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

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