Dirac Equation Research Articles

Unruh effect using Doppler shift method in DSR framework

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

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

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

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

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

A sharp criterion for zero modes of the Dirac equation

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

Split octonionic Dirac equation

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

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

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

Singular solutions for Lane–Emden equations with CKN operator

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

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

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

On the “contact” hyperfine interaction

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

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

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

Heun-type solutions for the Dirac particle on the curved background of Minkowski space-times

Heun-type solutions for the Dirac particle on the curved background of Minkowski space-times

A relaxation viewpoint to Unbalanced Optimal Transport: Duality, optimality and Monge formulation

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.

Dyonium induced fermion number violation

Dyonium induced fermion number violation is studied in a SU(2)L×U(1)Y gauge theory with a doublet Higgs field. Dyonium is a generalization of Nambu's monopolium, having finite sized pair of dyon and anti-dyon, connected with a thin string under the linear plus Coulomb force.1This is a follow-up of the author's paper on monopolium induced neutrino mass (including the lepton and baryon number violations), studied with a SO(10) Grand Unified model in 1983. To fulfill the requirement from the chiral anomaly or its index theorem, an electric field should be excited in parallel to the dipole magnetic field. This crucial dynamical problem, not fully answered then, has been solved, by considering not monopolium but dyonium.The fermionic zero modes in the dyonium background fields are necessary in evaluating the transition rate of the fermion number violation processes. Their Dirac equation can be reduced to a single component partial differential equation, similar to the renormalization group equation, which may be useful to estimate the reaction rates.

Fermion-antifermion pairs in a magnetized space-time with non-zero cosmological constant

We analyze a fermion-antifermion pair within the (2+1)-dimensional Bonnor-Melvin magnetic (BMM) spacetime featuring a nonzero cosmological constant (Λ>0). To do this, we rigorously investigate analytical solutions to the corresponding form of a fully-covariant two-body Dirac equation. We derive a non-perturbative wave equation and obtain an exact closed-form analytical solution for the system under consideration. In this magnetic background spacetime, described by a surface with positive Gaussian curvature, we noticed that fermion-antifermion pairs display a collective behavior resembling that of a single quantum oscillator, carrying a total rest mass of the particles and oscillating with a frequency (ωo∝Λ), even in the absence of mutual interaction between the particles. This indicates a compelling correlation between the cosmological constant and the energy spectrum of fermion-antifermion systems, implying connections between quantum realms and cosmology.

Ionization energies of Cu- and Ni-like ions with Z ≤ 92

A two-stage technique is developed to interpolate and extrapolate the ionization energies of Cu- and Ni-like ions along Z with accuracy to the fifth significant digit. I) The first stage consists of scaling the known ionization energies (IE) along Z, which brings the IE(Z)-function to a quasi-straight line. Function straightening detects points where the function’s smoothness breaks down. They are fitted to the IE(Z) curve in the same way as it is usually done in some experiments. Scaling on successive overlapping segments allows a confident extrapolation to the region Z=92. II) The verification of the obtained results in stage I is performed using the parameter of the relativistic model potential of a Dirac equation. The results of techniques I and II agree well up to Z∼92.

νe → νe scattering with massive Dirac or Majorana neutrinos and general interactions

We calculate the neutrino-electron elastic scattering cross section, extending the results previously obtained in ref. [1], in the presence of generic new interactions that take into account all the effects caused by finite neutrino masses. We address the potential significance of a heavy neutrino sector during precision measurements, particularly for tau neutrinos scattering with masses in the MeV range, for which the existing upper bounds on |Uτ4|2 would result in conceivably measurable contributions. Finally, we comment on the possibility to distinguish between Dirac and Majorana neutrinos, including the analysis of the new emerging parameters and its application to illustrative model-dependent scenarios.

Sedenion Algebra Model as an Extension of the Standard Model and Its Link to SU(5)

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.

Positive-Energy Dirac Particles and Dark Matter

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.

Asymmetric transport for magnetic Dirac equations

Asymmetric transport for magnetic Dirac equations

Can quantum statistics help distinguish Dirac from Majorana neutrinos?

Finding out if neutrinos are Dirac or Majorana particles is known to be extremely difficult due to the smallness of neutrino mass and the fact that in the limit mν = 0 both Dirac and Majorana neutrinos become Weyl fermions, i.e. are indistinguishable. There have been suggestions in the literature that in the case of processes with production of a neutrino-antineutrino pair (if neutrinos are Dirac particles) or two neutrinos (if they are of Majorana nature) quantum statistics may be of help. This is because for Majorana neutrinos quantum indistinguishability of identical particles requires the amplitude of the process to be antisymmetrized with respect to the interchange of the final-state neutrinos, whereas no such antisymmetrization must be done for Dirac neutrinos. It has been claimed that the resulting differences between the cross sections for Dirac and Majorana neutrinos persist even for arbitrarily small but not exactly vanishing neutrino mass. We demonstrate that, at least in the framework of the Standard Model, this is not the case. We also give a general proof that within the Standard Model quantum statistics does not help tell Dirac and Majorana neutrinos apart in the limit of negligibly small mν/E.