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We investigate $L^1\to L^\infty$ dispersive estimates for the one dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac12}$ decay rate, which may be improved to $t^{-\frac32}$ at the cost of spatial weights when the thresholds are regular. We classify the structure of threshold obstructions, showing that there is at most a one dimensional space at each threshold. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate, and satisfies the faster weighted bound except for a piece of rank at most two, one per threshold. Further, we prove high energy dispersive bounds that are near optimal with respect to the required smoothness of the initial data. To do so we use a variant of a high energy argument that was originally developed to study Kato smoothing estimates for magnetic Schr\"odinger operators. This method has never been used before to obtain $L^1 \to L^\infty$ estimates. As a consequence of our analysis we prove a uniform limiting absorption principle, Strichartz estimates and prove the existence of an eigenvalue free region for the one dimensional Dirac operator with a non-self-adjoint potential., Comment: This version corrects an error in Lemma~2.2 of the published version which also necessitates changes to the statement of Theorems~1.1 and 2.3 for high energies only. The authors thank Joseph Kraisler, Amir Sagiv and Michael Weinstein for pointing out the error in the published version

The asymptotic behavior of the solution of the Cauchy problem for the nonstationary Dirac equation with potential slowly dependent on time is studied. The construction of the asymptotic solution is based on the spectral decomposition of the solution at a current moment of time. It does not use the adiabatic theorem in scattering theory.

An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrodinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

Relativistic coupled channel analyses based on the Dirac equation are presented for polarized 800-MeV proton inelastic scatterings from an axially symmetric deformed nucleus $$^{92}{\text {Zr}}$$ by using an optical potential model and the first-order collective model. The sequential iteration method using a computer program is employed to solve the complicated Dirac coupled channel equations. The optical potential parameters of the Woods–Saxon shape and the deformation parameters of several low-lying excited states are determined phenomenologically and analyzed. The coupled channel effect between the excited states that belong to the ground-state rotational band is investigated.

There are many stars that are rotating spheroids in the Universe, and studying them is of very important significance. Since the times of Newton, many astronomers and physicists have researched gravitational properties of stars by considering the moment equations derived from Eulerian hydrodynamic equations. In this paper we study the scattering of spinors of the Dirac equation, and in particular investigate the scattering issue in the limit case of rotating Maclaurin spheroids. Firstly we give the metric of a rotating ellipsoid star, then write the Dirac equation under this metric, and finally derive the scattering solution to the Dirac equation and establish a relation between differential scattering cross-section, $\sigma$, and stellar matter density, $\mu$. It is found that the sensitivity of $\sigma$ to the change in $\mu$ is proportional to the density $\mu$. Because of weak gravitational field and constant mass density, our results are reasonable. The results can be applied to white dwarfs, main sequence stars, red giants, supergiant stars and so on, as long as their gravitational fields are so weak that they can be treated in the Newtonan approximations, and the fluid is assumed to be incompressible. Notice that we take the star's matter density to be its average density and the star is not taken to be compact. Obviously our results cannot be used to study neutron stars and black holes. In particular, our results are suitable for white dwarfs, which have average densities of about $10^{5}-10^{6}$\,g~cm$^{-3}$, corresponding to a range of mass of about $0.21-0.61 M_{\bigodot}$ and a range of radius of about $6000-10000$\,km., Comment: This version exactly corresponds to the published version with 12.pages

We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. One set includes new spin and orbital angular momentum (OAM) operators, which are derived from the parity-extended Poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. On the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.

The interaction of nucleon-nucleon (NN) has certain physical characteristics, indicated by nucleon, and meson degrees of freedom. The main purpose of this work is calculating the ground-state energies of 12H and 24He through the two-body system with the exchange of mesons (π, σ, and ω) that mediated between two nucleons. This paper investigates the NN interaction based on the quasirelativistic decoupled Dirac equation and self-consistent Hartree-Fock formulation. We construct a one-boson exchange potential (OBEP) model, where each nucleon is treated as a Dirac particle and acts as a source of pseudoscalar, scalar, and vector fields. The potential in the present work is analytically derived with two static functions of meson, the single-particle energy-dependent (SPED) and generalized Yukawa (GY) functions; the parameters used in meson functions are just published ones (mass, coupling constant, and cutoff parameters). The theoretical results are compared to other theoretical models and their corresponding experimental data; one can see that the SPED function gives more satisfied agreement than the GY function in the case of the considered nuclei.

Bound state solutions of a quantum Dirac particle with a tensor interactions terms are investigated by Asymptotic Iteration Method. The eigenvalues are calculated for tensor potentials and formulated into closed forms. Similarity Transformation is used to convert the Dirac equation into a generalized useful form. The s-wave analytical solutions for the relativistic problems of the Dirac-oscillator, Scarf, Rosen-Morse II, and Pöschl-Teller tensor potentials are obtained with an excellent agreement with previous works.

We show that if the solutions to the (2+1)-dimensional massless Dirac equation for a given 1D potential are known, then they can be used to obtain the eigenvalues and eigenfunctions for the same potential, orientated at an arbitrary angle, in a tilted anisotropic 2D Dirac material. This simple set of transformations enables all the exact and quasi-exact solutions associated with 1D quantum wells in graphene to be applied to the confinement problem in tilted Dirac materials such as borophene. We also show that smooth electron waveguides in tilted Dirac materials can be used to manipulate the degree of valley polarization of quasiparticles travelling along a particular direction of the channel. We examine the particular case of the hyperbolic secant potential to model realistic top-gated structures for valleytronic applications., 12 pages, 5 figures and 1 Supplementary Material

A quantum walk (QW) simulating the flat $$(1 + 2)$$ D Dirac equation on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group $${\mathrm{SO}}(3)$$ , but rather of its double cover $${\mathrm{SU}}(2)$$ , the QW can only be defined globally on an extended spacetime where the polar angle extends from 0 to $$4 \pi $$ . The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated by the QW.

In this work, the Dirac–Kratzer problem with spin and pseudo-spin symmetries in a deformed nucleus is analyzed. Thus, the Dirac equation in curved space–time was considered, with a line element given by [Formula: see text], where [Formula: see text] is a scalar potential, coupled to vector [Formula: see text] and tensor [Formula: see text] potentials. Defining the vector and scalar potentials of the Kratzer type and the tensor potential given by a term centrifugal-type term plus a term cubic singular at the origin, we obtain the Dirac spinor in a quasi-exact way and the eigenenergies numerically for the spin and pseudo-spin symmetries, so that these symmetries are removed due to the coupling of an Coulomb-type effective tensor potential coming from the curvature of space, however, when such potential is null the symmetries return. The probability densities were analyzed using graphs to compare the behavior of the system with and without spin and pseudo-spin symmetries.

Starting from an equation for causal propagator describing Dirac particles interacting with electromagnetic fields and weakly interacting with matter fields, we derive a path integral representation for the propagator. An effective gauge invariant action, which appears in the representation, is interpreted as a pseudoclassical action for the Dirac particles. Quantization of the action is nontrivial due to its gauge invariant nature as well as due to ordering problems that arise in course of a realization of commutation relations in a Hilbert space. The Dirac equation in the background under consideration appears as a quantum equation of motion in the constructed quantum mechanics, justifying, thus, the interpretation of the action. The path-integral representation allows one to calculate effectively the propagator and with its help emerging quantum currents. Studying these currents one may detect effects similar to the chiral magnetic effect. Pseudoclassical equations of motion in the nonrelativistic limit generalize nontrivially the Pauli quantum mechanics with external electromagnetic field.

Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle to perform a spatial search on a 2D grid of N points. As other quantum walks proposed for spatial search, these walks localise partially on the charge after a finite period of time. However, contrary to other walks, this localisation time scales as N for small values of N and tends asymptotically to a constant for larger Ns, thus offering a speed-up over conventional methods.

The ionization probabilities and photoelectron distributions of the hydrogen atom and hydrogenlike ions ${\mathrm{He}}^{+}$ and ${\mathrm{Ne}}^{9+}$ in superstrong electromagnetic fields are obtained by solving the time-dependent Dirac equation with the help of the generalized pseudospectral method in spherical coordinates. A simple transformation of the original radial Dirac Hamiltonian is suggested that removes the spurious states. The ionization probabilities are calculated both within and beyond the dipole approximation for two pulse durations and various peak field strengths scaled with respect to the nuclear charge for each target. Performance of two analytic parameters previously suggested to estimate the nondipole effects is assessed against the numerical data obtained in this study. The photoelectron energy and angular distributions exhibit the above-threshold ionization peaks as well as the low-energy structure. In the superstrong field regime, the low-energy structure builds up with increasing field strength while the above-threshold ionization peaks decrease. The nondipole effects in the photoelectron distributions are also revealed and analyzed.

A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper., 36 pages, 9 figures

Starting from a gauge-invariant Dirac Hamiltonian with noncommutativity of space sector in the presence of an external uniform magnetic field, the resulting Dirac equation has been solved for electrons and its corresponding Zitterbewegung (ZBW) phenomenon has been studied. The corresponding energy spectrum is shown to be different from previous studies wherein the non-gauge-invariant Dirac Hamiltonian has been used. The effects of noncommutativity alter the amplitude as well as the cyclotron and the ZBW frequencies of the average velocity of charge carriers. This result is contrary to previous studies wherein there was no magnetic field, and hence, neither the amplitude nor the frequency of the motion was affected. Moreover, all of the ZBW frequencies of the Landau energy levels appear in the results. Also, in weak magnetic fields, we have calculated the average velocity of charge carriers for two initially localized spin-up and spin-down cases. The plotted trajectories reveal difference between these two cases. In addition, the ZBW phenomenon has been shown that manifests itself as a circular motion whose direction is spin dependent while accompanied by the cyclotron motion.

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-\gamma} * \langle \psi, \beta \psi\rangle)\beta\psi$ with $c\in \mathbb R\setminus\{0\} $, $0 < \gamma < 2$. Our aim is to show the small data global well-posedness and scattering in $H^s$ for $s > \gamma-1$ and $1 < \gamma < 2$. The difficulty stems from the singularity of the low-frequency part $|\xi|^{-(2-\gamma)}\chi_{\{|\xi|\le 1\}}$ of potential. To overcome it we adapt $U^p-V^p$ space argument and bilinear estimates of \cite{yang, tes2d} arising from the null structure. We also provide nonexistence result for scattering in the long-range case $0 < \gamma \le 1$., Comment: 25 page, no figure, no table

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

We consider the Dirac equation and Maxwell’s electrodynamics in ℝ×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. To illustrate the suggested scheme of nonperturbative quantization, we write a simplified set of equations describing some physical situation. Additionally, we discuss the properties of quantum states and operators of interacting fields.

We show that using the multi-splitting algorithm as a preconditioner for the domain wall Dirac linear operator, arising in lattice QCD, effectively reduces the inter-node communication cost, at the expense of performing more on-node floating point and memory operations. Correctly including the boundary \textit{snake} terms, the preconditioner is implemented in the QUDA framework, where it is found that utilizing kernel fusion and the tensor cores on NVIDIA GPUs is necessary to achieve a sufficiently performant preconditioner. A reduced-dimension (reduced-$L_s$) strategy is also proposed and tested for the preconditioner. We find the method achieves lower time to solution than regular CG at high node count despite the additional local computational requirements from the preconditioner. This method could be useful for supercomputers with more on-node flops and memory bandwidth than inter-node communication bandwidth., Add DOI

We establish first-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The potential is allowed to depend arbitrarily on both variables. The systems we consider here include the scenario of a position-dependent mass as well as the massless case. Our Darboux transformations are more general than their existing counterparts (Pozdeeva and Schulze-Halberg in J Math Phys 51:113501, 2010).

Polarized 650-MeV proton inelastic scatterings from 206Pb are analyzed phenomenologically in the relativistic Dirac formalism using an optical potential model. The first-order collective model is used to obtain the transition potentials by considering several excited states at the inelastic scatterings. By reducing the Dirac equation to the Schrodinger-like second-order differential equation, we obtain and analyze the effective central and spin-orbit optical potentials. The deformation parameters for several excited states are calculated and compared with those obtained from nonrelativistic calculations.

Credible reasons are presented to reveal that many of the lingering century old enigmas, surrounding the behavior of at least an individual quantum particle, can be comprehended in terms of an objectively real specific wave function. This wave function is gleaned from the single particle energy-momentum eigenstate offered by the theory of space filling universal quantum fields that is an inevitable outcome of Dirac's pioneering masterpiece. Examples of these well-known enigmas are wave particle duality, the de Broglie hypothesis, the uncertainty principle, wave function collapse, and predictions of measurement outcomes in terms of probability instead of certainty. Paul Dirac successfully incorporated special theory of relativity into quantum mechanics for the first time. This was accomplished through his ingenious use of matrices that allowed the equations of motion to maintain the necessary first order time derivative feature necessary for positive probability density. The ensuing Dirac equation for the electron led to the recognition of the mystifying quantized spin and magnetic moment as intrinsic properties in contrast to earlier ad hoc assumptions. The solution of his relativistic equation for the hydrogen atom produced results in perfect agreement with experimental data available at the time. The most far reaching prediction of the celebrated Dirac equation was the totally unexpected existence of anti-particles, culminating in the eventual development of the quantum field theory of the Standard Model that reveals the deepest secrets of the universe known to date., 13 pages

Mathematical analysis on electromagnetic waves in photonic graphene, a photonic topological material which has a honeycomb structure, is one of the most important current research topics. By modulating the honeycomb structure, numerous topological phenomena have been observed recently. The electromagnetic waves in such media are generally described by the 2-dimensional wave equation. It has been shown that the corresponding elliptic operator with a honeycomb material weight has Dirac points in its dispersion surfaces. In this paper, we study the time evolution of the wave packets spectrally concentrated at such Dirac points in a modulated honeycomb material weight. We prove that such wave packet dynamics is governed by the Dirac equation with a varying mass in a large but finite time. Our analysis provides mathematical insights to those topological phenomena in photonic graphene.

This paper is a mathematical analysis of conduction effects at interfaces between insulators. Motivated by work of Haldane-Raghu , we continue the study of a linear PDE initiated in papers of Fefferman-Lee-Thorp-Weinstein. This PDE is induced by a continuous honeycomb Schrodinger operator with a line defect. This operator exhibits remarkable connections between topology and spectral theory. It has essential spectral gaps about the Dirac point energies of the honeycomb background. In a perturbative regime, Fefferman-Lee-Thorp-Weinstein construct edge states: time-harmonic waves propagating along the interface, localized transversely. At leading order, these edge states are adiabatic modulations of the Dirac point Bloch modes. Their envelops solve a Dirac equation that emerges from a multiscale procedure. We develop a scattering-oriented approach that derives all possible edge states, at arbitrary precision. The key component is a resolvent estimate connecting the Schrodinger operator to the emerging Dirac equation. We discuss topological implications via the computation of the spectral flow, or edge index., Comment: 61 pages; 11 figures

A sequence of necessary actions for the formulation of free leptonic equations (Dirac algorithm) is established. A complete and closed set of equations for the description of massive and massless, charged and neutral, stable and unstable leptons is obtained within a unified algorithmic approach. The Lorentz invariance and covariance of the newly derived equations is as rigorous as for the Dirac equation. All the established facts together explain the exceptional role of the structure of lepton equations that serves as a carrier of the properties of the observed particles.

In this work, we investigate the behaviour of Dirac particles in a class of Gödel-type space-time backgrounds in the presence of non-minimal coupling of the gravitational field with background curvature. We obtain the allowed energies for this relativistic system by solving analytically the Dirac equation in flat and curved space in a topologically trivial flat Gödel-type metric, and analyze the effects on the energy eiegnvalues.

The Generalized Uncertainty Principle and the related minimum length are normally considered in non-relativistic Quantum Mechanics. Extending it to relativistic theories is important for having a Lorentz invariant minimum length and for testing the modified Heisenberg principle at high energies.In this paper, we formulate a relativistic Generalized Uncertainty Principle. We then use this to write the modified Klein-Gordon, Schr\"odinger and Dirac equations, and compute quantum gravity corrections to the relativistic hydrogen atom, particle in a box, and the linear harmonic oscillator., Comment: 6 pages, Revtex

We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based on Duhamel's formula over supercritical range fails., Comment: v2; 18 pages, The error was corrected

In this paper, we study the massive Dirac equation with the presence of the Morse potential in polar coordinate. The Dirac Hamiltonian is written as two second-order differential equations in terms of two spinor wavefunctions. Since the motion of electrons in graphene is propagated like relativistic fermionic quasi-particles, then one is considered only with pseudospin symmetry for aligned spin and unaligned spin by arbitrary $k$. Next, we use the confluent Heun's function for calculating the wavefunctions and the eigenvalues. Then, the corresponding energy spectrum obtains in terms of $N$ and $k$. Afterward, we plot the graphs of the energy spectrum and the wavefunctions in terms of $k$ and $r$, respectively. Moreover, we investigate the graphene band structure by a linear dispersion relation which creates an energy gap in the Dirac points called gapped graphene. Finally, we plot the graph of the valence and conduction bands in terms of wavevectors., 15 pages, 5 figures

We extend the holographic Schwinger-Keldysh prescription introduced in arXiv:1812.08785 to charged black branes, with a view towards studying Hawking radiation in these backgrounds. Equivalently we study real-time fluctuations of the dual CFT held at finite temperature and finite chemical potential. We check our prescription using charged Dirac probe fields. We solve the Dirac equation in a boundary derivative expansion extending the results in arXiv:2011.07039. The Schwinger-Keldysh correlators derived using this prescription automatically satisfy the appropriate KMS relations with Fermi-Dirac factors., Comment: 10 pages + 1 figure. v2: minor edits, change in title

We consider the Dirac electron as a nonperturbative particle-like solution consistent with its own Kerr-Newman (KN) gravitational and electromagnetic field. We develop the earlier models of the KN electron regularized by Israel and López, and consider the non-perturbative electron model as a bag model formed by Higgs mechanism of symmetry breaking. The López regularization determines the unique shape of the electron in the form of a thin disk with a Compton radius reduced by 4π. In our model this disk is coupled with a closed circular string which is placed on the border of the disk and creates the caused by gravitation frame-dragging string tension produced by the vector potential of the Wilson loop. Using remarkable features of the Kerr-Schild coordinate system, which linearizes the Dirac equation, we obtain solutions of the Dirac equation consistent with the KN gravitational and electromagnetic field, and show that this solution takes the form of a massless relativistic string. Parallelism of this model with quantum representations in Heisenberg and Schrodinger pictures explains remarkable properties of the stringy electron model in the relativistic scattering processes.

Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics exists. Given the growing number of applications of high energy electron beams interacting with a variety of quantum systems in laser fields, novel methods for finding exact solutions to the Dirac equation are called for. We present a method for building up solutions to the Dirac equation employing a recently introduced approach for the description of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. We illustrate the method by developing several stationary as well as non-stationary solutions of the Dirac equation with well defined orbital angular momentum along the electron's propagation direction. The first set of solutions describe free electron beams in terms of Bessel functions as well as stationary solutions for both a homogeneous and an inhomogeneous magnetic field. The second set of solutions are new and involve a plane electromagnetic wave combined with a generally inhomogeneous longitudinal magnetic field. Moreover, the developed technique allows us to derive general physical properties of the dynamics in such field configurations, as well as provides physical predictions on the self-consistent electromagnetic fields induced by the dynamics., Comment: 21 pages,5 figures

This paper reviews how a two-state, spin-one-half system transforms under rotations. It then uses that knowledge to explain how momentum-zero, spin-one-half annihilation and creation operators transform under rotations. The paper then explains how a spin-one-half field transforms under rotations. The momentum-zero spinors are found from the way spin-one-half systems transform under rotations and from the Dirac equation. Once the momentum-zero spinors are known, the Dirac equation immediately yields the spinors at finite momentum. The paper then shows that with these spinors, a Dirac field transforms appropriately under charge conjugation, parity, and time reversal. The paper also describes how a Dirac field may be decomposed either into two 4-component Majorana fields or into a 2-component left-handed field and a 2-component right-handed field. Wigner rotations and Weinberg's derivation of the properties of spinors are also discussed., A pedagogical paper of 34 pages

In general, the relativistic wave equation considered to mathematically describe the so-called Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the so-called Majorana condition. Certainly, depending on the representation that one uses, the resulting differential equation changes. It could be a real or a complex system of coupled equations, or it could even be a single complex equation for a single component of the entire wave function. Any of these equations or systems of equations could be referred to as a Majorana equation or Majorana system of equations because it can be used to describe the Majorana particle. For example, in the Weyl representation, in (3+1) dimensions, we can have two non-equivalent explicitly covariant complex first-order equations; in contrast, in (1+1) dimensions, we have a complex system of coupled equations. In any case, whichever equation or system of equations is used, the wave function that describes the Majorana particle in (3+1) or (1+1) dimensions is determined by four or two real quantities. The aim of this paper is to study and discuss all these issues from an algebraic point of view, highlighting the similarities and differences that arise between these equations in the cases of (3+1) and (1+1) dimensions in the Dirac, Weyl, and Majorana representations. Additionally, to reinforce this task, we rederive and use results that come from a procedure already introduced by Case to obtain a two-component Majorana equation in (3+1) dimensions. Likewise, we introduce for the first time a somewhat analogous procedure in (1+1) dimensions and then use the results we obtain., Comment: 38 pages

We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulten plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudo-centrifugal) term. Using the Nikiforov-Uvarov and SUSYQM methods, we analytically develop the relativistic energy eigenvalues and associated Dirac spinor components of wave functions. We find that both methods give entirely the same results. Modifiable of our results into some particular potential cases, useful for other physical systems, are also discussed. We obtain complete agreement with the findings of previous works. The spin and pseudospin bound state energy spectra for various levels are presented in the absence as well as the presence of tensor coupling. Both energy spectrums are sensitive with regards to the quantum numbers $\kappa$ and $n$, as well as the parameter $\delta$. We also notice that the degeneracies between Dirac spin and pseudospin doublet eigenstate partners are completely removed by the tensor interaction. Finally, we present the parameter space of allowable bound state regions of potential strength $V_0$ with constants for both considered symmetry limits $C_S$ and $C_{PS}$., Comment: 21 pages, 9 figures, 2 tables

We consider non-periodic Dirac equations with nonlinearities which involve a combination of concave and convex terms. Using variational methods, we prove the existence of infinitely many large and small energy solutions. For small energy solutions, we establish a new critical point theorem which generalize the dual Fountain Theorem of Bartsch and Willen, by using the index theory and the $$\mathcal {P}$$ -topology. Some non-periodic conditions on the whole space $$\mathbb {R}^{3}$$ are given in order to overcome the lack of compactness.

We construct nonlinear coherent states for electrons in anisotropic 2D Dirac materials in a homogeneous magnetic field orthogonal to the sample. By solving the anisotropic Dirac equation in Landau gauge, we identify the ladder operators of the system. Nonlinear coherent states are obtained as eigenstates of a generalized annihilation operator with complex eigenvalues which depends on an arbitrary function f of the number operator. The anisotropy effects on these states are analyzed by using the probability density and the Heisenberg uncertainty relation for three different choices of f.

We present basic concepts of quantum mechanics from a nuclear-physics point of view. In particular, spin and isospin, the Heisenberg uncertainty relation and the Schrodinger and the Dirac equations are introduced. The analytical solution of the Schodinger equation for bound states in one-dimensional potential and quantum tunnelling are discussed. Quantum entanglement and the concept of Einstein–Podolsky–Rosen (EPR) pair are presented with the experimental evidence of nuclear observations. The importance of symmetries and symmetry breaking is illustrated.

We prove a classification result for ground state solutions of the critical Dirac equation on $\mathbb{R}^n$, $n\geq2$. By exploiting its conformal covariance, the equation can be posed on the round sphere $\mathbb{S}^n$ and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results., Final version to appear on Comm. Math. Phys

We investigate wavepacket dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first quantized setting. We further compare our semi-analytical wavepacket approach to exact finite-difference solutions of the relativistic wave equations., Comment: Some text moved from main body to Appendices; close to published version

As mentioned in the previous chapter a covariant transport theory requires a proper relativistic formulation of the particle dynamics which is based on the Dirac equation for fermions (of spin 1/2ħ) and the Klein-Gordon (or Proca) equation for the bosons. In the nuclear physics context a commonly used (and flexible) scheme is given by the Lagrangian of the isospin symmetric Quantum-Hadro-Dynamics (QHD) which consists of the free Dirac Lagrangian for the nucleons, the Lagrangian for a scalar field σ(x) and a vector field ωμ(x) with self-interactions and an interaction part between bosons and fermions of Yukawa type (in the stationary limit) [1, 2]. As pointed out in Appendix H the QHD model should be considered as an effective approach on the mean-field level that approximates the selfenergies provided by relativistic Dirac-Brueckner theory [3, 4, 5]. A reminder and short survey of quantum-hadro-dynamics is given in Appendix H as well as a proof of thermodynamic consistency. This is mandatory for transport theory since a system initially out-of equilibrium asymptotically (for t →∞) has to provide the correct equilibrium distribution functions (for localized systems).

We present a study of the thermodynamics of the massless free Dirac field at equilibrium with axial charge, angular momentum and external electromagnetic field to assess the interplay between chirality, vorticity and electromagnetic field in relativistic fluids. After discussing the general features of global thermodynamic equilibrium in quantum relativistic statistical mechanics, we calculate the thermal expectation values. Axial imbalance and electromagnetic field are included non-perturbatively by using the exact solutions of the Dirac equation, while a perturbative expansion is carried out in thermal vorticity. It is shown that the chiral vortical effect and the axial vortical effect are not affected by a constant homogeneous electromagnetic field.

Discussions which review through rigorous derivations the physics show the validity range of the analogy between solid state materials like graphene which possess K symmetry crystallographic points in k-space, and the relativistic solutions for massive and low mass particles associated with the Dirac equation. Both eigenenergies and eigenvectors are examined for the non-relativistic solutions of the Schrodinger equation using the tight-binding method, and the relativistic solutions of the Dirac equation. Implications for exploring new materials are drawn from the results. It is concluded Dirac materials are unlikely to fulfill the needs of transistor action materials, but two prime candidates which may satisfy those needs for 2D future electronics are proposed, graphyne and borophene.

The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron's anomalous magnetic moment is large enough. The spectrum of the subextremal self-adjoint Dirac operator with anomalous magnetic moment is purely absolutely continuous and consists of the whole real line; in particular, there are no eigenvalues. The same is true for the spectrum of any self-adjoint extension of the Dirac operator without anomalous magnetic moment interaction, in the subextremal black hole context. In the extremal black hole sector the point spectrum, if non-empty, consists of a single eigenvalue, which is identified., Comment: 21 pages; compared to v1, a short paragraph has been added in the final section, the bibliography has been updated. This is the preprint version of the published article

Atomic forces formulation based on the Dirac-Kohn-Sham equation and flexible (APW+lo/LAPW)+LO basis set is presented. The formulation was implemented in the code FlapwMBPT and allows a user to easily switch between different basis functions of the augmentation type (APW or LAPW) and between different kind of local orbitals. Similar to the work (Phys.Rev.B 91 (2015) 035105), the implementation takes into account small discontinuities of the wave functions, density, and potential at the muffin-tin sphere boundaries. Applications to the materials with strong relativistic effects, such as $\alpha$-Uranium, PuCoGa$_{5}$, and FePt, demonstrate robustness of the method. Comparison of the calculated forces with the ones obtained by numerical differentiation of the free energy shows close agreement with deviations about 0.1\% or less., Comment: 16 pages, 5 tables

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi periodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.

We provide a transfer matrix method for the Foldy-Wouthuysen representation of the Dirac equation. We develop the relationship between the reflection and transmission coefficients of the Dirac spinors and the wavefunction in the transformed representation. We develop a WKB approximation for Dirac fermions which has the same elegant form as the WKB solution to Schrodinger’s equation. Our WKB approximation is to all orders and includes the semi-classical turning point. We provide an extension to fully 2-dimensional periodic structures by Fourier methods for band engineering. We verify our methods for all energies by comparison with analytic solutions developed in the Dirac spinor representation. Also included is a rich appendix detailing our research into the Green’s functions of Dirac fermions.

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n&minus, 1, and n&minus, 2 respectively. We show that for n&ge, 3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheff&eacute, &rsquo, s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.