Dirac Equation Research Articles

Paraxial Dirac equation

In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.

Bound states of Dirac fermions in the presence of a Fermi velocity modulation

We investigate the effect of a position-dependent Fermi velocity on the electronic properties of two-dimensional Dirac materials. A physical Fermi velocity distribution, which approaches a finite value at infinity and experiences a modulation near x = 0, is considered. Such a position-dependent Fermi velocity could be realized in the curved graphene or by applying strain. It is shown that the bound states are absent in the presence of a pure Fermi velocity modulation without an electrostatic potential well. However, an extra electrostatic potential modulation could produce the bound states. A set of discrete energy level spectrum and the corresponding wave functions are obtained by solving the Dirac equation exactly. Local probes such as scanning tunnel microscopy should be able to observe the predicted bound states in two-dimensional materials.

A Mass-in-Mass Chain and the Generalization of the Dirac Equation with an Eight-Component Wave Function and with Optical and Acoustic Branches of the Dispersion Relation

A Mass-in-Mass Chain and the Generalization of the Dirac Equation with an Eight-Component Wave Function and with Optical and Acoustic Branches of the Dispersion Relation

A New Interpretation of the Negative Energy Solutions to Dirac Equation and a Configuration of the Electron

A New Interpretation of the Negative Energy Solutions to Dirac Equation and a Configuration of the Electron

Modification Study on Quantum Tunneling Radiation of Kinnersley Black Hole

In the spacetime of a linearly accelerating Kinnersley black hole, the Lorentz-breaking theory is used to modify the dynamical equations of Dirac particles by selecting gamma matrices and aether-like field vectors in the curved spacetime of this black hole. Using the WKB approximation and black hole quantum tunneling radiation theory, we investigate the characteristics of quantum tunneling radiation in this black hole.By solving the modified spinor field equations, we obtain expressions for the corrected quantum tunneling rate, Hawking temperature, and surface gravitation of the black hole. By studying the particle radial component of the general momentum in this curved spacetime, a new expression for the modified distribution of positive and negative energy levels of Dirac particles, as well as their maximum value of crossing energy level, is obtained. In order to further elucidate the physical significance of the research methodology employed in the article and a series of conclusions obtained, a detailed discussion of the corresponding results is provided in the later sections of this paper.

Hyperbolic Rotation with Euclidean Angle Illuminates Space Time Spinors

In this article it will be shown that by introducing a hyperbolic rotation as function of a Euclidean rotation angle, all possible Lorentz group spacetime rotations can be performed by Euclidean rotation parameters. This unification in Lorentz group rotations parameters (all Euclidean) allows for the calculation of the volume of a spacetime region bounded by the light cone of a past event, i.e., a causality-volume with the shape of a spacetime three-sphere. As will be calculated, the surface area of the spacetime three-sphere is the volume of our 3D space. Additionally, the unification in Lorentz group rotation parameters allows for the derivation of a Dirac spinor purely from the spacetime three-sphere symmetries. This derivation from the spacetime symmetries of a geometrical object yields the same result as solving the Dirac equation with quantum mechanical eigenvalue eigenvector complex matrix calculations.

Asymmetric particle-antiparticle Dirac equation: first quantization

We derive a Dirac-like equation, the asymmetric Dirac equation, where particles and antiparticles sharing the same wave number have different energies and momenta. We show that this equation is Lorentz covariant under proper Lorentz transformations (boosts and spatial rotations) and also determine the corresponding transformation law for its wave function. We obtain a formal connection between the asymmetric Dirac equation and the standard Dirac equation and we show that by properly adjusting the free parameters of the present wave equation we can make it reproduce the predictions of the usual Dirac equation. We show that the rest mass of a particle in the theoretical framework of the asymmetric Dirac equation is a function of a set of four parameters, which are relativistic invariants under proper Lorentz transformations. These four parameters are the analog to the mass that appears in the standard Dirac equation. We prove that in order to guarantee the covariance of the asymmetric Dirac equation under parity and time reversal operations (improper Lorentz transformations) as well as under the charge conjugation operation, these four parameters change sign in exactly the same way as the four components of a four-vector. The mass, though, being a function of the square of those parameters remains an invariant. We also extensively study the free particle plane wave solutions to the asymmetric Dirac equation and derive its energy, helicity, and spin projection operators as well as several Gordon’s identities. The hydrogen atom is solved in the present context after applying the minimal coupling prescription to the asymmetric Dirac equation, which also allows us to appropriately obtain its non-relativistic limit.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

Dirac equation in curved spacetime: the role of local Fermi velocity

We study the motion of charge carriers in curved Dirac materials, in the presence of a local Fermi velocity. An explicit parameterization of the latter emerging quantity for a nanoscroll cylindrical geometry is also provided, together with a discussion of related physical effects and observable properties.

Asymmetric particle-antiparticle Dirac equation: second quantization

We build the fully relativistic quantum field theory related to the asymmetric Dirac fields first presented in a prequel to this work. These fields are solutions of the asymmetric Dirac equation, a Lorentz covariant Dirac-like equation whose positive and ‘negative’ frequency plane wave solutions’ dispersion relations are no longer degenerate. At the second quantization level, we show that this implies that particles and antiparticles sharing the same wave number have different energies and momenta. In spite of that, we prove that by properly fixing the values of the relativistic invariants that define the asymmetric Dirac free field Lagrangian density, we can build a consistent, fully relativistic, and renormalizable quantum electrodynamics (QED) that is empirically equivalent to the standard QED. We discuss the reasons and implications of this non-trivial equivalence, exploring qualitatively other scenarios in which the asymmetric Dirac fields may lead to beyond the standard model predictions. We give a complete account of how the asymmetric Dirac fields and the corresponding annihilation and creation operators transform under improper Lorentz transformations (parity and time reversal operations) and under the charge conjugation operation. We also prove that the present theory respects the CPT theorem.

A Torsional Two-Component Description of the Motion of Dirac Particles at Early Stages of the Cosmic Evolution

A Torsional Two-Component Description of the Motion of Dirac Particles at Early Stages of the Cosmic Evolution

Discrete phase space-continuous time relativistic Klein–Gordon and Dirac equations, and a new non-singular Yukawa potential

This paper deals with the second quantization of interacting relativistic Fermionic and Bosonic fields in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. The corresponding Feynman diagrams and a new S^{#}-matrix theory is developed. In the special case of proton-proton Møller scattering via an exchange of a neutral meson, the explicit second order element langle f | S^{#}_{(2)} |i rangle is deduced. In the approximation of very low external three-momenta, a new Yukawa potential is explicitly derived from langle f | S^{#}_{(2)} |i rangle . Moreover, it is rigorously proved that this new Yukawa potential is divergence-free. The mass parameter of the exchanged meson may be set to zero to obtain a type of scalar Boson exchange between hypothetical Fermions. This provides a limiting case of a new Coulomb type potential directly from the new singularity free Yukawa potential. A divergence-free Coulomb potential between two Fermions at two discrete points is shown to be proportional to the Euler beta function. Within this relativistic discrete phase space continuous time, a single quanta is shown to occupy the hyper-tori S^{1}_{n^1} times S^{1}_{n^3} times S^{1}_{n^3} where S^{1}_{n} is a circle of radius sqrt{2n+1}.

The Implementation of MuDirac in Geant4: A Preliminary Approach to the Improvement of the Simulation of the Muonic Atom Cascade Process

Muonic Atom X-ray Emission spectroscopy (µ-XES) is a novel elemental technique that exploits the high-energy X-rays emitted from the muonic atom cascade process to characterize materials. At the ISIS Neutron and Muon Source, the technique is performed at Port4 of the RIKEN-RAL facility, with a user demand that is increasing every year. To cope with this demand, it is necessary to continue to improve the method, either for the hardware (detectors, acquisition, etc.) or software (data analysis and interpretation). In both cases, Monte Carlo codes play an important role: with a simulation, it is possible to reproduce the experimental setup and provide a reliable quantitative analysis. In this work, we investigate the capabilities of GEANT4 for such applications. From the results, we observed that the generation of X-rays, especially the kα and kβ transition for high Z atoms, are not in agreement with the experimental ones. A solution to this issue, other than an attempt with a small modification of the GEANT4 cascade class, could be provided by a database of transition energy calculated by a Dirac equation software called MuDirac. The software, developed by the UKRI scientific computing department and the ISIS muon group, can compute all the transition energy for a given nuclide. Here, preliminary results of the implementation of the MuDirac database in GEANT4 are reported.

Particle-pair creation by high-harmonic laser fields

We consider the dynamical pair-creation by intense electric fields in vacuum, based on exact solutions of the Dirac equation using the temporal Klein process. This approach gives a simple interpretation of the QED processes associated with arbitrary time-varying electric fields, and leads to the standard quantum kinetic equation. As an application, we consider the case of an intense laser field containing a large spectrum of coherent high-harmonics (HH). The role of the fast field oscillations is discussed. We show that HH pulses eventually lead to an enhancement of pair-creation and can be described as a sequence of dynamically equivalent Sauter pulses.

Gravitational analogue of radiation reaction in STVG

We have studied influence of STVG in circular motion of massive particle around the Schwarzschild-MOG black hole and discussed the Innermost Stable Circular Orbit (ISCO) and Marginally Bound Orbit (MBO) position of massive test particle. This research explores dynamics of massive particles within the framework of Scalar–Tensor–Vector Gravity (STVG) when navigating the vicinity of Schwarzschild-MOG black holes. The study extends its investigation to the angular velocity of massive particles orbiting Schwarzschild-MOG black holes, comparing it to predictions from general relativity. Finally, we have investigated gravitational analogue of radiation reaction by considering Landau–Abraham–Dirac equation in STVG. We have shown that because of radiation reaction massive test particle falls down to the black hole.

Quantum tunneling of spin-1/2 and spin-3/2 particles in the black holes of asymptotically safe gravity

Hawking initiated study of radiations from black holes. The black hole thermodynamics holds a significant importance in understanding of quantum nature of gravity. In this article, Hawking radiations have been studied by employing the semi-classical perspective of quantum tunneling. Dirac and Rarita-Schwinger equations have been employed to determine the tunneling probabilities of spin-1/2 and spin-3/2 particles, respectively, in the background of black holes developed in the asymptotically safe gravity. Hawking temperature determined in both cases is in agreement.

Geometric post-Newtonian description of massive spin-half particles in curved spacetime

We consider the Dirac equation coupled to an external electromagnetic field in curved four-dimensional spacetime with a given timelike worldline γ representing a classical clock. We use generalised Fermi normal coordinates in a tubular neighbourhood of γ and expand the Dirac equation up to, and including, the second order in the dimensionless parameter given by the ratio of the geodesic distance to the radii defined by spacetime curvature, linear acceleration of γ, and angular velocity of rotation of the employed spatial reference frame along γ. With respect to the time measured by the clock γ, we compute the Dirac Hamiltonian to that order. On top of this ‘weak-gravity’ expansion we then perform a post-Newtonian expansion up to, and including, the second order of , corresponding to a ‘slow-velocity’ expansion with respect to γ. As a result of these combined expansions we give the weak-gravity post-Newtonian expression for the Pauli Hamiltonian of a spin-half particle in an external electromagnetic field. This extends and partially corrects recent results from the literature, which we discuss and compare in some detail.

Different Aspects of Spin in Quantum Mechanics and General Relativity

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum mechanical picture. On the other hand, it seems very unlikely that the similarities should be just coincidental. The interpretation of the author is that this can be a golden opportunity to investigate the interplay between these two theories.

Visualizations and Analyses of Quantum Behavior, Spacetime Curvature, and Metric Relationships in Relativistic Physics

This study aims to investigate essential concepts in quantum mechanics and theoretical physics, with an emphasis on the 1+1 dimension. We examine the Dirac equation for relativistic spin-1/2 particles, the Time-Dependent Schrödinger Equation in 1+1 spacetime with flat conformal metric, and connect them to the Dirac equation. Additionally, we explore the Alcubierre Metric related to warp drive, particle modeling in a harmonic potential using the Schrödinger Equation, and the Gödel Metric Solution to depict the peculiarities of spacetime. The research aims to deepen the understanding of these concepts, identify theoretical implications, and their potential applications. This research aims to enhance the understanding of fundamental physics, assist in the development of future technologies, and provide deeper insights into the universe. Its benefits lie in contributing to theoretical understanding in physics, which can spark the development of new theories. This study is limited to physics concepts in the 1+1 dimensions, without empirical experiments or practical applications. The primary focus is on the theoretical analysis of these concepts. The results of this research have potential theoretical implications in understanding basic physics and spacetime phenomena. The simplification and connections between these concepts can aid in the development of new theories in theoretical physics. The uniqueness of this research lies in its integrative approach to quantum mechanics and theoretical physics concepts in the 1+1 dimension, which may not have been extensively explored previously. Through this research, we have investigated several key concepts in quantum mechanics and theoretical physics in the 1+1 dimension. These findings can make a significant contribution to our understanding of the universe and the potential development of new theories in physics.

The Dirac Equation on Metrics of Eguchi-Hanson Type II with Negative Constant Scalar Curvature

The Dirac Equation on Metrics of Eguchi-Hanson Type II with Negative Constant Scalar Curvature