Dirac Equation Research Articles

Minimally coupled fermion–antifermion pairs via exponentially decaying potential

In this study, we explore how a fermion–antifermion (ff¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\overline{f}$$\\end{document}) pair interacts via an exponentially decaying potential. Using a covariant one-time two-body Dirac equation, we examine their relative motion in a three-dimensional flat background. Our approach leads to coupled equations governing their behavior, resulting in a general second-order wave equation. Through this, we derive analytical solutions by establishing quantization conditions for pair formation, providing insights into their dynamics. Notably, we find that such interacting ff¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\overline{f}$$\\end{document} systems are unstable and decay over time, with the decay time depending on the Compton wavelength of the fermions.

Rotational influence on fermions within negative curvature wormholes

In this research, we examine relativistic fermions within the rotating frame of negative curvature wormholes. Initially, as is typical in our context, we introduce the wormholes by embedding a curved surface into a higher-dimensional flat Minkowski spacetime. Subsequently, we derive the spacetime metric that characterizes the rotating frame of these wormholes. We then investigate analytical solutions of the generalized Dirac equation within this framework. Through exploring a second-order non-perturbative wave equation, we seek exact solutions for fermions within the rotating frame of hyperbolic and elliptic wormholes, also known as negative curvature wormholes. Our analysis provides closed-form energy expressions, and we generalize our findings to Weyl fermions. By considering the impact of the rotating frame and curvature radius of wormholes, we discuss how these factors affect the evolution of fermionic fields, offering valuable insights into their behavior.

Dirac fermions in a spinning conical Gödel-type spacetime

Abstract In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in (2+1)-dimensions, where we work with the Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels, where such levels are quantized in terms of the quantum numbers n and m j, and explicitly depends on the spin parameter s, spinorial parameter u, curvature and rotation parameters α and β, and on the vorticity parameter Ω. In particular, the quantization is a direct result of the existence of Ω (i.e., Ω acts as a kind of ``external field or potential''). We see that for m j>0, the energy levels do not depend on s and u; however, depend on n, m j, α, and β. In this case, α breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit 4Ωβ/α→1. Already for m j<0, we see that the energy levels depend on s, u and n; however, it no longer depends on m j, α and β. In this case, it is as if the fermion ``lives only in a flat Gödel-type spacetime''. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of Ω, α, and β for three different values of n.

Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

AbstractWe 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 , and 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 , while the spatial oscillations have a wavelength of , with a wave speed of . 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 , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.

Pauli-type coupling of spinors and curved spacetime

Abstract In this study we prove that the Pauli interaction---which is associated with a length parameter---emerges when the minimal coupling recipe is applied to the non-degenerate Dirac Lagrangian.
The conventional Dirac Lagrangian is rendered non-degenerate if supplemented by a particular term quadratic in the derivatives of the spinors.
For dimensional reasons, this non-degenerate Dirac Lagrangian is associated with a length parameter $\ell$.
It yields---just as the conventional degenerate Dirac Lagrangian---the standard free Dirac equation in Minkowski space as the $\ell$ terms cancel when setting up the Euler-Lagrange equations.
However, if the Dirac spinor is minimally coupled to gauge fields, then the length parameter~$\ell$ becomes a physical coupling constant yielding novel interactions.
For the U($1$) symmetry the Pauli coupling of fermions to electromagnetic fields arises, modifying the fermion's magnetic moment.
We discuss the impact of these findings on electrodynamics, and estimate the upper bound of the length parameter~$\ell$ from the yet existing discrepancy between the (SM) theory and measurement of the anomalous magnetic moment of light leptons.
This, and also recent studies of the renormalization theory, suggest that the Pauli coupling of leptons to the electromagnetic field is a necessary ingredient in QED, supporting the notion of a fundamental nature of the non-degenerate Dirac Lagrangian.

In a second step we then investigate how analogous ``Pauli-type'' couplings of gravity and matter arise if fermions are embedded in curved spacetime.
The diffeomorphism and Lorentz invariance of that system leads to an anomalous spin-torsion interaction and a curvature-dependent mass correction.
We calculate the mass correction in the De~Sitter geometry of vacuum with the cosmological constant $\Lambda$.
Possible implications for the existence of effective non-zero rest masses of neutrinos are addressed.
Finally, an outlook on the impact of mass correction on the physics of ``Big Bang'' cosmology, black holes, and of neutron stars is provided.

Non-compact gauge groups, tensor fields and Yang-Mills-Einstein amplitudes

Scattering amplitudes in Yang-Mills-Einstein theories have been investigated mostly for compact gauge groups. While non-compact gauge groups are not physically viable in Yang-Mills theory, non-compact gaugings feature prominently in the supergravity literature, where any choice of perturbative vacuum spontaneously breaks the gauge group to a compact subgroup. In this paper, we formulate double-copy constructions for several five-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supergravities with non-compact gauge groups. On one side of the double copy, we employ amplitudes from a super-Yang-Mills theory with a massive hypermultiplet. On the other, we use amplitudes from particular non-supersymmetric Yang-Mills-scalar theories with massive fermions, chosen to obey constraints coming from color/kinematics duality. Supergravities with massive self-dual tensors in five dimensions are also considered, showing that tensors are straightforwardly realized as double copies of gauge-theory fermions with suitable choices of signs in the corresponding solutions of the Dirac equation. We present several examples of these constructions, noting in particular the appearance of Heisenberg groups in the supergravity gauge symmetry and, in some cases, the possibility of exotic tensor-vector matter couplings.

Massive Dirac equation in static Bumblebee black hole space-times, detailed derivations and novel exact solutions

In this work, we construct and investigate the relativistic spin-12 fermionic fields quantum dynamics in static spherically symmetric Bumblebee black hole background. The derivation of the Dirac equation in a general static spherically symmetric black hole space-time is carried out in detail via tetrad formalism. With the help of total angular momentum operator, the angular equation can be separated from the radial part where the solution is given in terms of the spinor harmonics. The radial Dirac equation has a well-known problem due to the presence of the square root terms appearing simultaneously with the rest mass that prevents us to find exact solutions. In this work, we present exact solutions of light mass fermion's wave function and energy levels bound in the static Bumblebee black hole. We discover the exact solutions of the massive Dirac's radial equation in terms of the Confluent Heun functions. Moreover, thanks to the well-known polynomial condition of the Confluent Heun functions, we also derive the energy quantization.

Complex partition function of 1-dimensional Dirac particle confined in Woods-Saxon interaction

We present the bound state solutions of one-dimensional Schrödinger and Dirac equations for a particle confined in Woods-Saxon potential. In each case the wavefunctions and the energy eigenvalues have been obtained in terms of the Jacobi polynomials by using the Nikiforov-Uvarov method. The complex eigenenergy obtained from the Dirac equation is explicitly shown to contain the nonrelativistic energies plus complex relativistic correction terms The quantum partition functions obtained using the relativistic and the nonrelativistic energies have been used to generate important thermodynamic properties of the system in the high temperature limit. In particular, the complex partition function obtained in the relativistic regime is taken to indicate some important physical effects in the system.

Anisotropic effects in the nondipole relativistic photoionization of hydrogen

In the nonrelativistic and dipole regime of multiphoton ionization, spherical symmetry in all but the polarization direction of the laser pulse ensures that directional dependency in the photoelectron spectra is limited to the laser polarization direction, with the final distribution exhibiting no asymmetry along the propagation direction of the laser. When relativistic effects and spatial dependency in the external potential are accounted for however, the addition of time dilation and radiation pressure both impose anisotropic effects. Previously we have found that nondipole effects induce a redshift in the photoelectron energy distribution, while conversely relativistic effects induce a blueshift, with the net effect of an apparent near-cancellation of the two. In this work we study these effects further. By examining photoelectron momentum distributions acquired from simulations with the time-dependent Dirac equation we propose explanatory models for both phenomena and present a simplified model of the shifts as a function of the angle relative to the propagation direction of the laser pulse. It is found that both nondipole and relativistic effects must be accounted for on an equal footing in order to correctly describe the photoelectron momentum distribution in the high-intensity regime.

Schwarzschild deformed supergravity background: Possible geometry origin of fermion generations and mass hierarchy

The problem of fermion masses hierarchy in the Standard Model is considered on a toy model of a 10-dimensional space-time with a IIA supergravity type background. The Dirac equation written on this background, after compactification of extra four- and one-dimensional subspaces, gives the spectrum of Fermi fields which profiles in five dimensions and corresponding Higgs generated masses in four dimensions depend on the eigenvalues of Dirac operator on the named compact subspaces. Schwarzschild Euclidean deformation of the supergravity throat with the “apple-shaped” conical singularity permits to leave only three nondivergent angular modes interpreted as three generations of the down-type quarks. The resulting expressions for the quark masses are a geometry version of the Froggatt-Nielsen mechanism. Published by the American Physical Society 2024

Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave

The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.

Chirality-dependent topological edge states in photonic metacrystal.

Topological edge state, a unique mode for manipulating electromagnetic waves (EMs), has been extensively studied in both fundamental and applied physics. Up to now, the work on topological edge states has focused on manipulating linearly polarized waves. Here, we realize chirality-dependent topological edge states in one-dimensional photonic crystals (1DPCs) to manipulate circularly polarized waves. By introducing the magneto-electric coupling term (chirality), the degeneracy Dirac point (DP) is opened in PCs with symmetric unit cells. The topological properties of the upper and lower bands are different in the cases of left circularly polarized (LCP) and right circularly polarized (RCP) waves by calculating the Zak phase. Moreover, mapping explicitly 1D Maxwell's equations to the Dirac equation, we demonstrate that the introduction of chirality can lead to different topological properties of bandgaps for RCP and LCP waves. Based on this chirality-dependent topology, we can further realize chirality-dependent topological edge states in photonic heterostructures composed of two kinds of PCs. Finally, we propose a realistic structure for the chirality-dependent topological edge states by placing metallic helixes in host media. Our work provides a method for manipulating topological edge states for circularly polarized waves, which has a broad range of potential applications in designing optical devices including polarizers, filters, and sensors with robustness against disorder.

Solar neutrinos and leptonic spin forces

We quantify the effects of light spin-zero particles with pseudoscalar couplings to leptons and scalar couplings to nucleons on the evolution of solar neutrinos. In this scenario the matter potential sourced by the nucleons in the Sun’s matter gives rise to spin precession of the relativistic neutrino ensemble. As such the effects in the solar observables are different if neutrinos are Dirac or Majorana particles. For Dirac neutrinos the spin-flavour precession results into left-handed neutrino to right-handed neutrino (i.e., active-sterile) oscillations, while for Majorana neutrinos it results into left-handed neutrino to right-handed antineutrino (i.e., active-active) oscillations. In both cases this leads to distortions in the solar neutrino spectrum which we use to derive constraints on the allowed values of the mediator mass and couplings via a global analysis of the solar neutrino data. In addition for Majorana neutrinos spin-flavour precession results into a potentially observable flux of solar electron antineutrinos at the Earth which we quantify and constrain with the existing bounds from Borexino and KamLAND.

Pseudo-supersymmetric approach to the Dirac operator in the Schwarzschild spacetime

Abstract 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 ± ∞ .

Edge-dependent Majorana corner modes in an s-wave superconductor

In contrast to Majorana edge modes in first-order topological superconductors, Majorana corner modes are usually related to spatial symmetries and further rely on specific corner of the sample. In this paper, we study Majorana corner modes in an extended Kane-Mele model with arbitrary phase. Majorana corner mode exists at both corner between zigzag edge and armchair edge and corner between zigzag edge and zigzag edge for the phase at π/2. However, while the Majorana corner mode at the corner between zigzag edge and armchair edge remains as the phase deviates from π/2, Majorana mode at the corner between zigzag edge and armchair edge disappears. These Majorana corner modes are edge dependent and originate from the sign change of Dirac mass or Dirac velocity for edge states at adjacent edges tuned by the phase.

℘ -adic quantum mechanics, the Dirac equation, and the violation of Einstein causality

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.

Dirac Theory in Noncommutative Phase Spaces

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.

A conservative stochastic Dirac-Klein-Gordon system

Considered herein is a particular nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations. They are coupled by nonlinear terms due to the Yukawa interaction. We consider a case of homogeneous multiplicative noise that seems to be very natural from the perspective of the least action formalism. We are able to show existence and uniqueness of a corresponding Cauchy problem in Bourgain spaces. Moreover, the regarded model implies charge conservation, known for the deterministic analogue of the system, and this is used to prove a global existence result for suitable initial data.

Recent advances in relativistic excitation of atomic hydrogen

Within the framework of the relativistic regime ( E k > 2700 eV) and the first Born approximation, we investigate the inelastic excitation of atomic hydrogen by electron impact from 1 s to 2 p state. The incoming and outgoing electrons are described by free Dirac spinors, while the hydrogen atom target is described by the exact solutions of Dirac equation with a central potential. A detailed study has been devoted to the non-relativistic regime as well as the relativistic regime of the process, and the results obtained show the differences between the two regimes. These differences have revealed that the differential cross section (DCS) is influenced by the parameters of the 1 s and 2 p states, along with significant effects on diffusion during small pulse transfers. The relativistic effects are responsible for the significant changes in the DCS as a function of diffusion angles which depend on the chosen geometry.

Dirac mass induced by optical gain and loss.

Mass is commonly considered an intrinsic property of matter, but modern physics reveals particle masses to have complex origins1, such as the Higgs mechanism in high-energy physics2,3. In crystal lattices such as graphene, relativistic Dirac particles can exist as low-energy quasiparticles4 with masses imparted by lattice symmetry-breaking perturbations5-8. These mass-generating mechanisms all assume Hermiticity, or the conservation of energy in detail. Using a photonic synthetic lattice, we show experimentally that Dirac masses can be generated by means of non-Hermitian perturbations based on optical gain and loss. We then explore how the spacetime engineering of the gainandloss-induced Dirac mass affects the quasiparticles. As we show, the quasiparticles undergo Klein tunnelling at spatial boundaries, but a local breaking of a non-Hermitian symmetry can produce a new flux non-conservation effect at the domain walls. At a temporal boundary that abruptly flips the sign of the Dirac mass, we observe a variant of the time-reflection phenomenon: in the non-relativistic limit, the Dirac quasiparticle reverses its velocity, whereas in the relativistic limit, the original velocity is retained.