Nonrelativistic Limit Research Articles

Non-relativistic ten-dimensional minimal supergravity

We construct a non-relativistic limit of ten-dimensional mathcal{N} = 1 supergravity from the point of view of the symmetries, the action, and the equations of motion. This limit can only be realized in a supersymmetric way provided we impose by hand a set of geometric constraints, invariant under all the symmetries of the non-relativistic theory, that define a so-called ‘self-dual’ Dilatation-invariant String Newton-Cartan geometry. The non-relativistic action exhibits three emerging symmetries: one local scale symmetry and two local conformal supersymmetries. Due to these emerging symmetries the Poisson equation for the Newton potential and two partner fermionic equations do not follow from a variation of the non-relativistic action but, instead, are obtained by a supersymmetry variation of the other equations of motion that do follow from a variation of the non-relativistic action. We shortly discuss the inclusion of the Yang-Mills sector that would lead to a non-relativistic heterotic supergravity action.

Bound state solutions of the hyper-radial Klein-Gordon equation under the Deng-Fan potential by WKB and SWKB methods

The energy levels of the Klein–Gordon equation in hyper-radial space under the Deng-Fan potential energy function are studied by the SWKB and WKB approximation methods. We obtained the analytic solution of the energy spectra and the ground state wave function in closed form. Furthermore, we obtained the energy equation corresponding to the Schrodinger equation by invoking the non-relativistic limit. The variations of the non-relativistic N-dimensional energy spectra with the potential parameters and radial quantum number are investigated. The energy levels are degenerate for and increase with the dimensionality number. The ground state wave function and its gradient are continuous at the boundary Our results for the energy spectra are in excellent agreement with the ones obtained by other analytical methods where similar centrifugal approximations were applied.

Three-boson stability for boosted interactions towards the zero-range limit

We study the three-boson bound-state mass and wave functions for ground and excited states within the three-body relativistic framework with Kamada and Glöcke boosted potentials in the limit of a zero-range interaction. We adopt a nonrelativistic short-range separable potential, with Yamaguchi and Gaussian form factors, and drive them towards the zero-range limit by letting the form factors' momentum scales go to large values while keeping the two-body binding fixed. We show that the three-boson relativistic masses and wave functions are model-independent towards the zero-range limit, and the Thomas collapse is avoided, while the nonrelativistic limit kept the Efimov effect. Furthermore, the stability in the zero-range limit is a result of the reduction of boosted potential with the increase of the virtual pair center of mass momentum within the three-boson system. Finally, we compare the present results with Light-Front and Euclidean calculations.

Is there any Nambu monopolium out there?

Magnetic monopoles have been a subject of study for more than a century since the first ideas by A. Vaschy and P. Curie, circa 1890. In 1974, Y. Nambu proposed a model for magnetic monopoles exploring a parallelism between the broken symmetry Higgs and the superconductivity Ginzburg-Landau theories in order to describe the pions quark-antiquark confinement states. There, Nambu described an energetic string where its end points behave like two magnetic monopoles with opposite magnetic charges —quark and antiquark. Consequently, not only the interaction among monopole and antimonopole, mediated by a massive vector boson (Yukawa potential), but also the energetic string (linear potential) contribute to the effective interaction potential. We propose here a monopole-antimonopole non-confining attractive interaction of the Nambu type, and then investigate the formation of bound states, the monopolium. Some necessary conditions for the existence of bound states to be fulfilled by the proposed Nambu-type potential, Kato weakness, Setô and Bargmann conditions, are verified. In the following, ground-state energies are estimated for a variety of monopolium reduced mass, from 102 MeV to 102 TeV, and Compton interaction lengths, from am to pm, where discussion about non-relativistic and relativistic limits validation is carried out. Dedicated to the memory of Dr. César Augusto Parga Rodrigues (1939–2021).

Quantum kinetic equation for fluids of spin-1/2 fermions

Fluid of spin-1/2 fermions is represented by a complex scalar field and a four-vector field coupled both to the scalar and the Dirac fields. We present the underlying action and show that the resulting equations of motion are identical to the (hydrodynamic) Euler equations in the presence of Coriolis force. As a consequence of the gauge invariances of this action we established the quantum kinetic equation which takes account of noninertial properties of the fluid in the presence of electromagnetic fields. The equations of the field components of Wigner function in Clifford algebra basis are employed to construct new semiclassical covariant kinetic equations of the vector and axial-vector field components for massless as well as massive fermions. Nonrelativistic limit of the chiral kinetic equation is studied and shown that it generates a novel three-dimensional transport theory which does not depend on spatial variables explicitly and possesses a Coriolis force term. We demonstrated that the three-dimensional chiral transport equations are consistent with the chiral anomaly. For massive fermions the three-dimensional kinetic transport theory generated by the new covariant kinetic equations is established in small mass limit. It possesses the Coriolis force and the massless limit can be obtained directly.

Electromagnetic quantum shifts in relativistic Bose-Einstein condensation

We compute deviations from ideal gas behavior of the pressure, density, and Bose-Einstein condensation temperature of a relativistic gas of charged scalar bosons caused by the current-current interaction induced by electromagnetic quantum fluctuations treated via scalar quantum electrodynamics. We obtain expressions for those quantities in the ultra-relativistic and nonrelativistic limits, and present numerical results for the relativistic case.

Molecular energies of a modified and deformed exponential-type potential model

The solutions of a one-dimensional relativistic Klein-Gordon equation for a modified and deformed exponential-type potential are obtained using supersymmetric approach and parametric Nikiforov-Uvarov method respectively. The numerical values for lithium molecule were obtained using the energy equations from the two methods. The non-relativistic limit of the relativistic Klein-Gordn equation was obtained for each of the energy equation using a popular transformation invoked on the solution of the Klein-Gordon equation. Numerical values for lithium and sodium molecules were also generated based on the two methods. The generated values for both relativistic and non-relativistic were compared with the experimental values. These results agreed perfectly with the experimental values of the studied molecules.

The large scale structure bootstrap: perturbation theory and bias expansion from symmetries

We investigate the role played by symmetries in the perturbative expansion of the large-scale structure. In particular, we establish which of the coefficients of the perturbation theory kernels are dictated by symmetries and which not. Up to third order in perturbations, for the dark matter density contrast (and for the dark matter velocity) only three coefficients are not fixed by symmetries and depend on the particular cosmology. For generic biased tracers, where number/mass and momentum conservation cannot be imposed in general, this number rises to seven in agreement with other bias expansions discussed in the literature. A crucial role in our analysis is provided by extended Galilean invariance, which follows from diffeomorphism invariance in the non-relativistic limit. We identify a full hierarchy of extended Galilean invariance constraints, which fix the analytic structure of the perturbation theory kernels as the sums of an increasing number of external momenta vanish.Our approach is especially relevant for non-standard models that respect the same symmetries as ΛCDM and where perturbation theory at higher orders has not been exhaustively explored, such as dark energy and modified gravity scenarios.In this context, our results can be used to systematically extend the bias expansion to higher orders and set up model independent analyses.

A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry

We consider a non-relativistic limit of the bosonic sector of eleven-dimensional supergravity, leading to a theory based on a covariant ‘membrane Newton-Cartan’ (MNC) geometry. The local tangent space is split into three ‘longitudinal’ and eight ‘transverse’ directions, related only by Galilean rather than Lorentzian symmetries. This generalises the ten-dimensional stringy Newton-Cartan (SNC) theory. In order to obtain a finite limit, the field strength of the eleven-dimensional four-form is required to obey a transverse self-duality constraint, ultimately due to the presence of the Chern-Simons term in eleven dimensions. The finite action then gives a set of equations that is invariant under longitudinal and transverse rotations, Galilean boosts and local dilatations. We supplement these equations with an extra Poisson equation, coming from the subleading action. Reduction along a longitudinal direction gives the known SNC theory with the addition of RR gauge fields, while reducing along a transverse direction yields a new non-relativistic theory associated to D2 branes. We further show that the MNC theory can be embedded in the U-duality symmetric formulation of exceptional field theory, demonstrating that it shares the same exceptional Lie algebraic symmetries as the relativistic supergravity, and providing an alternative derivation of the extra Poisson equation.

Propagatory dynamics of nucleus-acoustic waves excited in gyrogravitating degenerate quantum plasmas electrostatically confined in curved geometry

A theoretic model to investigate the dynamics of the longitudinal nucleus-acoustic waves (NAWs) in gyrogravitating electrostatically confined degenerate quantum plasma (DQP) system in spherically symmetric geometry is constructed. The model setup consists of non-degenerate heavy nuclear species (HNS), lighter nuclear species (LNS), and quantum degenerate electronic species (DES). It specifically considers the influences of the Bohm potential, Coriolis rotation, viscoelasticity, and electrostatic confinement pressure (ECP, scaling quadratically in density). A standard normal spherical mode analysis gives a generalized dispersion relation (septic). It highlights the dependency of various atypical instability response on the equilibrium plasma parameters. A numerical illustrative platform portrays that the relative nuclear charge-to-mass coupling parameter (beta) acts as a destabilizing agency and the heavy-to-light nuclear charge density ratio (mu) acts as a stabilizing agency in both the non-relativistic (NR) and ultra-relativistic (UR) limits. Another interesting conjuncture is that the Coriolis rotation introduces a destabilizing influence on the system in both the limits. The progressive analysis presented herein has correlations and consistencies in the dynamic growth backdrop of various compact astro objects and their circumvent atmospheres, such as white dwarfs, neutron stars, etc.

Relativistic electron-impact ionization of hydrogen atom from its metastable 2S-state in the symmetric/asymmetric coplanar geometries

In this paper, we present a detailed analytical computation of the triple differential cross sections, in the first Born approximation, for the relativistic electron-impact ionization of hydrogen atom in the metastable 2S-state in the symmetric and asymmetric coplanar geometries. The process is investigated by using the relativistic Dirac-formalism where the effects of spin and relativity are taken into account. It is shown that the nonrelativistic limit is accurately reproduced when applying low incident kinetic energies. At high energies, relativistic and spin effects are found to significantly affect the triple differential cross sections. The numerical results obtained for the triple differential cross section are compared with some other theoretical results in the nonrelativistic regime for asymmetric coplanar geometry. For this particular process and in the absence of any experimental data and theoretical models at high energies for the triple differential cross sections, the results of our calculated relativistic total cross section are compared with available theoretical and experimental results. We hope that the present study will provide significant contribution to future experiments.

Current-conserving relativistic linear response for collisional plasmas

We investigate the response of a relativistic plasma to electromagnetic fields in the framework of the Boltzmann equation incorporating a collision term in the relaxation rate approximation selected in a form assuring current conservation. We obtain an explicit solution for the linearized perturbation of the Fermi–Dirac equilibrium distribution in terms of the average relaxation rate κ. We study the resulting covariant, gauge invariant, and current conserving form of the polarization tensor in the ultrarelativistic and non-relativistic limits. We evaluate the susceptibility in the ultrarelativistic limit and explore their dependence on κ. Finally, we study the dispersion relations for the longitudinal and transverse poles of the propagator. We show that for κ>2ωp, where ωp is the plasma frequency, the plasma wave modes are overdamped. In the opposite case, κ≪ωp, the propagating plasma modes are weakly damped.

Angular anisotropy parameters for photoionization delays

Anisotropy parameters describing the angular dependence of the photoionization delay are defined. The formalism is applied to results obtained with the relativistic random phase approximation with exchange for photoionization delay from the outermost $s$-orbitals in selected rare-gas atoms. Any angular dependence in the Wigner delay is induced here by relativistic effects, while the measurable atomic delay exhibits such a dependence also in the nonrelativistic limit. The contributions to the anisotropy from the different sources are disentangled and discussed. For the heavier rare gases, it is shown that measurements of the delay for electrons ejected in specific angles, relative to, e.g., those ejected along the laser polarization, are directly related here to the Wigner delay. For a considerable range of angles, the contributions from the second photon largely get canceled when the results in different angles are compared, and this angle-relative atomic delay is then close to the corresponding Wigner delay.

Direct detection of pseudo-Nambu-Goldstone dark matter with light mediator

It has been found that a pseudo-Nambu-Goldstone boson dark matter suppresses the amplitude for elastic scattering with nuclei in non-relativistic limit, and thus can naturally evade the strong constraint of dark matter direct detection experiments. In this paper, we show that non-zero elastic scattering cross section can be induced if the mediator mass is as small as momentum transfer. The predicted recoil energy spectrum can differ from that for usual thermal dark matter. Together with the relevant constraints such as thermal relic abundance, indirect detection and Higgs decays, we investigate the detectability through the current and future dark matter direct detection experiments.

Effective theory of nuclear scattering for a WIMP of arbitrary spin

We introduce a systematic approach to characterize the most general non-relativistic WIMP-nucleus interaction allowed by Galilean invariance for a WIMP of arbitrary spin $j_\chi$ in the approximation of one-nucleon currents. Five nucleon currents arise from the nonrelativistic limit of the free nucleon Dirac bilinears. Our procedure consists in (1) organizing the WIMP currents according to the rank of the $2 j_\chi+1$ irreducible operator products of up to $2 j_\chi$ WIMP spin vectors, and (2) coupling each of the WIMP currents to each of the five nucleon currents. The transferred momentum $q$ appears to a power fixed by rotational invariance. For a WIMP of spin $j_\chi$ we find a basis of 4+20$j_\chi$ independent operators that exhaust all the possible operators that drive elastic WIMP-nucleus scattering in the approximation of one-nucleon currents. By comparing our operator basis, which is complete, to the operators already introduced in the literature we show that some of the latter for $j_\chi=1$ were not independent and some were missing. We provide explicit formulas for the squared scattering amplitudes in terms of the nuclear response functions, which are available in the literature for most of the targets used in WIMP direct detection experiments.

Relativistic quantum bouncing particles in a homogeneous gravitational field

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein–Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schrödinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein–Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.

Interaction of ion acoustic solitons for Zakharov Kuznetsov equation in relativistically degenerate quantum magnetoplasmas

Nonlinear electrostatic waves on the ion time scale have been studied in a quantum magnetoplasma in the presence of relativistically degenerate electrons. In this regard, Zakharov Kuznetsov (ZK) equation has been derived using the reductive perturbation technique. The single and two soliton solutions of the ZK equation have been obtained by employing Hirota formalism. The nonrelativistic and ultrarelativistic limits of the relativistically degenerate electrons have been studied and discussed in detail for important plasma parameters both for the single and two soliton solutions of the ZK equation. We have applied our study for the plasma parameters that are found in the white dwarfs. It has been noticed that the interaction of the ZK solitons depends on the propagation vectors in the predominant direction of propagation. Importantly, it has been found that the spatial and temporal scales over which the interaction of solitons occur are different for the nonrelativistic and ultrarelativistic cases.

Capture of Massless and Massive Particles by Parameterized Black Holes

We study an influence of the leading coefficient of the parameterized line element of the spherically symmetric, static black hole on the capture of massless and massive particles. We have shown that negative (positive) values of ϵ decreases (increases) the radius of characteristic circular orbits and consequently, increases (decreases) the energy and decreases (increases) the angular momentum of the particle moving along these orbits. Moreover, we have calculated and compared the capture cross section of the massive particle in the relativistic and non-relativistic limits. It has been shown that in the case of small deviation from general relativity the capture cross section for the relativistic and nonrelativistic particle has an additional term being linear in the small dimensionless deviation parameter ϵ.

Non-relativistic limits and stability of composite wave patterns to the relativistic Euler equations

We are concerned with the structural stability of composite wave patterns to the Cauchy problem for the relativistic full Euler equations consisting of a large 1-shock, a large 2-contact discontinuity, and a large 3-shock. When the initial data are bounded but possibly large total variations, approximate solutions have been constructed via the wave front tracking scheme, a weighted Glimm functional has been introduced and its monotonicity has been proved on the basis of the local wave interaction estimates, and then the global stability of these wave patterns has been established. Moreover, the non-relativistic limits of such solutions can be obtained as the light speed c → +∞.

Relativistic Extended Thermodynamics of Polyatomic Gases with Rotational and Vibrational Modes

In a recent article an infinite set of balance equations has been proposed to modelize polyatomic gases with rotational and vibrational modes in the non-relativistic context. To obtain particular cases, it has been truncated to obtain a model with 7 or 15 moments. Here the following objectives are pursued: 1) to obtain the relativistic counterpart of this model which, at the non-relativistic limit, gives the same balance equations as in the known classical case; 2) to obtain the previous result for the model with an arbitrary but fixed number of moments, 3) to obtain the closure of the resulting relativistic model so that all the functions appearing in the balance equations are expressed in terms of the independent variables. To achieve these goals, the following methods are used: 1) The Entropy Principle is imposed. As a result is obtained that the closure is determined up to a single 4-vectorial function usually called 4-potential. 2) To determine this last function, a more restrictive principle is imposed, namely the Maximum Entropy Principle (MEP). 3) Since all the functions involved must be expressed in the covariant form, so as not to depend on the observer, the Representation Theorems are used. Findings of this article are exactly the goals outlined earlier. They are clearly novelty because they had never been achieved before. They can be considered also improvements because, if the aforementioned arbitrary number of moments is restricted to 16, the present work coincide with that already known in literature. Doi: 10.28991/HIJ-2021-02-03-04 Full Text: PDF