Massless states and negative mass states of the coupled electron-positron system with completely symmetric representation of the particles

Abstract

We argue that the free electron and positron can be considered as different, independent particles, each of which is characterized by the complete set of the Dirac plane waves. This completely symmetric representation of the particles makes it necessary to choose another solution of the Dirac equation for the free particle propagator as compared to that currently used in QED. The Bethe-Salpeter equation is studied in the ladder approximation with using these free propagators. A new branch of electron-positron bound states which represent the massless composite bosons, have been found for the actual coupling equal to the fine structure constant. We have obtained that: 1) the massless boson states have the normalized complex wave functions; 2) the average distance between the electron and positron diverges as the boson kinetic energy goes to zero; 3) the spatial contraction of the wave function of the transverse motion of strongly coupled electron-positron pair is continuously occurred with increasing the boson kinetic energy. Unlike the usual annihilation process in which nothing remains from the electron and positron, a similar annihilation-like process in which the reaction products are two or three gamma quanta and the massless boson, is predicted. In this symmetric representation one could expect states, which have a certain symmetry relative to Ps states, but have negative masses. For these states the equal-time bound-state equation was derived neglecting the interaction retardation and interaction through the vector potential. It turned out that the wave functions of the negative mass boson states are not normalized. Beyond these assumptions, the existence of these negative mass states remains unclear.