Relativistic inverse scattering problem for a superposition of a nonlocal separable and a local quasipotential

Within the relativistic quasipotential approach to quantum field theory, the relativistic inverse scattering problem is solved for the case where the total quasipotential describing the interaction of two relativistic spinless particles having different masses is a superposition of a nonlocal separable and a local quasipotential. It is assumed that the local component of the total quasipotential is known and that there exist bound states in this local component. It is shown that the nonlocal separable component of the total interaction can be reconstructed provided that the local component, an increment of the phase shift, and the energies of bound states are known.     

Solving a relativistic quasipotential equation for a superposition of a nonlocal separable and a local quasipotential

Within the relativistic quasipotential approach to quantum field theory, a method is developed for solving a finite-difference quasipotential equation for the case where a total quasipotential describing the interaction of two relativistic spinless particles of unequal masses is a superposition of a nonlocal separable and a local quasipotential. The cases are investigated where the local component of the total interaction—it is assumed to be known—either admits or does not admit the existence of bound states. This makes it possible to obtain an exact expression for the increment of the phase shift, to determine the conditions of the existence of bound states, and to give a generalization of the Levinson theorem.     

Relativistic bound-state equations for fermions with instantaneous interactions

In thispaper three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary spin, parity, and charge parity are derived. For the special case of spinless states bound by interactions with a Coulomb-type potential the properties of the ground-state solutions of the three equations are investigated both analytically and numerically. The coupling-constant spectrum turns out to depend strongly on the spinor structure of the fermion interaction. If the latter is chosen such that the nonrelativistic limits of the equations coincide, an analogous spectrum is found for the instantaneous Bethe-Salpeter and the quasi-potential equations, whereas the two-particle Dirac equation yields qualitatively different results.     

Exact flow equation for bound states

We develop a formalism to describe the formation of bound states in quantum field theory using an exact renormalization group flow equation. As a concrete example we investigate a nonrelativistic field theory with instantaneous interaction where the flow equations can be solved exactly. However, the formalism is more general and can be applied to relativistic field theories, as well. We also discuss expansion schemes that can be used to find approximate solutions of the flow equations including the essential momentum dependence.     

Relativistic studies of the decay constants of S wave and P wave mesons

Relativistic corrections are important in hadronic physics since even for the heavy hadrons there are sizable relativistic corrections. Therefore one should use a relativistic model to describe the higher excited states. This note summarizes our predictions for the decay constants of the S wave and P wave heavy mesons by means of the instantaneous relativistic Bethe Salpeter equation （Salpeter equation）.     

Relativistic studies of the decay constants of S wave and P wave mesons

Relativistic corrections are important in hadronic physics since even for the heavy hadrons there are sizable relativistic corrections. Therefore one should use a relativistic model to describe the higher excited states. This note summarizes our predictions for the decay constants of the S wave and P wave heavy mesons by means of the instantaneous relativistic Bethe Salpeter equation (Salpeter equation).     

Einstein-Podolsky-Rosen constraints on quantum action at a distance: The Sutherland paradox

Assuming that future experiments confirm Aspect's discovery of nonlocal interactions between quantum pairs of correlated particles, we analyze the constraints imposed by the EPR reasoning on the said interactions. It is then shown that the nonlocal relativistic quantum potential approach plainly satisfies the Einstein causality criteria as well as the energy-momentum conservation in individual microprocesses. Furthermore, this approach bypasses a new causal paradox for timelike separated EPR measurements deduced by Sutherland in the frame of an approach by means of space-time zigzags with advanced potentials. It is finally demonstrated that this inherent quantum causal direct interaction establishes permanent EPR correlations which are always restricted to spacelike separations and are instantaneous only in the center-of-mass rest frame of the two-particle system.     

Nonlocality in Relativistic Dynamics

Recent experiments have renewed interest in nonlocal interpretations of quantum mechanics. The experimental observation of the violation of Bell's inequalities implies the existence of nonlocality. Bohm expressed the nonlocal connection between quantum particles through the wave function and the quantum potential. This paper shows that a similar connection exists in a relativistic dynamical theory known as parametrized relativistic quantum theory (PRQT). We present an introduction to PRQT, derive the quantum potential for a system of relativistic scalar particles, and discuss alternative interpretations of nonlocality.     

Static observables of relativistic three-fermion systems with instantaneous interactions

We show that static properties like the charge radius and the magnetic moment of relativistic three-fermion bound states with instantaneous interactions can be formulated as expectation values with respect to intrinsically defined wave functions. The resulting operators can be given a natural physical interpretation in accordance with relativistic covariance. We also indicate how the formalism may be generalized to arbitrary moments. The method is applied to the computation of static baryon properties with numerical results for the nucleon charge radii and the baryon octet magnetic moments. In addition, we make predictions for the magnetic moments of some selected nucleon resonances and discuss the decomposition of the nucleon magnetic moments in contributions of spin and angular momentum, as well as the evolution of these contributions with decreasing quark mass.     

A relativistic formulation of the Einstein-Podolsky-Rosen paradox

The Einstein-Podolsky-Rosen (EPR) paradox and the correlated states it introduced comprise one of the central interpretive problems of quantum mechanics. Because of the apparent nonlocal character of this paradox, it should be given a relativistic treatment. The purpose of this paper is to provide such a treatment.     

Variational wave equations of two fermions interacting via scalar,pseudoscalar, vector,pseudovector and tensor fields

We consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.     

The Dirac-Oscillator Green's Function

We obtain the two-point Green's function for the relativistic Dirac oscillator problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a map of the latter into the former. The relativistic bound states energy spectrum is obtained by locating the energy poles of this Green's function in a simple and straightforward manner.     

Instantaneous Bethe-Salpeter Equation and Its Exact Solution

We present an approach to solve Bethe-Salpeter (BS) equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and well-determined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium, the exact solutions precisely present corrections to those of the corresponding Schrödinger equation in order v¹ (v is the relative velocity) for eigenfunctions, in order v² for eigenvalues, and the mixing between S and D components in J^PC=1^-- states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the O(v) corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states.     

Bound states of spatial optical dark solitons in nonlocal media

It is shown that multiple dark solitons can form bound states in a series of balance distances in nonlocal bulk media. Dark solitons can either attract or repel each other depending on their separated distance. The stability of such bound states are studied numerically. There exist unstable degenerate bound states decaying in different ways and having different lifetimes.     

Lagrangian approach for dark soliton in nonlocal nonlinear media

We theoretically investigate the dynamics of dark solitons as well as their interaction in nonlocal media. Approximate equations describing the evolution of the beams are obtained via suitable trial functions of amplitude u and refractive index n in an averaged Lagrangian. Our results reveal that out-of-phase dark solitons can evolve into stable bound states in nonlocal materials. Moreover, it is found that the separations in the bound state monotonically increase with the degree of nonlocality in nonlocal limit. These results are in excellent agreement with the numerical simulations.     

Instantaneous Formulation for Transitions Between Two Instantaneous Bound States and Its Gauge Invariance

We have precisely derived a ＂rigorous instantaneous formulation＂ for transitions between two bound states when the bound states are well-described by instantaneous Bethe-Salpeter （BS） equation （i.e. the kernel of the equation is instantaneous ＂occasionally＂）. The obtained rigorous instantaneous formulation, in fact, is expressed as an operator sandwiched by two ＂reduced BS wave functions＂ properly, while the reduced BS wave functions appearing in the formulation are the rigorous solutions of the instantaneous BS equation, and they may relate to Schroedinger wave functions straightforwardly. We also show that the rigorous instantaneous formulation is gauge-invariant with respect to the Uem（1） transformation precisely, if the concerned transitions are radiative. Some applications of the formulation are outlined.     

Gray spatial solitons in nonlocal nonlinear media

We study gray solitons in nonlocal nonlinear media and show that they are stable and can form bound states. We reveal that the gray soliton velocity depends on the nonlocality degree and that it can be drastically reduced in highly nonlocal media. This is in contrast with the case of local media, where the maximal velocity is dictated solely by the asymptotic soliton amplitude.     

Relativistic Nonlocal Quantum Field Theory

A theory is defined to be relativistic if its Hamiltonian, total momenta, and boost's generators satisfy commutation relations of the Poincaré group. Field theories with usual local interactions are known to be relativistic. A simple example of a relativistic nonlocal theory is found. However, it has divergences. Some conditions are obtained which are necessary in order that a nonlocal theory be relativistic and divergenceless.