Abstract
Two particles interacting by photon exchange, form the bound states predicted by the non-relativistic Schrödinger equation with the Coulomb potential (Balmer series). More than 60 years ago, in the solutions of relativistic Bethe-Salpeter equation, in addition to the Balmer series, were found another series of energy levels. These new series, appearing when the fine structure constant α is large enough (α > π/4), are not predicted by the Schrödinger equation. However, this new (non-Balmer) states can hardly exist in nature, since in order to create a strong e.m. field with α > π/4 a point-like charge Z > 107 is needed. The nuclei having this charge, though exist starting with bohrium, are far from to be point-like. In the present paper, we analyze the more realistic case of a strong interaction created by exchange of a massive particle. It turns out that in the framework of the Bethe-Salpeter equation this interaction still generates a series of new relativistic states, which are similar to those of the massless exchange case, and which are absent in the Schrödinger equation. The properties of these solutions are studied. Their existence in nature seems possible.
Highlights
The Bethe-Salpeter (BS) equation [1] is a relativistic counterpart of the Schrödinger equation
This homogeneous equation has another infinite spectrum Mnk, distinct from the ordinary relativistic generalization of the Balmer series, corresponding to bound states gnk with binding energies
The aim of the present research is to answer the question: can the abnormal states exist in the nature or not? In the case of the massless exchange considered by Wick and Cutkosky and sketched above, the answer seems to be negative
Summary
The Bethe-Salpeter (BS) equation [1] is a relativistic counterpart of the Schrödinger equation. This homogeneous equation has another infinite spectrum Mnk, distinct from the ordinary relativistic generalization of the Balmer series, corresponding to bound states gnk with binding energies. In addition to the Balmer series, the BS equation predicts for each n, the existence of another series of states with binding energies Bnk given – in the limit B/m
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