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Wavelets and multiwavelets have lately been adopted in quantum chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few), they provide a framework that narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for multiwavelet calculations (algebra and integral and differential operator application) and exposes the required functionality to write a simple Python code to solve, among others, the Hartree–Fock equations, the generalized Poisson equation, the Dirac equation, and the time-dependent Schrödinger equation up to any predefined precision. In this study, we will outline the main features of multiresolution analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms. [ABSTRACT FROM AUTHOR]

In the present work, two quasi-molecular compounds each involving one antiproton and one electron ( p ̄) , H e + − p ̄ and H − p ̄ , are investigated. Using completely relativistic calculations within the finite-basis method adapted to systems with axial symmetry, the adiabatic potential curves are constructed by numerically solving the two-center Dirac equation. The binding energies of electron are obtained as a function of the inter-nuclear distance and compared with the corresponding nonrelativistic values and relativistic leading-order corrections calculated in the framework of other approaches. A semantic analysis of antiproton quasi-molecular ions with compounds containing a proton (p) instead of an antiproton is given. The advantages of the A-DKB method are demonstrated. [ABSTRACT FROM AUTHOR]

We consider the L 2 -supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. In that case, there are obvious non-scattering global solutions, i.e., the standing waves. In this paper, we show a scattering and blow-up dichotomy for threshold even solutions in the low-frequency case. We emphasize that this dichotomy still holds at the critical frequency between high and low. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-relativistic limit of the Dirac equation for relativistic spin-1/2 particles within the framework of the conformable fractional derivative (CFD) using the Foldy–Wouthuysen (FW) transformation. This approach leads to the derivation of a conformable fractional Schrödinger–Pauli equation. We propose and employ a conformable fractional version of the FW transformation, thoroughly examining its efficacy and behavior in the non-relativistic limit. Additionally, based on perturbation theory, we compute the energy shifts within the context of CFD and derive a conformable fractional fine structure of the hydrogen spectrum. [ABSTRACT FROM AUTHOR]

To describe the four fundamental interactions observed in the universe, we used the Dirac equation containing an interaction of the screened Kratzer and the Hulthen potential with a ring-shaped term. We study the energy spectra of spin- 1 / 2 particles in noncompact extra dimensions. The polar and angular parts of the N-dimensional Dirac equation are solved using the Nikiforov–Uvarov method. The transcendental energy equation and the associated two components spinor wavefunctions of the spin- 1 / 2 particles are obtained. Moreover, we obtained the nonrelativistic limit of our results by mean of a specific mapping. Some special cases of the proposed potential have been discussed, and their corresponding eigenvalue energies were determined. Finally, we proposed that our results can be used in many branches of physics, to investigate scenarios in extra dimensions where spin- 1 / 2 particles are involved. To test the reliability of our model, we computed the numerical values of the energy spectra of lithium hydride and hydrogen chloride diatomic molecules using a special case of the proposed potential. The energy spectra are calculated for different quantum numbers (n , l) at different values of the dimensionality of the space. The results of this research are overall in good agreement with the theoretical studies of similar investigation. [ABSTRACT FROM AUTHOR]

Pseudomodes of non-self-adjoint Schrödinger operators corresponding to large pseudoeigenvalues are constructed. The approach is non-semiclassical and extendable to other types of models including the damped wave equation and Dirac operators. [ABSTRACT FROM AUTHOR]

The Hamiltonian mapping of the Dirac equation with a quantum perturbation in D black hole models are investigated. We derive the Hamiltonians of the modified Rabi model for the point matter black hole and noncommutative black hole, and obtain perturbed expressions for the velocity and acceleration of the simulating Dirac particles. The fluctuation equations for the charge–current density in two types of black holes are derived based on the solutions of the Dirac equation. The results indicate that the Hamiltonians contain the correction terms with different powers of the creation and annihilation operators, and the accelerations have some additional Dirac Zitterbewegung terms. In addition, the coordinate operators in the fluctuation equations of the charge–current density have different powers. These characteristics can be used to distinguish different types of black holes in the analogical gravity models. [ABSTRACT FROM AUTHOR]

We present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where 0<ε≤1$$ 0<\varepsilon \le 1 $$, and ε$$ \varepsilon $$ is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of O(ε)$$ O\left(\varepsilon \right) $$, while the spatial oscillations have a wavelength of O(1)$$ O(1) $$, with a wave speed of O(ε−1)$$ O\left({\varepsilon}^{-1}\right) $$. We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size h$$ h $$, the time step τ$$ \tau $$ and the small dimensionless parameter ε$$ \varepsilon $$. The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (ε$$ \varepsilon $$‐scalability) necessitates setting h=O(1)$$ h=O(1) $$ and τ=O(ε)$$ \tau =O\left(\varepsilon \right) $$ when 0<ε≪1$$ 0<\varepsilon \ll 1 $$. Finally, some numerical examples are provided to validate the error bounds. [ABSTRACT FROM AUTHOR]

The paper is concerned with the asymptotic expansion of solutions to the following 2 × 2 Dirac type system: L y = - iB - 1 y ′ + Q x y = λ y , B = b 1 0 0 b 2 , y = col y 1 , y 2 , 0.1 with a smooth matrix potential Q ∈ W 1 n 0 , 1 ⊗ C 2 × 2 and b1 < 0 < b2. If b2 = −b1 = 1, this equation is equivalent to one dimensional Dirac equation. We apply these formulas to get the asymptotic expansion of the characteristic determinant of the boundary value problem associated with the above equation subject to the general two-point boundary conditions. This expansion directly yields new completeness result for the system of root functions of such BVP with nonregular boundary conditions. [ABSTRACT FROM AUTHOR]

We construct a nonstandard Lagrangian, called the multiplicative form, for a homogeneous scalar field and a fermion field via the inverse calculus of variations with the equations of motion that still satisfy the respective Klein–Gordon and Dirac equations. By employing the nonuniqueness of the Lagrangian, we show that the Lagrangians can be written as linear combinations of the standard and nonstandard Lagrangians. The stability of the ghost field, an unnatural smallness of the cosmological constant, and the chiral condensate are discussed by using these new Lagrangians. [ABSTRACT FROM AUTHOR]

Context: The diatomic molecules of potassium 39 K 2 a 3 ∑ u + is widely used in industrial chemicals and alternative energy. Besides that, 39 K 2 a 3 ∑ u + 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. Methods: The Dirac equation has been solved for an arbitrary spin-orbit quantum number κ using the path integral technique with the q -deformed generalized Pöschl-Teller potential (D G P T) . 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 n and κ . With the constraint λ ~ > η ~ + 1 , 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 39 K 2 a 3 ∑ u + molecule were computed and compared with the RKR data. The average absolute percentage deviations from the RKR data obtained for the potassium molecule are 0.5018 % . This demonstrates that the (D G P T) model is a very consistent model to study and characterize diatomic molecules. [ABSTRACT FROM AUTHOR]

In Gérard et al. (Ann Sci Ecole Norm Sup 56:127–196, 2023), the Unruh state for massless fermions on a Kerr spacetime was constructed and the authors showed its Hadmard property in the case of very slowly rotating black holes a ≪ M . In this note, we extend this result to the full non extreme case a < M . [ABSTRACT FROM AUTHOR]

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

We consider the nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional Euclidean space. We classify the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. [ABSTRACT FROM AUTHOR]

The energies of an electron in a one-dimensional crystal are studied with both the Schrödinger and Dirac equations using the plane wave expansion method. The crystalline potential sensed by the electron in a cell was calculated by accounting for the Coulombic (electrostatic) interaction between the electron and the surrounding cores (immobile positive ions at the center of the crystal cells). The energies and wave functions of the electron were calculated as a function of four parameters: the period a p of the lattice, the dimension n d i m of the matrix in the momentum space, the partition number l p a in which the unit cell is divided to calculate the potential and the number of cores n c o that affect the electron. It was found that 8000 cores (surrounding the electron) were needed to reach our convergence criterion. An analytical equation that accurately describes the behavior of the energies in function of the cores that affect the electron was also found. As case studies, the energies for pseudo-lithium and pseudo-graphene were obtained as a first approximation for one-dimensional lattices. Subsequently, the energies of an isolated dimer nanoparticle were also calculated using the supercell method. [ABSTRACT FROM AUTHOR]

Based on the position and momentum of noncommutative relations with a noncanonical map, we study the Dirac equation and analyze its parity and time reversal symmetries in a noncommutative phase space. Noncommutative parameters can be endowed with the Planck length and cosmological constant such that the noncommutative effect can be interpreted as an effective gauge potential or a (0,2)-type curvature associated with the Plank constant and cosmological constant. This provides a natural coupling between dynamics and spacetime geometry. We find that a free Dirac particle carries an intrinsic velocity and force induced by the noncommutative algebra. These properties provide a novel insight into the Zitterbewegung oscillation and the physical scenario of dark energy. Using perturbation theory, we derive the perturbed and nonrelativistic solutions of the Dirac equation. The asymmetric vacuum gaps of particles and antiparticles reveal the particle–antiparticle symmetry breaking in the noncommutative phase space, which provides a clue to understanding the physical mechanisms of particle–antiparticle asymmetry and quantum decoherence through quantum spacetime fluctuation.

In noncommutative phase space (NCPS), we investigate the pair creation rate for the problem of scalar and spinorial particles emerging from a vacuum under the influence of uniform external electromagnetic fields using the Schwinger formal technique. It is demonstrated that under specific conditions, where the speed of light c approaches the Fermi velocity υ F and ω ̄ → ω c ∕ 2 (with ω c = | e | ℬ m c representing the cyclotron frequency), the behavior of the Dirac oscillator problem in a uniform electromagnetic field within NCPS is akin to the monolayer graphene problem in NCPS. In addition, the production probability per unit volume per unit time is discussed, along with the limits of parameter deformations. [ABSTRACT FROM AUTHOR]

The tunneling of scalar particles and fermions across the event horizon of KN-AdS black hole surrounded by perfect fluid in Rastall theory are investigated in the frame dragging coordinate systems by applying modified Klein–Gordon equation and modified Dirac equation under the influence of Lorentz symmetry violation in curved spacetime. The Hawking temperature and the heat capacity of the black hole are modified due to the influence of Lorentz symmetry violation. The Hawking temperature and the Bekenstein–Hawking entropy of the black hole beyond the semiclassical approximation are also derived by taking the effects of Planck constant ℏ. The modified Hawking temperature and entropy correction of the black hole are dependent on the Lorentz violation parameter λ , the ether-like vectors u α and Planck constant ℏ. [ABSTRACT FROM AUTHOR]

In this paper, we concern the isolated singular solutions for Lane–Emden equations involving the CKN operator 0.1−Δu+μ1|x|2x·∇u+μ2|x|2u=upinΩ∖{0},u=0on∂Ω.$$\begin{equation} -\Delta u+\frac{\mu _1}{|x|^2}x\cdot \nabla u+\frac{\mu _2}{|x|^2} u =u^p\quad {\rm in}\quad \Omega \setminus \lbrace 0\rbrace,\qquad u=0\quad {\rm on}\quad \partial \Omega. \end{equation}$$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. [ABSTRACT FROM AUTHOR]

Cornerstones of the Cellular Automaton Interpretation of Quantum Mechanics are its ontological states that evolve by permutations, in this way never creating would-be quantum mechanical superposition states. We review and illustrate this with a classical Ising spin chain. It is shown that it can be related to the Weyl equation in the continuum limit. Yet, the model of discrete spins or bits unavoidably becomes a model of qubits by generating superpositions, if only slightly deformed. We study modifications of its signal velocity which, however, do not relate to mass terms. To incorporate the latter, we consider the Dirac equation in 1+1 dimensions and sketch an underlying discrete deterministic "necklace of necklaces" automaton that qualifies as ontological. [ABSTRACT FROM AUTHOR]

Based on the position and momentum of noncommutative relations with a noncanonical map, we study the Dirac equation and analyze its parity and time reversal symmetries in a noncommutative phase space. Noncommutative parameters can be endowed with the Planck length and cosmological constant such that the noncommutative effect can be interpreted as an effective gauge potential or a (0 , 2) -type curvature associated with the Plank constant and cosmological constant. This provides a natural coupling between dynamics and spacetime geometry. We find that a free Dirac particle carries an intrinsic velocity and force induced by the noncommutative algebra. These properties provide a novel insight into the Zitterbewegung oscillation and the physical scenario of dark energy. Using perturbation theory, we derive the perturbed and nonrelativistic solutions of the Dirac equation. The asymmetric vacuum gaps of particles and antiparticles reveal the particle–antiparticle symmetry breaking in the noncommutative phase space, which provides a clue to understanding the physical mechanisms of particle–antiparticle asymmetry and quantum decoherence through quantum spacetime fluctuation. [ABSTRACT FROM AUTHOR]

We have discussed the Dirac equation in Schwarzschild spacetime using pseudo-supersymmetric quantum mechanics and have obtained the partner Hamiltonian of the initial Hamiltonian operator. We demonstrate that the partner metric tensors, corresponding to these Hamiltonians, can be derived using the intertwining relations inherent in pseudo-supersymmetric approaches. We have seen that the Schwarzschild metric may be expanded into Schwarzschild–Tangherlini metric using the aspects of pseudo-supersymmetry. Furthermore, we have derived approximate solutions for the radial component within the framework of N = 2 supersymmetry with the corresponding radial potential graphs presented. Subsequently, our discussion extends to the quasinormal modes, focusing particularly on the asymptotic limits as r approaches ± ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we study Feynman checkers, one of the most elementary models of electron motion. It is also known as a one-dimensional quantum walk or an Ising model at an imaginary temperature. We add the simplest non-trivial electromagnetic field and find the limits of the resulting model for small lattice step and large time, analogous to the results by Narlikar from 1972 and Grimmet–Jason–Scudo from the 2000s. It turns out that the limits in the model with the added field are obtained from the ones without field by mass renormalization. Also, we find an exact solution of the resulting model. [ABSTRACT FROM AUTHOR]

The spin-torsion theory is a gauge theory approach to gravity that expands upon Einstein's general relativity (GR) by incorporating the spin of microparticles. In this study, we further develop the spin-torsion theory to examine spherically symmetric and static gravitational systems that involve free-falling macroscopic particles. We posit that the quantum spin of macroscopic matter becomes noteworthy at cosmic scales. We further assume that the Dirac spinor and Dirac equation adequately capture all essential physical characteristics of the particles and their associated processes. A crucial aspect of our approach involves substituting the constant mass in the Dirac equation with a scale function, allowing us to establish a connection between quantum effects and the scale of gravitational systems. This mechanism ensures that the quantum effect of macroscopic matter is scale-dependent and diminishes locally, a phenomenon not observed in microparticles. For any given matter density distribution, our theory predicts an additional quantum term, the quantum potential energy (QPE), within the mass expression. The QPE induces time dilation and distance contraction, and thus mimics a gravitational well. When applied to cosmology, our theory yields a static cosmological model. The QPE serves as a counterpart to the cosmological constant introduced by Einstein to balance gravity in his static cosmological model. The QPE also offers a plausible explanation for the origin of Hubble redshift (traditionally attributed to the universe's expansion). The predicted luminosity distance–redshift relation aligns remarkably well with SNe Ia data from the cosmological sample of SNe Ia. In the context of galaxies, the QPE functions as the equivalent of dark matter. The predicted circular velocities align well with rotation curve data from the SPARC (Spitzer Photometry and Accurate Rotation Curves database) sample. Importantly, our conclusions in this paper are reached through a conventional approach, with the sole assumption of the quantum effects of macroscopic matter at large scales, without the need for additional modifications or assumptions. [ABSTRACT FROM AUTHOR]

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional ℘ -adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1 / ℘ , where ℘ is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new ℘ -adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the ℘ -adic space imposes substantial restrictions on the solutions of the new equation. This equation admits localized solutions, which is impossible in the standard case. We show that an isolated quantum system whose evolution is controlled by the ℘ -adic Dirac equation does not satisfy the Einstein causality, which means that the speed of light is not the upper limit for the speed at which conventional matter or energy can travel through space. The new ℘ -adic Dirac equation is not intended to replace the standard one; it should be understood as a new version (or a limit) of the classical equation at the Planck length scale. [ABSTRACT FROM AUTHOR]

Because the electron transport mechanism in graphene is heavily impacted by the strain and the ferromagnetic metal stripe as well as several other avenues, in this paper we investigate the effects of the strained barrier induced by the strain and the magnetic field generated by the ferromagnetic metal stripe on the valley polarization through numerical calculation. When the strength and the width of the strained barrier as well as the magnitude of the magnetic field are changed, the rapid variation of the valley polarization is observed. This study will be helpful for devising and manufacturing new-style valleytronic devices. [ABSTRACT FROM AUTHOR]

Starting from the pseudo‐differential decomposition D=(−Δ)12H of the Dirac operator D=∑j=1nej∂xj in terms of the fractional operator (−Δ)12 of order 1 and of the Riesz–Hilbert type operator H we will investigate the fundamental solutions of the space‐fractional Dirac equation of Lévy–Feller type ∂tΦα(x,t;θ)=−(−Δ)α2expiπθ2HΦα(x,t;θ)involving the fractional Laplacian −(−Δ)α2 of order α, with 2m≤α < 2m + 2 (m∈ℕ), and the exponentiation operator expiπθ2H as the hypercomplex counterpart of the fractional Riesz–Hilbert transform carrying the skewness parameter θ, with values in the range |θ|≤min{α−2m,2m+2−α}. Such model problem permits us to obtain hypercomplex counterparts for the fundamental solutions of higher‐order heat‐type equations ∂tFM(x,t)=κM(∂x)MFM(x,t)(M=2,3,...) in case where the even powers resp. odd powers D2m=(−Δ)m (M=α=2m) resp. D2m+1=(−Δ)m+12H (M=α=2m+1) of D are being considered. [ABSTRACT FROM AUTHOR]

We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, Q 8 , 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. [ABSTRACT FROM AUTHOR]

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Abstract The approximate solutions of the Dirac equation for spin symmetry and pseudospin symmetry are studied with a coshine Yukawa potential model via the traditional supersymmetric approach (SUSY). To remove the degeneracies in both the spin and pseudospin symmetries, a coshine Yukawa tensor potential is proposed and applied to both the spin symmetry and the pseudospin symmetry. The proposed coshine tensor potential removes the energy degenerate doublets in both the spin symmetry and pseudospin symmetry for a very small value of the tensor strength (H = 0.05). This shows that the coshine Yukawa tensor is more effective than the real Yukawa tensor. The non-relativistic limit of the spin symmetry is obtained by using certain transformations. The results obtained showed that the coshine Yukawa potential and the real Yukawa potential has the same variation with the angular momentum number but the variation of the screening parameter with the energy for the two potential models differs. However, the energy eigenvalues of the coshine Yukawa potential model, are more bounded compared to the energies of the real Yukawa potential model.

The influence of a a rectangular potential barrier on the quantum transport of fermions in silicene is explored. Specifically, analytical solutions are presented to derive transmission and reflection probabilities together with conductance. It is shown that the transmission is highly sensitive to both the barrier height and incident energy. As a result, the occurrence of Klein and resonant tunnelings is observed, with a significant dependence on the barrier width. Notably, it is found that perfect transmission extends beyond normal incidence, occurring at various oblique angles. Moreover, the transmission pattern exhibits a more fragmented structure with increasing barrier width, reminiscent of Fabry‐Pérot resonances. In contrast, the conductance displays a non‐monotonic dependence on incident energy and features rapid oscillations with a rising barrier height. However, at a constant barrier height, there is a minimal disparity among conductance profiles for high incident energy values. When incident energy equals the barrier height, the conductance experiences a local minimum. For a thin barrier, a substantial reduction in conductance is observed, unlike the oscillatory behavior seen with a thicker barrier. These findings underscore the progress in silicene research and offer a fresh perspective on the relativistic applications of tunneling in this material. [ABSTRACT FROM AUTHOR]

Discontinuum physics (DCP) provides the non-arbitrary requirement for excluding the antimatter solutions to Dirac's equation, resulting in a matter-antimatter asymmetry. A matter-antimatter asymmetrical universe then results from coupling the matter-antimatter asymmetry with (1) the prior DCP proposal that the universe came into existence from a quantum fluctuation of a type of vacuum state with a zero net total energy, (2) allowing this quantum fluctuation to constitute a discontinuity containing the non-arbitrary requirement, with this occurring similar to how DCP showed the atomic absorption and emission process (AE-process) of Bohr's atomic model constitutes a discontinuity, and (3) allowing this discontinuity to be conserved when the universe comes into existence, with this occurring similar to how DCP showed the AE-process discontinuity of Bohr's atomic model is conserved as the emitted electromagnetic radiation (i.e., photon) discontinuity, where (a) a photon spends a large amount of time in the nonzero state during which it can interact with another photon in a collision, becoming a matter-antimatter pair, and (b) a photon spends a tiny amount of time in the zero state during which it cannot interact with any particles and allowing it to be capable of spontaneously become a matter particle. [ABSTRACT FROM AUTHOR]

We introduce a new equation (a class of equations) to be considered as a candidate for the equation for a nonzero-mass neutrino. [ABSTRACT FROM AUTHOR]

In topological insulators, massive surface states resulting from local symmetry breaking were thought to exhibit a half-quantized Hall conductance, obtained from the low-energy effective model in an infinite Brillouin zone. In a lattice model, the surface band is composed of a combination of surface states and bulk states. The massive surface states alone may not be enough to support an exact one-half quantized surface Hall conductance in a finite Brillouin zone and the whole surface band always gives an integer quantized Hall conductance as enforced by the TKNN theorem. To explore this, we investigate the band structures of a lattice model describing the magnetic topological insulator film that supports the axion insulator, Chern insulator, and semi-magnetic topological insulator phases. We reveal that the gapped and gapless surface bands in the three phases are characterized by an integer-quantized Hall conductance and a half-quantized Hall conductance, respectively. We propose an effective model to describe the three phases and show that the low-energy dispersion of the surface bands inherits from the surface Dirac fermions. The gapped surface band manifests a nearly half-quantized Hall conductance at low energy near the center of Brillouin zone, but is compensated by another nearly half-quantized Hall conductance at high energy near the boundary of Brillouin zone because a single band can only have an integer-quantized Hall conductance. The gapless band hosts a zero Hall conductance at low energy but is compensated by another half-quantized Hall conductance at high energy, and thus the half-quantized Hall conductance can only originate from the gapless band. Moreover, we calculate the layer-resolved Hall conductance of the system. The conclusion suggests that the individual gapped surface band alone does not support the half-quantized surface Hall effect in a lattice model. [ABSTRACT FROM AUTHOR]

We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It is a modification of the model known under the names Feynman checkers or one-dimensional quantum walk. It can be viewed as a six-vertex model with certain complex weights of the vertices. The discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero. It is exactly solvable in terms of hypergeometric functions. We introduce interaction resembling Fermi’s theory and establish perturbation expansion. [ABSTRACT FROM AUTHOR]

This article delves into the Ramsauer-Townsend effect regarding the realm of quaternionic relativistic quantum mechanics. The investigation centers on studying the Dirac equation within this framework. The article presents a thorough examination of various facets of the subject matter, including a comprehensive analysis of quaternionic potential wells, wave functions, probability currents, reflection coefficients, and transmission coefficients. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann's eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow's SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model's elementary particles, which obey SU(3) ⊗ SU(2) ⊗ U(1) symmetry. [ABSTRACT FROM AUTHOR]

We use the semiclassical approach to solve the Klein–Gordon and Dirac equations in the presence of a time-varying electric field. Our objective is to calculate the density of particle creation in a cosmological anisotropic Bianchi- I space–time. We demonstrate that when the electric interaction is proportional to the Ricci scalar of curved space–time, the distribution of particles subjected to the electric field transforms into a thermal state. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonrelativistic limit of normalized solutions to a nonlinear Dirac equation as given below: - i c ∑ k = 1 3 α k ∂ k u + m c 2 β u - Γ ∗ (K | u | κ ) K | u | κ - 2 u - P | u | s - 2 u = ω u , ∫ R 3 | u | 2 d x = 1. Here, c > 0 represents the speed of light, m > 0 is the mass of the Dirac particle, ω ∈ R emerges as an indeterminate Lagrange multiplier, Γ , K, P are real-valued function defined on R 3 , also known as potential functions. Our research first confirms the presence of normalized solutions to the Dirac equation under high-speed light conditions. We then illustrate that these solutions converge to normalized ground states of nonlinear Schrödinger equations, and we also show uniform boundedness and exponential decay of these solutions. Our results form the first discussion on nonrelativistic limit of normalized solutions to nonlinear Dirac equations. This not only aids in the study of normalized solutions of the nonlinear Schrödinger equations, but also physically explains that the normalized ground states of high-speed particles and low-speed motion particles are consistent. [ABSTRACT FROM AUTHOR]

The approximate solutions of the Dirac equation for spin symmetry and pseudospin symmetry are studied with a coshine Yukawa potential model via the traditional supersymmetric approach (SUSY). To remove the degeneracies in both the spin and pseudospin symmetries, a coshine Yukawa tensor potential is proposed and applied to both the spin symmetry and the pseudospin symmetry. The proposed coshine tensor potential removes the energy degenerate doublets in both the spin symmetry and pseudospin symmetry for a very small value of the tensor strength (H = 0.05). This shows that the coshine Yukawa tensor is more effective than the real Yukawa tensor. The non-relativistic limit of the spin symmetry is obtained by using certain transformations. The results obtained showed that the coshine Yukawa potential and the real Yukawa potential has the same variation with the angular momentum number but the variation of the screening parameter with the energy for the two potential models differs. However, the energy eigenvalues of the coshine Yukawa potential model, are more bounded compared to the energies of the real Yukawa potential model. [ABSTRACT FROM AUTHOR]

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F ˜ λ , we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F ˜ λ. [ABSTRACT FROM AUTHOR]

In this paper, we have considered an inverse scattering problem for discrete Dirac operator L with eigenparameter dependent boundary condition. At the outset, the Jost solution and the Jost function of L are given. Then the zeros of the Jost function and some properties of the scattering function are examined. After finding the scattering dataset and the main equation for the operator L, the uniqueness of the kernel, continuity of the scattering function and the appropriate Levinson type formula are obtained. [ABSTRACT FROM AUTHOR]

F. Wilf's results on the need to consider the rules of quantum-mechanical correspondence in the Dirac equation for an electron are used to show that the equation obtained by giving physical meaning to α‑Dirac operators should be considered a phenomenological equation for a particle of non-zero size (the EM polaron introduced earlier by the author). This allows a solution to be found to the inherent paradox of the Dirac equation, which consists of the equality of the velocity of the moving particles to the speed of light in a vacuum, which is a priori unobtainable, and to understand the physical essence of spin as the intrinsic mechanical moment of an EM polaron. It is also shown that the Dirac–Wilf equation for a single spatial dimension can be considered a generalization of the Schrödinger equation for relativistic energies. [ABSTRACT FROM AUTHOR]

We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ⁢ (1 / c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger–Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger–Poisson equation also arises as the mean field limit of the 푁-body Schrödinger equation with Coulomb interaction. We propose that the Pauli–Poisson equation arises as the mean field limit N → ∞ of the linear 푁-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli–Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in 1 / c of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb–Thirring estimates. [ABSTRACT FROM AUTHOR]

In this work, we investigate the behavior of massless spin 12 field by working out the Dirac equation in a curved static spherically symmetric Schwarzschild space-time. Detailed derivation of novel exact massive and massless scalar quasibound state in Schwarzschild black hole background is presented in this work. After decoupling the Dirac equation, we successfully solve both of the angular and radial parts, respectively in terms of spin weighted Spherical Harmonics and Confluent Heun functions. With the exact radial wave solution in hand, applying its polynomial condition leads to the discovery of the quantized energy levels expression. We found that the massless spinor field around the Schwarzschild black hole has complex valued energy levels, in contrast to the purely imaginary for a massless boson around the same black hole. In the last section, the Hawking radiation distribution function is derived via Damour-Ruffini method and the Hawking temperature is obtained.