The quantum mechanical transform of the Bethe-Salpeter equation (original) (raw)

Transformation of the spinless Salpeter equation

arXiv: High Energy Physics - Phenomenology, 2019

Spinless Salpeter equation for two bound particles is analyzed. We use the fact that in relativistic kinematics the spatial two particle relative momentum is relativistic invariant. Free particle hypothesis for the bound state is developed: comstituents move as free particles inside of the system. The Shr\"odinger-type wave equation is derived. Three equivalent forms of the eigenvalue equation are given. Relative motion of quarks in eigen states is described by the asymptotic solution in the form of the standing wave of cos(kx+a)\cos(kx+a)cos(kx+a) for each spatial degree of freedom. To test the model the spin center-of-gravity energy levels for the hydrogen atom are calculated and compared with the NIST data. Complex eigenmasses for the HHH atom are obtained.

Facets of the spinless Salpeter equation

arXiv (Cornell University), 2004

The spinless Salpeter equation represents the simplest, and most straightforward, generalization of the Schrödinger equation of standard nonrelativistic quantum theory towards the inclusion of relativistic kinematics. Moreover, it can be also regarded as a well-defined approximation to the Bethe-Salpeter formalism for descriptions of bound states in relativistic quantum field theories. The corresponding Hamiltonian is, in contrast to all Schrödinger operators, a nonlocal operator. Because of the nonlocality, constructing analytical solutions for such kind of equation of motion proves difficult. In view of this, different sophisticated techniques have been developed in order to extract rigorous analytical information about these solutions. This review introduces some of these methods and compares their significance by application to interactions relevant in physics.

A Novel Approach in Solving the Spinor-Spinor Bethe-Salpeter Equation

2008

To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the non relativistic ones and to the results given by light front dynamics.

Solutions of the Bethe-Salpeter Equation in Minkowski Space: A Comparative Study

Few-Body Systems, 2014

The Bethe-Salpeter equation for a two-scalar, S-wave bound system, interacting through a massive scalar, is investigated within the ladder approximation. By assuming a Nakanishi integral representation of the Bethe-Salpeter amplitude, one can deduce new integral equations that can be solved and quantitatively studied, overcoming the analytic difficulties of the Minkowski space. Finally, it is shown that the Light-front distributions of the valence state, directly obtained from the Bethe-Salpeter amplitude, open an effective window for studying the two-body dynamics.

Schrödinger models for solutions of the Bethe–Salpeter equation in Minkowski space

Physical Review D, 2012

In view of the obstacles encountered in any attempts to solve the Minkowski-space Bethe-Salpeter equation for bound states of two fermions, we study the possibility to model the bound-state features, at least at a qualitative level, by a Schrödinger description. Such a nonrelativistic potential model can be constructed by applying, to any given Bethe-Salpeter spectral data, 'geometric spectral inversion' in its recently extended form, which tolerates also singular potentials. This leads to the adaptation of explicit models that provide an overview accounting for the Bethe-Salpeter formalism's complexities.

Instantaneous Bethe-Salpeter equation: Utmost analytic approach

Physical Review D, 2001

The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.

Quantitative studies of the homogeneous Bethe-Salpeter equation in Minkowski space

Physical Review D, 2014

The Bethe-Salpeter Equation for a bound system, composed by two massive scalars exchanging a massive scalar, is quantitatively investigated in ladder approximation, within the Nakanishi integral representation approach. For the S-wave case, numerical solutions with a form inspired by the Nakanishi integral representation, have been calculated. The needed Nakanishi weight functions have been evaluated by solving two different eigenequations, obtained directly from the Bethe-Salpeter equation applying the Light-Front projection technique. A remarkable agreement, in particular for the eigenvalues, has been achieved, numerically confirming that the Nakanishi uniqueness theorem for the weight functions, demonstrated in the context of the perturbative analysis of the multi-leg transition amplitudes and playing a basic role in suggesting one of the two adopted eigenequations, can be extended to a non perturbative realm. The detailed, quantitative studies are completed by presenting both probabilities and Light-Front momentum distributions for the valence component of the bound state.

On solving nonhomogeneous Bethe-Salpeter equations

Physics of Atomic Nuclei, 2005

We develop an advanced method of solving homogeneous and inhomogeneous Bethe-Salpeter equations by using the expansion over the complete set of 4-dimensional spherical harmonics. We solve Bethe-Salpeter equations for bound and scattering states of scalar and spinor particles for the case of one meson exchange kernels. Phase shifts calculated for the scalar model are in agreement with the previously published results. We discuss possible manifestations of separability for one meson exchange interaction kernels.

Bethe-salpeter equation with instantaneous harmonic oscillator exchange

Annals of Physics, 1980

The Bethe-Salpeter equation in the form due to Cung ef al. (Ann. Phys. (N. Y.) 98 (1976), 516) is investigated for the special case of instantaneous harmonic oscillator exchange. An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the 3(J f lb modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker's equation. This similarity to Whittaker's equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a %S, state in terms of the Bethe-Salpeter wave function is derived and applied to the &family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a 3O, state to couple directly to a virtual photon.