance. We point out the indirect observational evidence of the continuing generation of new baryonic DM. We hope that our results will stimulate a further research in this direction.

A Causal Net approach to Relativistic Quantum Mechanics and Gravitation

In this paper we discuss a causal network model to describe quantum mechanics and gravittation, where space–time is built solely from point events and connecting probabilities. A causal net provides a representation which combines the quantum complementary features of both partticles and waves as each vertex on the causal net represents a possible point event or particle observation. Simultaneity on the causal net is defined by hyperplanes of equivalent proper time. The causal net model when constructed in Minkowksi space–time is shown to lead to a formulation for relativistic quantum mechanics including the Dirac equation. In a curved Riemann space–time the causal paths of the causal net are geodesics and in the local Lorentz frame the net probabilities and the Dirac formalism are preserved. The variation of space–time density of events in the causal paths modifies the metric and provides a space–time curvature leading to the Hilbert action associated with general relativity. Consideration of the Schwarzchild metric shows that in classical limit the perturbation in the density of possible events is proportional to the Newtonian gravitational potential.

Relativistic Bohmian mechanics revisited: A covariant reformulation for spin-1/2 particles

We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.

Electron–ion collision and polarization of X-ray fluorescence radiation under hot quantum plasma conditions

In the current article, the spectral properties and electron collision (total and magnetic) excitation cross sections of ions taking placed in quantum plasmas are investigated. These cross sections are further used to study the polarization and angular distribution characteristics of the de-excitation radiation X-ray spectra, which play an important role in basic theoretical research, the diagnosis of the plasma environment, and the design of optical devices. To do so, a distorted wave method within the relativistic Dirac–Coulomb atomic structure scheme is suggested. The effective interaction potential between electrons and particles in hot quantum plasmas in the method is determined using a quantum approach that incorporates the influence of effective plasma screening effects caused by collective plasma oscillations. This potential replaces the traditional Coulomb interaction potential and is used in solving the modified Dirac equation to obtain the bound and continuum electron wave functions. Higher-order relativistic effects, such as the Breit interaction and the dominant quantum electrodynamics corrections, are added to enhance the accuracy of the method. Detailed calculations for the relativistic atomic structure and collision excitation dynamics process are carried out, taking the highly stripped H-like O7+ ion of astrophysical importance as an illustrative example. Detailed investigations are also conducted on the variation of energies, collision cross sections, and fluorescence polarizations as functions of the plasma parameters. Our results suggest that the joint effects of shielding and plasma coupling lead the energies, cross sections and fluorescence polarizations decrease (compared with the isolated case). The angular distribution of the X-ray fluorescence emission shows large change, suggesting their sensitivity to these effects. This study not only offers a valuable approach to investigating the plasma shielding and plasmon coupling effects in quantum plasmas but also holds significant relevance for applications in controlled nuclear fusion, astrophysical plasmas and so on.

Does quark mixing play a role in the lepton sector?

We suggest a simple relation between the quark and the lepton mixing within the framework of type-I seesaw mechanism. We show that within our ansatz the empirical King–Mohapatra–Smirnov relation, which suggests a connection between the CKM and PMNS mixing where |V13PMNS|≈sinθC/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|V^\ ext {PMNS}_{13}|\\approx \\sin \ heta _C/\\sqrt{2} $$\\end{document}, can be derived. This is possible within a restricted region of the Dirac and Majorana mass parameters.

Quaternion Spin

We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, Q8, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is to introduce a bivector into the spin algebra, which complexifies the Dirac field. It then separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin, and the latter its helicity, generated by a unit quaternion.

Nuclear deformation effects in the spectra of highly charged ions

With the increasing precision of spectroscopic measurements, there is a growing demand for more accurate theoretical predictions, which requires the estimation of various higher-order effects, such as the nuclear deformation effect. Here, we present the results for nuclear deformation correction for the widest range of nuclei. The effects are investigated in terms of electronic transition energies, g factors, and hyperfine splitting constants, by implementing the deformed Fermi nuclear model to the Dirac equation. Based on the improved methodology and appropriate data classification, we present the results for over 1100 nuclei on figures, showing the general trends and anomalies. Moreover, it can serve as a simple tool providing estimations for specific planned research. In addition, we examine the connections and significance of deformation effects in the search of new physics with singly charged ions. Published by the American Physical Society 2024

Unruh effect using Doppler shift method in DSR framework

We study the Unruh effect in doubly special relativity (DSR) framework by generalising the Doppler-shift method to DSR. For both the scalar and Dirac particles, we observe a deviation in the power spectrum of Unruh radiation from the standard Bose–Einstein and Fermi–Dirac distributions, respectively, due to the presence of the frame independent length scale of DSR. We further show that this deviation results in the modification of Unruh temperature which then depends non-linearly on the proper acceleration in DSR.

A numerical algorithm for solving the coupled Schrödinger equations using inverse power method

The inverse power method is a numerical algorithm to obtain the eigenvectors of a matrix. In this work, we develop an iteration algorithm, based on the inverse power method, to numerically solve the Schrödinger equation that couples an arbitrary number of components. Such an algorithm can also be applied to the multi-body systems. To show the power and accuracy of this method, we also present an example of solving the Dirac equation under the presence of an external scalar potential and a constant magnetic field, with source code publicly available.

Vacuum instability in QED with an asymmetric x step: New example of an exactly solvable case

We present a new exactly solvable case in strong-field QED with a one-dimensional step potential (x-step). The corresponding x-step is given by an analytic asymmetric with respect to the axis x reflection function. The step can be considered as a certain analytic “deformation” of the symmetric Sauter field. Moreover, it can be treated as a new regularization of the Klein step field. We study the vacuum instability caused by this x-step in the framework of a nonperturbative approach to strong-field QED. Exact solutions of the Dirac equation used in the corresponding nonperturbative calculations are represented in the form of stationary plane waves with special left and right asymptotics and identified as components of initial and final wave packets of particles. We show that in spite of the fact that the symmetry with respect to positive and negative bands of energies is broken, the distribution of created pairs and other physical quantities can be expressed via elementary functions. We consider the processes of transmission and reflection in the ranges of the stable vacuum and study physical quantities specifying the vacuum instability. We find the differential mean numbers of electron-positron pairs created from the vacuum, the components of current density and energy-momentum tensor of the created electrons and positrons leaving the area of the strong field under consideration. Besides, we study the particular case of the particle creation due to a weakly inhomogeneous electric field and obtain explicitly the total number, the current density and energy-momentum tensor of created particles. Unlike the symmetric case of the Sauter field the asymmetric form of the field under consideration causes the energy density and longitudinal pressure of created electrons to be not equal to the energy density and longitudinal pressure of created positrons. Published by the American Physical Society 2024

Complete solutions of the Dirac equation with q-deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential

Under the condition of the spin symmetry, we rigorously solve the Dirac equation with the q − deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential. According to the values of the deformation parameter q, four different cases are considered. For the two cases q ≥ 1 and q ≤ − 1 with 12αlnq<r<+∞ , the analytical energy spectra and the spinor wave functions associated with the l d − wave bound states are obtained using a suitable approximation to the centrifugal potential term. When 0 < q < 1 or −1 < q < 0, the spinor wave functions for the s − wave bound states are derived and we find that the quantization conditions are transcendental equations which can be solved numerically. The special case q = 0 is also discussed.

A sharp criterion for zero modes of the Dirac equation

It is shown that \Vert A \Vert_{L^d}^2\ge \frac{d}{d-2}S_{d} is a necessary condition for the existence of a nontrivial solution \psi of the Dirac equation \gamma \cdot (-i\nabla -A)\psi = 0 in d dimensions. Here, S_{d} is the sharp Sobolev constant. If d is odd and \Vert A \Vert_{L^d}^2= \frac{d}{d-2}S_{d} , then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.

Split octonionic Dirac equation

The novel forms of the split octonionic Dirac equation and its corresponding Lagrangian are derived using symbolic computing techniques.

Amplification of genuine tripartite nonlocality and entanglement in the Schwarzschild spacetime under decoherence

We investigate the amplification of the genuine tripartite nonlocality (GTN) and the genuine tripartite entanglement (GTE) of Dirac particles in the background of a Schwarzschild black hole by a local filtering operation under decoherence. It is shown that the physically accessible GTN will be completely destroyed by decoherence, and the local filtering operation can cause the physically accessible GTN to be generated in the system coupled with the environment, which has not been discovered before and is beneficial for quantum information processing. Furthermore, we also find that the physically accessible GTE approaches a stable value in the limit of infinite Hawking temperature for most cases, and the nonzero stable value of GTE can be increased by performing the local filtering operation, even in the presence of decoherence. In particular, if the decoherence parameter p is less than 1, the “sudden death” of GTE will take place when the decoherence strength is large enough. Finally, we have shown that physically inaccessible GTE can be enhanced with the operation of the local filter, whereas inaccessible GTN are not allowed to occur in the presence of decoherence.

Singular solutions for Lane–Emden equations with CKN operator

AbstractIn this paper, we concern the isolated singular solutions for Lane–Emden equations involving the CKN operator We classify the isolated singularities by building the connection with the equations involving the Dirac mass in a weighted distributional sense. Then, we obtain the minimal singular solution with given singularities at the origin, analyze the stability of the minimal singular solutions. Finally, we can derive a second solution by the variational method with the given singularities.

Relativistic energies for the q-deformed Scarf potential with Feynman path integrals formulation

In this paper, the Dirac equation with the q-deformed Scarf potential for spin symmetry is solved for an arbitrary spin-orbit quantum number κ, in the presence of Coulomb-like potential tensor. Using the Feynman path integral formalism and the Pekeris approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the associated spinor of the Dirac particle. Furthermore, this method is used to determine the spectrum of two diatomic molecules Li 2(61Π u ) and KRb(B −1Π). The obtained results are compared to the experimental and numerical ones.

Flat-band engineering of quasi-one-dimensional systems via supersymmetric transformations

We introduce a systematic method to spectrally design quasi-one-dimensional crystal models described by the Dirac equation in the low-energy regime. The method is based on the supersymmetric transformation applied to an initially known pseudo-spin-1/2 model. This allows one to extend the corresponding supersymmetric partner so that the new model describes a pseudo-spin-1 system. The spectral design allows the introduction of a flat band and discrete energies at will into the new model. The results are illustrated in three examples where the Su-Schriefer-Heeger chain is locally converted into a stub lattice. Published by the American Physical Society 2024

On the “contact” hyperfine interaction

When Enrico Fermi analyzed the magnetic interaction between the electron and Na, Cs, and Rb nuclei in 1930, he used the Dirac equation to compute the energy of an electron interacting with a point charge and magnetic dipole (the nucleus) fixed at the origin. After the mathematical dust had settled, he found an interaction that appeared to act only at a single point: the center of the electron wavefunction; it has been called a contact interaction. In 1936, Casimir analyzed the magnetic interaction of proton and neutron spins in the ground state of the deuteron using the Schrödinger equation and classical electrodynamics. Using symmetries appropriate only to s states, and performing an integration by parts, he found that they, too, seemed to interact only at a single common point of the proton and neutron: their center-of-mass. When applied to hydrogen, his result agreed with Fermi's. We present an expanded version of Casimir's important but little-known calculation.

Thermodynamic properties of an interacting fermion-antifermion pair in a magnetized spacetime with a non-zero cosmological constant

In a magnetized three-dimensional Bonnor-Melvin spacetime with a non-zero cosmological constant, we explore the dynamics of a fermion-antifermion pair interacting through an attractive Coulomb potential. To analyze the relativistic behavior, we seek an analytical solution for the fully covariant two-body Dirac equation derived from quantum electrodynamics. The resulting equation provides a second-order wave equation that governs the relative motion of the interacting pair. Obtaining an exact solution to this wave equation seems not possible; however, we notice solubility, especially when we consider particles to be closely spaced, meaning as the distance between them approaches zero. At that rate, we determine the energy eigenvalues and wave functions utilizing well-known special functions. By employing these solutions, we determine the thermal properties of this system. Despite the divergence observed in the partition function, we effectively tackle this issue by applying a regularization technique based on the mathematical zeta Hurwitz function. This method facilitates the computation of various thermal quantities, such as free energy, total energy, entropy function, and specific heat. Consequently, we provide an in-depth analysis of the thermodynamic characteristics of the system under consideration.