On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.     

Space-Like Form of Bethe-Salpeter Equation: Scatter State

According to Dirac, and based on our space-like form of Bethe-Salpeter equation of bound state, we extend the definition of the space-like wavefunctions and derive out the spacelike form of Bethe-Salpeter equation for scatter state. The solution of the inhomogeneous term is also obtained in detail so as to clarify the significance of inhomogeneous space-like form of Bethe-Salpeter equation for scatter state. Moreover the translation method is finally discussed.     

Bethe-Salpeter Equation With Self- Energy Graphs Ⅱ——Spectral Analysis for Scalar BS Equation

Bethe-Salpeter equation in ladder approximation with self-energy graphs is diecussed for the model problem of two dftinguiahable equal-mass scalar particles exchanging a single maesive ecalar boson. The spectral features are analyzed.     

Diffuse waves in a random medium layer with rough boundaries

The problem of scattering from a random medium layer with rough boundaries is formulated as an integral equation in which the random fluctuations are represented as a zero-mean random operator. The analysis for the diffuse fields is based on the ladder-approximated Bethe-Salpeter equation. An integral equation for the diffuse intensities thus derived displays the various multiple-scattering processes involved in our problem. Transport equations are also derived and several special cases are considered to illustrate the characteristics of the results and to compare them with those in the literature.     

Space-Like Form of Bethe-Salpeter Equation: Many Particles

The Gell-Mann and Low's technology and Bethe-Salpeter equation for manyparticles are first discussed. Then the space-like form of Bethe-Salpeter equation involving both the bound state and scatter state are extended to many-particle case. Consequently we build the formalism of the space-like equation with explicit Lorentz covariant form and without the difficulty of ghost states automatically.     

相对论类空方程

根据Dirac类空自洽性条件的思想,通过引入类空因子定义了类空波函数.它的物理部分与Bethe-Salpeter波函数相一致.利用普适的相互作用核的重排技术,导出了对于束缚态和散射态的相对论类空方程,并且将它们推广到多粒子的情形.也得到了束缚态类空波函数的归一化条件和散射态类空方程的非齐次项的解.因此,建立起了相对论类空方程体系.     

Dynamical instability of the vacuum in the lagrangian formalism of the Bethe-Salpeter bound states

By using the auxiliary field method, the stability problem of the vacua corresponding to the various solutions of the Schwinger-Dyson equation is discussed. The appearance of a tachyon bound state in the Bethe-Salpeter equation is shown to lead to the instability of the vacuum. This instability is resolved by the condensation into the tachyon mode which means the transition to another solution of the SD equation. The critical coupling of this instability is given explicitly.     

Angular momentum analysis of a composite particle scattering problem and its cross channel analog

We investigate the solutions, by a determinental method, for a partial wave Bethe-Salpeter equation describing composite particle scattering and for its cross channel analog. We compare the behavior at s = ?∞ of the leading angular momentum singularity from the Bethe-Salpeter equation with that of the leading Regge singularity of the cross channel equation. We mention the effect of the general properties of the kernel of the Bethe-Salpeter equation on the angular momentum structure of the solution.     

Bound state spectrum from the light-cone two-body equation in φ3-theories

The relativistic bound state problem with the light-cone two-body equation (i.e. Bethe-Salpeter equation in the infinite momentum frame) is investigated in the light-cone ladder approximation to φ³-theories. A variational principle is found for the eigenvalues. Numerical results are compared with some existing calculations and with some new analytic results.     

Application of spectral representations to the nonrelativistic and the relativistic Bethe-Salpeter equation

The eigenvalue problem of the scalar Bethe-Salpeter equation is solved by application of the vertical Dyson representation. The method of solution is developed in complete analogy to the solution of Schrödinger's equation by a Stieltjes representation in the case of a Yukawa potential. The eigenvalues are zeros of a characteristic determinant, which can be understood as a generalization of the nonrelativistic Jost function.     

Numerical Method for Solving Two-Body Bound-State Bethe-Salpeter Equations for Unequal Constituent Masses

 A theoretical technique is developed for obtaining finite-energy numerical solutions to a class of two-body, bound-state Bethe-Salpeter equations in the ladder approximation when the constituent masses are unequal. The class of equations is restricted to those for which the Bethe-Salpeter equation can be written as a differential equation and to situations where the coupling constant is real. Such equations can result when the binding force is created by the exchange of a massless quanta. The theoretical technique is tested numerically by obtaining finite-energy solutions of the partially-separated Bethe-Salpeter equation describing the unequal-mass Wick-Cutkosky model in the ladder approximation. Received February 19, 1997; Revised April 2, 1998; accepted for publication October 30, 1998     

Bound-state,Bethe—Salpeter solutions with conjugate pairs of complex coupling constants

Solutions are obtained to the Bethe-Salpeter equation describing bound states of two massive scalars interacting via the exchange of a third, massive scalar. Covariance of the equation implies that the interaction is retarded, and in part because the energy appears more than once in the equation, a Hamiltonian for the bound state does not exist. Thus in contrast to the Schrodinger equation, the Bethe-Salpeter equation is solved by specifying the energy and solving for the coupling constant as an eigenvalue. Although the Bethe-Salpeter equation is derived from a Lagrangian with real coupling constants, depending on the value of the energy and the masses of the scalars, some values of the coupling constant that satisfy the Bethe-Salpeter equation are complex and always occur in conjugate pairs. The unexpected existence of solutions with real energy and a complex coupling constant raises the possibility that there are also resonance solutions with real values of the coupling constant and complex energy. Supported by a grant from the Ohio Supercomputer Center. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Stability of Covariant Relativistic Quantum Theory

In this paper we study the relativistic quantum-mechanical interpretation of the solution of the inhomogeneous Euclidean Bethe-Salpeter equation. Our goal is to determine conditions on the input to the Euclidean Bethe-Salpeter equation so the solution can be used to construct a model Hilbert space and a dynamical unitary representation of the Poincaré group. We prove three theorems that relate the stability of this construction to properties of the kernel and driving term of the Bethe-Salpeter equation. The most interesting result is that the positivity of the Hilbert space norm in the non-interacting theory is not stable with respect to Euclidean covariant perturbations defined by Bethe-Salpeter kernels. The long-term goal of this work is to understand which model Euclidean Green functions preserve the underlying relativistic quantum theory of the original field theory. Understanding the constraints imposed on the Green functions by the existence of an underlying relativistic quantum theory is an important consideration for formulating field-theory motivated relativistic quantum models.This work supported in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286     

Solutions of a bethe-salpeter equation

We study systematically and exhibit the Lorentz symmetry of the Bethe-Salpeter equation in the light-like case for two scalar quarks of different masses interacting via the exchange of a scalar photon. We develop a new approach for solving the eigenvalue problem of the equation for the general case (not only the light-like). Our method permits accurate analytic expressions for the spectrum and the wave functions.     

Bound states of two surface phonons at a crystal surface with an adsorbed monolayer

We obtain the Bethe-Salpeter equation for the Green's function describing the propagation of two surface phonons interacting via cubic and quartic anharmonicity. We solve that equation by keeping only the latter contribution to its kernel. The two-dimensional nature of the problem is reflected in the existence of a bound state of two high-frequency surface phonons associated with the presence of an overlayer of light atoms, for all values of the coupling strength. This is illustrated with the detailed solution of a simple model.     

Relation between Bethe-Salpeter and Schrödinger wave functions for many-particel systems

The connection is made between a many-time approach to S-matrix elements and energy eigenvalues, which naturally arises from a field theoretical point of view, and the single time Schrödinger- and Breit-like formalism often used in detailed calculations for many-particle systems, such as many-electron atoms. Specifically, the many-particle Bethe-Salpeter equation is expressed in terms of the corresponding Schrödinger equation for the non-relativistic case in which the Bethe-Salpeter kernel consists of only two-particle local static interactions. Also, the one-photon transition matrix element for this case in the Bethe-Salpeter formalism is shown to be equivalent to the corresponding well-known Schrödinger result. The treatment developed is well suited to systematic relativistic generalization.     

Parametrization of realistic Bethe-Salpeter amplitude for the deuteron

The parametrization of the realistic Bethe-Salpeter amplitude for the deuteron is given. Eight components of the amplitude in the Euclidean space are presented as an analytical fit to the numerical solution of the Bethe-Salpeter equation in the ladder approximation. The applicability of the parametrization to the observables of the deuteron is briefly discussed.     

Solving the Bethe-Salpeter equation for positronium

We propose a Coulomb-like kernel for the relativistic two-fermion Bethe-Salpeter equation, to be used as the lowest-order approximation in systematic perturbative calculation of bound-state energy levels in QED. The kernel is symmetric in the two fermions and for the exchange of in and out momenta. The resulting equation is exactly soluble, unlike previously considered unperturbed kernels. We give explicitly the Green function and eigen-functions. We also discuss the problem of the behaviour of the wave functions at zero relative coordinate in connection with the contribution to energy levels from the one-photon annihilation channel in QED.     

The relativistic two- and three-quark problems

For theqqq problem, the corresponding covariant-instantaneity-ansatz dynamics is formulated through the introduction of new Lorentz-invariant momenta, which again are closely parallel to the older null-plane-ansatz treatment for the same on a two-tier basis, but facilitate a derivation of the 3-dimensional Bethe-Salpeter equation in a more general form for unequal-mass kinematics than seem to have been attempted previously.