Exact Treatment of the Bound State Problems in the Non-Relativistic Quantum Field Theory

Abstract

An exact equation for the bound state of nucleons in the static neutral scalar model is obtained by using the field equation for Heisenberg operator and the method of canonical transformaticn. The formulation of the eigenvalue problem is completed by the analyticity consideration. In particular, a boundary condition at -r=O ( -i-r being the "negative imaginary" relative time) turns out that the derivative of the "wave function" with respect to -r must be zero at -r=O. The abnormal wlutions exist in this case as well as in the ladder approximation, but there is difference in nature between them for the zero meson mass case. The appearance of abnormal solutions is mainly due to the structure of the fundamental equation itself. This is demonstrated by handling a first order equation for the static neutral scalar model and the Bethe-Salpeter equation for the Lee model, in both cases of which the abnormal solutions do not appear.

A new method for decoupling Bogolyubov’s chains for quantum models

“…In Section 4 we study the interaction between two nucleons in the model by Yukawa. It has turned out that our method readily leads to an exact result which reproduces those obtained by N. Mugibayashi [12] with the help of a much more laborious method (using Feynman diagrams, see [13]).…”

“…One way to solve this problem is our method of approximate pair correlations which enables one to solve the problem exactly. It is convenient to base ourselves on paper [12] where an exact equation for the transition amplitude is obtained. The main aim of this section is to test once again the quality of our approximation on various models.…”

Bethe-Salpeter equation: 3D reductions, heavy mass limits and abnormal solutions

“…Wick suggested that these solutions could be an artefact due to the ladder approximation, and would disappear if the higher-order irreducible graphs could be included [3]. Mugibayashi showed however that the abnormal solutions remain present in the exact static model [7], which is simple enough to be completely solved. Recently, the conclusions of Mugibayashi were questioned by Jallouli and Sazdjian [8], so that the road to Wick's interpretation is open again.…”

There are no abnormal solutions of the Bethe - Salpeter equation in the static model

“…In this connection the exact static model was analyzed by Mugibayashi [9]. Using hamiltonian formalism he derived a second order differential equation that he identified with the Bethe−Salpeter equation.…”