Non-relativistic Case Research Articles

Does relativistic motion always degrade quantum Fisher information?

We investigate the ultimate estimation precision, characterized by the quantum Fisher information, of a two-level atom as a detector which is coupled to massless scalar field in the Minkowski vacuum. It has been shown that for an inertial detector moving with a constant velocity, its quantum Fisher information is completely unaffected by the velocity, however, it still decays over time due to the decoherence caused by the interaction between the atom and the field. In addition, for a uniformly accelerated detector ($w=0$) moving along spatially straight line, the accelerated motion will reduce the quantum Fisher information in the estimation of state parameters. However, when the detector trajectory is generated by a combination of the linear accelerated motion and a component of the four-velocity $w=dy/d\tau$, we find quite unlike the previous results that, for the non-relativistic case $(w\ll1)$, the acceleration could degrade the quantum Fisher information, while the four-velocity component will suppress the degradation of the quantum Fisher information, and thus could enhance the precision of parameters estimation. Furthermore, in the case for ultra-relativistic velocities $(w\rightarrow\infty)$, although the detector still interacts with the environment, it behaves as if it were a closed system as a consequence of relativity correction associated to the velocity, and the quantum Fisher information in this case can be shield from the effect of the external environment, and thus from the relativistic motion.

High precision methods for solving a system of cold plasma equations taking into account electron–ion collisions

Abstract High precision simulation algorithms are proposed and justified for modelling cold plasma oscillations taking into account electron–ion collisions in the non-relativistic case. The specific feature of the approach is the use of Lagrangian variables for approximate solution of the problem formulated initially in Eulerian variables. High accuracy is achieved both through the use of analytical solutions on trajectories of particles and due to sufficient smoothness of the solution in numerical integration of Cauchy problems. Numerical experiments clearly illustrate the obtained theoretical results. As a practical application, a simulation of the well-known breaking effect of multi-period relativistic oscillations is carried out. It is shown that with an increase in the collision coefficient one can observe that the breaking process slows down until it is completely eliminated.

On the reformulation of the Thomas–Fermi model to make it compatible with the Planck-scale

Thomas–Fermi model is considered here to make it cogent to capture the Planck-scale effect with the use of a generalization of uncertainty relation. Here generalization contains both linear and quadratic terms of momentum. We first reformulate the Thomas–Fermi model for the non-relativistic case. We have shown that it can also be reformulated for taking into account the relativistic effect. We study the dialectic screening for both the non-relativistic and relativistic cases and find out the Fermi length for both the cases explicitly.

Relativistic Corrections to Phase Shift Calculation in the GNXAS Package

Modern XAFS (X-ray Absorption Fine Structure) data-analysis is based on accurate multiple-scattering (MS) calculations of the X-ray absorption cross-section. In this paper, we present the inclusion and test of relativistic corrections for the multiple-scattering calculations within the GnXAS suite of programs, which is relevant to the treatment of the XAFS signals when atoms with high atomic number are contained into the system. We present a suitable strategy for introducing relativistic corrections without altering the basic structure of the programs. In particular, this is realized by modifying only the Phagen program calculating the atomic absorption cross sections and scattering t-matrices for the selected cluster. The modification incorporates a pseudo-Schrödinger Equation (SE) replacing the Dirac relativistic form. The phase-shift calculations have been put to a test in two known molecular and crystalline cases: molecular bromine Br2 and crystalline Pb. Calculations in an extended energy range have been shown to be very close to the non-relativistic case for Br2 (Br K-edge) while corrections have been found to exceed 25% for amplitude and phases of the XAFS multiple-scattering signals (Pb L3-edge). Benefits in the structural refinement using relativistic corrections are discussed for crystalline Pb at room temperature.

Convergence rate towards the fractional Hartree equation with singular potentials in higher Sobolev trace norms

In this work, we show convergence of the [Formula: see text]-particle bosonic Schrödinger equation towards the Hartree equation. Hereby, we extend the results of [I. Anapolitanos and M. Hott, A simple proof of convergence to the Hartree dynamics in Sobolev trace norms, J. Math. Phys. 57(12) (2016) 122108; I. Anapolitanos, M. Hott and D. Hundertmark, Derivation of the Hartree equation for compound Bose gases in the mean field limit, Rev. Math. Phys. 29(07) (2017) 1750022]. We first consider the semi-relativistic Hartree equation in the defocusing and the focusing cases. We show that Pickl’s projection method [P. Pickl, Derivation of the time dependent Gross–Pitaevskii equation without positivity condition on the interaction, J. Statist. Phys. 140(1) (2010) 76–89; P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97(2) (2011) 151–164; P. Pickl, Derivation of the time dependent Gross–Pitaevskii equation with external fields, Rev. Math. Phys. 27(1) (2015) 1550003], can be adapted to this problem. Next, we extend this result to the case of fractional Hartree equations with potentials that are more singular than the Coulomb potential. Finally, in the non-relativistic case, we derive the Hartree equation assuming only [Formula: see text] initial data for potentials with a quantitative bound on the convergence rate.

Cosmology of strongly interacting fermions in the early universe

In view of growing interest in long range scalar forces in the early universe to generate primordial black holes, we study in detail the general relativistic formulation of a Fermi gas interacting with a scalar field in cosmology. Motivated by long range forces, we mainly focus on the cosmological massless limit of the scalar field. In this limit, our main finding is that the Yukawa interaction leads to a solution where the scalar field oscillates around zero fermion mass and all energy densities decay as radiation. On one hand, we show that if the Fermi gas starts relativistic, it could stay relativistic. On the other hand, if the fermions are initially non-relativistic, they remain non-relativistic for all practical purposes. We find that in both cases the energy density of the fermions and the scalar field decays as radiation. In the non-relativistic case, this is due to an oscillating and decaying effective mass. Such background dynamics questions whether there is a substantial enhancement of the fermion density fluctuations in the non-relativistic case. Our work can be easily extended to more general field dependent fermion mass and to general scalar field potentials. The analysis of the cosmological perturbations will be presented in a follow-up work.

Geometrizing non-relativistic bilinear deformations

We define three fundamental solvable bilinear deformations for any massive non-relativistic 2d quantum field theory (QFT). They include the mathrm{T}overline{mathrm{T}} deformation and the recently introduced hard rod deformation. We show that all three deformations can be interpreted as coupling the non-relativistic QFT to a specific Newton-Cartan geometry, similar to the Jackiw-Teitelboim-like gravity in the relativistic case. Using the gravity formulations, we derive closed-form deformed classical Lagrangians of the Schrödinger model with a generic potential. We also extend the dynamical change of coordinate interpretation to the non-relativistic case for all three deformations. The dynamical coordinates are then used to derive the deformed classical Lagrangians and deformed quantum S-matrices.

TB2J: A python package for computing magnetic interaction parameters

We present TB2J, a Python package for the automatic computation of magnetic interactions, including exchange and Dzyaloshinskii–Moriya, between atoms of magnetic crystals from the results of density functional calculations. The program is based on the Green’s function method with the local rigid spin rotation treated as a perturbation. As input, the package uses the output of either Wannier90, which is interfaced with many density functional theory packages, or of codes based on localized orbitals. One of the main interests of the code is that it requires only one first-principles electronic structure calculation in the non-relativistic case (or three in the relativistic case) and from the primitive cell only to obtain the magnetic interactions up to long distances, instead of first-principles calculations of many different magnetic configurations and large supercells. The output of TB2J can be used directly for the adiabatic magnon band structure and spin dynamics calculations. A minimal user input is needed, which allows for easy integration into high-throughput workflows. Program summaryProgram Title: TB2JCPC Library link to program files:https://doi.org/10.17632/dm45fcn69d.1Developer’s repository link:https://github.com/mailhexu/TB2JCode Ocean capsule:https://codeocean.com/capsule/6486145Licensing provisions: BSD 2-clauseProgramming language: PythonNature of problem: TB2J is a package for the computing of parameters in the extended Heisenberg model of the magnetic interaction, including the isotropic exchange, anisotropic exchange and Dzyaloshinskii–Moriya interactions from first principles result. It can make use of the Wannier function Hamiltonian, which can be constructed from many first principles codes, or localized orbital based codes.Solution method: It uses the magnetic force theorem and takes the rigid spin rotation as a perturbation to the electronic structure. The energy variation is calculated from the Green’s functions from tight-binding like Hamiltonian based on Wannier functions or localized orbitals.Additional comments including restrictions and unusual features: Isotropic exchange, anisotropic exchange, and Dzyaloshinskii–Moriya interactions can all be computed with the input of many DFT codes through the interface of Wannier90, or directly from localized orbital codes. The magnetic interaction parameters up to any distance can be computed from one DFT calculation. A minimum user-input is required which provides a black-box like experience. It generates output for several spin dynamics codes and thus bridges the first principles electronic structure simulation with the large scale spin dynamics simulation.

A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations

We introduce a new family of paraxial asymptotic models that approximate the Vlasov-Maxwell equations in non-relativistic cases. This formulation is $n$-th order accurate in a parameter $\eta$, which denotes the ratio between the characteristic velocity of the beam and the speed of light. This family of models is interesting, first because it is simpler than the complete Vlasov-Maxwell equation, then because it allows us to choose the model complexity according to the expected accuracy.

Non-Extensive Thermodynamics Effects in the Cosmology of f(T) Gravity

Using f(T) gravitational theory, we construct modified cosmological models via the first law of thermodynamics by using the non-extensive thermodynamics framework, the effects of which are captured by the parameter δ. The resulting cosmological equations are modified compared to the standard Einstein-Hilbert ones, with the modifications coming from the f(T) gravitational theory and from the non-extensive parameter which quantifies the non-extensive thermodynamics effects quantified by the parameter δ, which when is set equal to unity, one recovers the field equations of f(T) gravity. We study in detail the cosmological evolution of the model in the presence of collisionless non-relativistic matter case, and we derive the exact forms of the dark energy density parameter and of the dark energy equation of state parameter, from which we impose constraints on the non-extensive thermodynamics parameter, δ, by using the Planck 2018 data on cosmological parameters. Accordingly, we repeat our calculations after including the relativistic matter along with the non-relativistic one, and we derive the new forms of the dark energy density parameter and of the dark energy equation of state parameter. Our study shows that the inclusion of non-extensive thermodynamic effects, quantified by the parameter δ, for a flat Friedmann-Robertson-Walker Universe, has measurable differences compared with the normal thermodynamics case. We confront our results with Type Ia supernovae observations for z≥0.4 and we obtain reasonably agreement with the observational data.

Strong Lagrangian Solutions of the (Relativistic) Vlasov--Poisson System for NonSmooth, Spherically Symmetric Data

We prove a local existence and uniqueness result for the non-relativistic and relativistic Vlasov-Poisson system for data which need not even be continuous. The corresponding solutions preserve all the standard conserved quantities and are constant along their pointwise defined characteristic flow so that these solutions are suitable for the stability analysis of not necessarily smooth steady states. They satisfy the well-known continuation criterion and are global in the non-relativistic case. The only unwanted requirement on the data is that they be spherically symmetric.

Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low-energy regime

I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (non-relativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the non-relativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.

Caloric curves of self-gravitating fermions in general relativity

We study the nature of phase transitions between gaseous and condensed states in the self-gravitating Fermi gas at finite temperature in general relativity. The condensed states can represent compact objects such as white dwarfs, neutron stars, or dark matter fermion balls. The caloric curves depend on two parameters: the system size R and the particle number N. When N < NOV, where NOV is the Oppenheimer–Volkoff limit, there exists an equilibrium state for any value of the temperature T and energy E as in the nonrelativistic case [P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)]. Gravitational collapse is prevented by quantum mechanics (Pauli’s exclusion principle). When N > NOV, there is no equilibrium state below a critical energy and below a critical temperature. In that case, the system is expected to collapse toward a black hole. We plot the caloric curves of the general relativistic Fermi gas, study the different types of phase transitions that occur in the system, and determine the phase diagram in the (R, N) plane. The nonrelativistic results are recovered for N ≪ NOV and R ≫ ROV with NR3 fixed. The classical (non quantum) results are recovered for N ≫ NOV and R ≫ ROV with N∕R fixed. We discuss the commutation of the limits c → +∞ and ℏ → 0. We study the relativistic corrections to the nonrelativistic caloric curves and the quantum corrections to the classical caloric curves. We highlight a situation of physical interest where a self-gravitating Fermi gas, by cooling, first undergoes a phase transition toward a compact object (white dwarf, neutron star, dark matter fermion ball), then collapses into a black hole. This situation occurs in the microcanonical ensemble when NOV < N < 3.73 NOV. We also relate the phase transitions from a gaseous state to a core-halo state in the microcanonical ensemble to the onset of red-giant structure and to the supernova phenomenon.

Derivation of Classical Mechanics in an Energetic Framework via Conservation and Relativity

The notions of conservation and relativity lie at the heart of classical mechanics, and were critical to its early development. However, in Newton’s theory of mechanics, these symmetry principles were eclipsed by domain-specific laws. In view of the importance of symmetry principles in elucidating the structure of physical theories, it is natural to ask to what extent conservation and relativity determine the structure of mechanics. In this paper, we address this question by deriving classical mechanics—both nonrelativistic and relativistic—using relativity and conservation as the primary guiding principles. The derivation proceeds in three distinct steps. First, conservation and relativity are used to derive the asymptotically conserved quantities of motion. Second, in order that energy and momentum be continuously conserved, the mechanical system is embedded in a larger energetic framework containing a massless component that is capable of bearing energy (as well as momentum in the relativistic case). Imposition of conservation and relativity then results, in the nonrelativistic case, in the conservation of mass and in the frame-invariance of massless energy; and, in the relativistic case, in the rules for transforming massless energy and momentum between frames. Third, a force framework for handling continuously interacting particles is established, wherein Newton’s second law is derived on the basis of relativity and a staccato model of motion-change. Finally, in light of the derivation, we elucidate the structure of mechanics by classifying the principles and assumptions that have been employed according to their explanatory role, distinguishing between symmetry principles and other types of principles (such as compositional principles) that are needed to build up the theoretical edifice.

Simplified calculations of plasma oscillations with non-extensive statistics

We use the exponential parametrization of the nonextensive distribution to calculate the dielectric constant in an electron gas obeying the nonextensive statistics. As we show, the exponential parametrization allows us to make such calculations in a straightforward way, bypassing the use of intricate formulas obtained from integral tables and/or numerical methods. For illustrative purposes, we apply first the method to the calculation of the permittivity and the corresponding dispersion relation in the ultrarelativistic limit of the electron gas, and verify that it reproduces in a simple way the results that had been obtained previously by other authors using the standard parametrization of the nonextensive distribution. In the same spirit we revisit the calculation of the same quantities for a non-relativistic gas, in the high frequency limit, which has been previously carried out, first by Lima, Silva and Santos, and subsequently revised by Chen and Li. Our own results agree with those obtained by Chen and Li. For completeness, we also apply the method the low frequency limit in the non-relativistic case, which has been previously considered by Dai, Chen and Li in the context of the stream plasma instability. We discuss some features of the results obtained in each case and their interpretation of terms of generalized nonextensive quantities, such as the Debye length $\lambda_{D}^{(q)}$, the plasma frequency $\omega_{p}^{(q)}$ and the ultra-relativistic frequency $\Omega^{(q)}_{e,rel}$. In the limit $q \rightarrow 1$ such quantities reduce to their classical value and the classical result of the dispersion relations are reproduced.

Bound Electron Transitions under the Influence of Electromagnetic Wave in Constant Magnetic Field

Motion and radiative transitions of an electron in a magnetic field under the influence of an external electromagnetic wave are studied for various confining conditions in semiconductor, graphene, in quantum wells, and relativistic generalization in terms of the Klein–Gordon equation are considered. In particular, the following problems are discussed. The so-called cyclotron resonance, which may appear in graphene, is studied with indication for appearance of the so-called frequency-halving. The problem is solved for two-dimensional massless charged particle, whose gapless nature is protected by sublattice symmetry. The exact classical calculation of this effect is undertaken in the framework of a 2D classical equation for a zero-mass electron. We also find an exact solution of the Schrödinger equation for charge carriers in semiconductors under the influence of an external magnetic field and in the field of electromagnetic wave with an account for their radiative transitions. Solutions of the relativistic Klein–Gordon equation in this configuration of electromagnetic fields are found as a certain generalization of the results obtained for the non-relativistic case. These results may serve as a first step for further efforts to find exact solutions of wave equations for quasiparticles in solid state structures in external fields.

Expression of Relativistic Expectation Values of Powers of r in Terms of Clebsch–Gordan Coefficients

It is shown that, in the relativistic hydrogenic approximation, expectation values of the type $\langle n\ell j|r^k|n\ell j\rangle$ can be expressed in terms of Clebsch-Gordan coefficients or $3jm$ symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1-5 (2015); Opt. Spektrosk. 118, 3-7] in the non-relativistic hydrogenic case.

Mass–radius relation and dynamical stability of strongly magnetized white dwarfs in anisotropic configuration using Bowers and Liang model

This paper is concerned with the effect of anisotropic pressure in the mass–radius relation and dynamical stability for strongly magnetized white dwarfs using Bowers and Liang model. Strongly magnetized white dwarfs is hot topic where it is believed that it can be a progenitor for some peculiar type Ia supernovae that was observed recently. In order to get the equation of state we assumed that the internal magnetic field strength of the star is constant and strong enough to oblige the electrons to occupy the first Landau level. We dealt with the problem by using both relativistic and non-relativistic equilibrium equations. In non-relativistic formalism, the mass–radius relation of the star was investigated and it is found that the dynamical stability of the star does not depend on the radius but on the mass of the star. In relativistic formalism, it is found that the radius of the star decreases in comparison with non-relativistic case. Also the maximum mass limit of the star does not exceed the non-relativistic one in isotropic case. Unfortunately, in both cases the predicted internal magnetic field of the star always exceeds the magnetic field strength needed for the stability. This leads to weaken the stability of the star and making the star deviates from the spherical configuration to oblate spheroidal shape. Thus according to this model the magnetic white dwarfs configurations are unstable and unbound.

Ion acoustic solitary and shock waves in strongly coupled degenerate plasma with relativistic effects

The ion acoustic solitary and shock waves are studied in strongly coupled nonrelativistic and relativistic plasma. The wave profile has been discussed for the kinetic and hydrodynamic regimes. The ions are considered to be strongly coupled, and electrons as degenerated and relativistic to deal with nonlinear waves using continuity and Poisson’s equations together with generalized hydrodynamical (GH) and quantum hydrodynamical (QH) models. The reductive perturbation method is used to formulate Korteweg–de Vries (KdV) and Korteweg–de Vries Burgers (KdVB) equations in both nondegenerate and degenerated cases. The effects of relativistic, degeneracy parameter and longitudinal viscosity coefficient on the profile of nonlinear waves are discussed. The amplitude and width of a shock in both nonrelativistic and ultrarelativistic cases increase with an increase in the viscosity coefficient, while with an increase in the electron diffraction parameter, the amplitude and width of the shock wave increase, and for a solitary wave decreases in both the nonrelativistic and ultra-relativistic cases. It is also shown that solitary ion acoustic wave propagates with more energy in nonrelativistic plasma than ultrarelativistic. The results of the work will be useful, for example, for astrophysics to understand the process of wave propagation in dense astrophysical bodies.

Generalized Uncertainty Principle in Astrophysics from Fermi Statistical Physics Arguments

The Heisenberg’s uncertainty principle is one of the most important concepts in quantum mechanics. One of its important outcomes is that enormously high momentum is required to look at extremely small length-scales. Nevertheless, quantum gravity and Grand Unified models taught us that a minimal observable length cutoff which is of the order of the Planck length is required. Such an important feature suggests that the conventional uncertainty principle must be generalized and accordingly the Heisenberg’s uncertainty principle is replaced by a generalized uncertainty relation. Although the effects of the generalized uncertainty relation can only be found in the context of extremely high energies/short distances and not at any scale currently probed in the laboratory, we discuss in this study its impacts on white dwarfs using the quantum mechanical statistical properties of a degenerate Fermi gas in D-spatial dimensions. Both the non-relativistic and extreme relativistic cases were discussed. It was observed that the effect of the generalized uncertainty relation is to shrink the radius of proton, a phenomenon which was observed recently using high precision laser spectroscopy of atomic hydrogen. Several points were discussed and a number of quantum statistical properties were derived.