Relativistic Representation Research Articles

Leptonic Decay Widths for the Composite System of Two Relativistic Fermions

The new relativistic semiclassical leptonic decay widths of vector mesons as the relativistic systems of two quarks with arbitrary masses interacting by means of the funnel-type potentials with Coulomb interaction are obtained. The behavior of the relativistic leptonic decay widths of vector mesons is investigated. Comparison of the behavior for new expression with its relativistic spinless analog is given. Consideration is conducted within the framework of completely covariant quasipotential approach in the Hamiltonian formulation of quantum field theory via a transition to the relativistic configurational representation in the case of two relativistic spin particles of arbitrary masses.

Form Factor of the Two Fermions Bound State: The Case of Equal Masses and Vector Currentl

New form factor components of two relativistic with equal masses fermions bound state in the case of a vector current are obtained. Consideration is performed within the framework of the relativistic quasipotential approach on the basis of covariant Hamiltonian formulation of quantum field theory by transition to three-dimensional relativistic configurational representation in the case of two relativistic particles with equal masses and spin 1/2.

Consistency Proof for Multi-time Schr\xf6dinger Equations with Particle Creation and Ultraviolet Cut-Off

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

Propagation of Lp estimates for the spatially homogeneous relativistic Boltzmann equation

In this paper, we prove the propagation of Lp upper bounds for the spatially homogeneous relativistic Boltzmann equation for any 1<p<∞. We consider the case of relativistic hard ball with Grad's angular cutoff. Our proof is based on a detailed study of the inter-relationship between the relative momenta, the regularity and the Lp estimates for the gain operator, the development of the relativistic Carleman representation, and several estimates on the relativistic hypersurface Ev′−vv⁎. We also derive a Pythagorean theorem for the relative momentag(v,v⁎), g(v,v′), and g(v′,v⁎), which has a crucial role in the reduction of the momentum singularity.

Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

Approximate analytical and numerical solutions of the three-dimensional Logunov–Tavkhelidze equation are found for the spherically symmetric case. Solutions are obtained in momentum and relativistic configuration representations. The wave functions in the relativistic configuration representation have additional zeroes compared to the wave functions of the nonrelativistic harmonic oscillator in the coordinate representation.

Road bowling in Ireland: social space and the context of context

ABSTRACTConsumer research has offered a multitude of understandings of space. While these insights have contributed both to absolute and relativistic appreciations, the discourse has tended more often towards absolute representations. Through an examination of Irish road bowling, built from a four-year ethnography, we position Henri Lefebvre’s triadic model of social space as a heuristic device that may be used to further relativistic representations of space. In doing so we expose how Irish road bowlers produce space on public roads. We find that such space and the actions of road bowlers within it are deeply influenced by both historic and contemporary socio-cultural discourses. In this way, we highlight how Lefebvre can be used to get at the context of context and offer an alternative understanding of normative and existential communitas.

Self-consistent tensor effects on nuclear matter systems within a relativistic Hartree-Fock approach

With the relativistic representation of the nuclear tensor force that is included automatically by the Fock diagrams, we explored the self-consistent tensor effects on the properties of a nuclear matter system. The analysis was performed within the density-dependent relativistic Hartree-Fock (DDRHF) theory. The tensor force is found to notably influence the saturation mechanism, the equation of state, and the symmetry energy of nuclear matter, as well as the neutron star properties. Without introducing any additional free parameters, the DDRHF approach is a natural way to reveal the tensor effects on the nuclear matter system.

The Sherman function in highly relativistic elastic electron–atom scattering

The Sherman function, which is a measure of the spin asymmetry in the elastic scattering of transversely polarized electrons from heavy targets, is calculated within the relativistic partial-wave representation. For collision energies above 40 MeV, oscillations of the Sherman function develop at the backward scattering angles which mirror the influence of the nuclear potential and which scale inversely with the nuclear size. We give predictions for 20–200 MeV electrons colliding with 64Zn, 208Pb and 238U. We propose the measurement of the diffraction structures in the angular distribution of the Sherman function for a 208Pb target at a beam energy near 80 MeV. The feasibility of such an experiment is addressed.

Relativistic chiral representation of the [formula omitted] scattering amplitude I: The Goldberger–Treiman relation

In this work, we study the πN scattering process within the Baryon Chiral Perturbation Theory framework in the covariant scheme of Extended-On-Mass-Shell (EOMS). We compare the description obtained in this scheme with the previously obtained using the Infrared Regularization scheme and show that EOMS accomplishes the best convergence, being able to extract from partial wave analyses reliable values of important quantities as the Goldberger–Treiman deviation. In regard to the latter, we solve the long-standing problem concerning to the extraction of the Goldberger–Treiman deviation with covariant ChPT that jeopardized the applicability of ChPT to the πN system. We also show the potential of the unitarization techniques applied to the perturbative calculation in the EOMS scheme, that allow us to increase the range of validity of our description up to ≈200 MeV in s.

Information transmittal, Newton’s law of gravitation, and tensor approach to general relativity

The special relativity considered in [A. Einstein, Zur Elektrodynamik der bewegte Körper. Ann. Physik, 17 (1905) 891–921] is based on the concept of finite speed of information transmittal by the available signals (rays of light). It is demonstrated that the same concept applies to Newton’s law of universal gravitation since the magnitude of distances between attracting masses can be physically defined (carried, accounted in acting forces of gravity) only by signals (physical processes) propagating at finite velocities. It follows that the speed of propagation of gravity is finite. The linear transformations of special relativity are applied to Newton’s law of gravitation to take into account the relativistic effects of information transmittal in a field of central forces of attraction. Relativistic representations of Newton’s law are obtained with respect to the center of gravity exposing illusory effects that appear at high velocities. It is verified that in atomic physics the effect of Newtonian gravitation on the motion of elementary particles at high velocities is negligible also in relativistic consideration. Computational methods are developed to measure the intensity of gravitation at a distant space–time location using a body that travels in space, emitting uniform pulses of light that are received by the observer at a different space–time location. It is demonstrated that the tensor approach to the general relativity and the united theory of space, time and gravitation in which the geometrical properties (metric) of the four-dimensional space–time continuum depend on the distribution of gravitating masses in space and their motion represent a transformed Lorentz invariant with a new type of inertia in the field of forces changing in space and time. Real physical processes evolve according to the forces represented in the tensor form by this invariant which is equivalent to the coordinate-free local invariant of relativistic dynamics that defines the field and the motion of a body whose velocities and accelerations can be measured by relativistic identification methods at a point, time and direction of interest. The results open new avenues for research in the general relativity and can be used for software development, field measurements and experimental studies in application to distant or fast moving systems.

Gravitational modulation of the optical length of long-baseline laser interferometers

The effect of quasistatic variations of the Earth’s gravitational field on the output signal of long-baseline gravitational-wave interferometers has been considered. The relativistic representation of the gravitational field in the form of a varying refractive index is used. An analytical calculation with numerical estimates of the tidal effect has been performed to explain the recent results observed on the LIGO interferometers.

Recent progress in the EOSTA model

The EOSTA model combines the STA and INFERNO models to calculate opacities and EOS on the same footing. A central point in this model is the quantum treatment of the plasma continuum giving rise to shape resonances that are essential for accurate EOS and opacity while avoiding unphysical irregularities. In this work we present the recent progress in EOSTA. For opacity calculations, we have connected the Local Density Approximation (LDA) potential, which includes the quantum continuum effect, with the STA optimized parametric potentials allowing for bound orbital relaxation. The results display regular behavior with respect to density and temperature variation and are in agreement with experiments. For EOS calculations we have advanced the relativistic virial theorem Lower Component Representation (LCR) method, utilizing the equivalence between the LCR pressures with the volume derivative of the free energy. The use of the LCR method enables handling low temperatures plasmas fluids and even solids in addition to high temperature and high density plasmas, warm dense matter, etc. Excellent agreement is obtained with experiments covering a wide range of densities and temperatures. The code is now used to generate EOS and opacity databases for the use of hydrodynamic simulations.

Quasi-exact solution of the problem of relativistic bound states in the (1+1)-dimensional case

We investigate the problem of bound states for bosons and fermions in the framework of the relativistic configurational representation with the kinetic part of the Hamiltonian containing purely imaginary finite shift operators e±ihd/dx instead of differential operators. For local (quasi)potentials of the type of a rectangular potential well in the (1+1)-dimensional case, we elaborate effective methods for solving the problem analytically that allow finding the spectrum and investigating the properties of wave functions in a wide parameter range. We show that the properties of these relativistic bound states differ essentially from those of the corresponding solutions of the Schrodinger and Dirac equations in a static external potential of the same form in a number of fundamental aspects both at the level of wave functions and of the energy spectrum structure. In particular, competition between ℏ and the potential parameters arises, as a result of which these distinctions are retained at low-lying levels in a sufficiently deep potential well for ℏ ≪ 1 and the boson and fermion energy spectra become identical.

Wave packet approach to the ionization of high-Z, hydrogen-like ions

The projectile ionization of high-Z, hydrogen-like ions is studied for relativistic collisions with low- and medium-Z target atoms. For such collisions, the ionization cross-sections at intermediate and high energies are calculated within the framework of first-order perturbation theory and by utilizing a wave packet approach to the continuum. Emphasis in this study is given in particular to the relativistic representation of the continuum and bound state wave functions which, until now, has seldom been used for the description of total ionization cross-sections in relativistic collisions. The results are compared with the more commonly applied semi- and ultra-relativistic approximations of Anholt and Becker [Phys. Rev. A 36 (1987) 4628] as well as with available experimental data.

Relativistic wave and Green's functions for hydrogen-like ions

The Greens library is presented which provides a set of C++ procedures for the computation of the (radial) Coulomb wave and Green's functions. Both, the nonrelativistic as well as relativistic representations of these functions are supported by the library. However, while the wave functions are implemented for all, the bound and free-electron states, the Green's functions are provided only for bound-state energies ( E<0). Apart from the Coulomb functions, moreover, the implementation of several special functions, such as the Kummer and Whittaker functions of the first and second kind, as well as a few utility procedures may help the user with the set-up and evaluation of matrix elements.

Relativistic field-theoretical representation of the Schrödinger equation and field-theoretical generalization of the inverse scattering method

The relativistic three-dimensional equation ( 2 E p ′ − 2 E p ) 〈 p ′ / Ψ p 〉 = − ∫ V ( t ) d 3 p ″ 〈 p ″ / Ψ p 〉 with E p = m 2 + p 2 and t = ( E p′ − E p″ ) 2−( p′− p″) 2 is transformed into Schrödinger equation (Δ r + k 2)〈 r|ϕ k 〉=− v( r)〈 r|ϕ k 〉 with E p = k 2/2 m+ m using the Fourier transformation. The potential v( r) and the wave function 〈 r|ϕ k 〉 differ from the corresponding Fourier images of the relativistic potential V( t) and the relativistic wave function 〈 p′| Ψ p 〉 with the kinematical factors only. In the framework of the standard field-theoretical approach the three-dimensional relativistic equations for the NN scattering amplitude with the total NN potential V tot = V( t)+ V nonl are derived. The nonlocal part of this NN potential V nonl is constructed from the πNN vertex functions which are determined from the πN phase shifts. The other part of the NN potential V( t) consists of the π-, σ-, ρ-, ω-, … meson exchange diagrams and for the NN contact (overlapping) terms. In the case of the renormalizable Lagrangians V( t) reproduces exactly the One Boson Exchange (OBE) Bonn model of the NN potential. The inverse scattering problem for this V( t) potential is reduced to the construction of the potential v( r) of the ordinary Schrödinger equation from the NN phase shifts.

Scalar Charged Particle in Weyl–Wigner–Moyal Phase Space. Constant Magnetic Field

A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables) is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with the peculiarities of the constraints on the initial conditions. An effective increase in coherence between eigenstates of the Hamiltonian is found and possibilities of its experimental observation are discussed.

Dispersion approach to quark-binding effects in weak decays of heavy mesons

Dispersion approach based on the constituent quark picture and its applications to weak decays of heavy mesons are reviewed. Meson interaction amplitudes are represented within this approach as relativistic spectral integrals over the mass variables in terms of the meson wave functions and spectral densities of the corresponding Feynman diagrams. Various applications of this approach are discussed: Relativistic spectral representations for meson elastic and transition form factors at spacelike momentum transfers are constructed. Form factors at $q^2>0$ are obtained by the analytical continuation. As a result of this procedure, form factors are given in the full $q^2$ range of the weak decay in terms of the wave functions of the participating mesons. The 1/m_Q expansion of the obtained spectral representations for the form factors for the particular limits of the heavy-to-heavy and heavy-to-light transitions are analysed. Their full consistency with the constraints provided by long-distance QCD for these limits is demonstrated. Predictions for form factors for B, B(s), D, and D(s) decays to light mesons are given. The B\to\gamma l\nu decay and the weak annihilation in rare radiative decays are considered. Nonfactorizable corrections to the B^0-\bar B^0 mixing are calculated. Inclusive weak B-decays are analysed and the differential distributions are obtained in terms of the B-meson wave function.

Relativistic two-particle one-dimensional scattering problem for superposition of delta-potentials

The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.

Lorentz covariant representation of the NN interaction at energies above 500 MeV

Using the formalism developed in the paper by Maxwell [Nucl. Phys. A 600 (1996) 509] a relativistic Love-Franey representation of the NN t-matrix has been obtained over the laboratory energy range from 500 to 800 MeV. The real and imaginary parts of this t-matrix are expanded separately in complete sets of on-shell Lorentz invariant amplitudes, which are represented by isoscalar and isovector “meson” exchanges. The masses associated with these exchanges are assumed to be independent of energy, while the coupling strengths and form factor masses are allowed to vary quadratically and linearly with energy, respectively. Direct and exchange contributions to the interaction are handled explicitly in the model. The model parameters are fit to the Summer 1997 Arndt amplitudes [Arndt and Roper, VPI and SU scattering analysis interactive dial-in program and data base (SAID)] and results given for various np and pp observables. The spin-averaged differential cross section and the vector asymmetry are generally reproduced quite well. The results for the other observables considered are not quite as good, especially at higher energy.