  It is known that in the ladder approximation the relativistic two-fermion
bound-state equation of Bethe and Salpeter has solutions corresponding to the
binding energy equal to the total mass of the particles. The study of these
massless states has been carried out only for the bound system at rest. Of
course, such composite boson can not be in the state of rest. But it is more
importantly that this approach for the massless boson can not be interpreted as
the limiting case of a nonzero mass system because the phase velocity of the
boson wave must equal to the speed of light. Using the Bethe-Salpeter equation
in the ladder approximation, we have obtained the wave equation for the
massless bound states of two fermions with equal masses and the electromagnetic
interaction between them. Neglecting retardation of the interaction, solutions
corresponded to the stationary wave states of the composite boson, have been
found. The boson wave function can be represented as an infinite, straight
thread, the transverse radius of which is more than the Compton wavelength of
the fermion. Two energy branches of the bosons with quantized energies have
been determined. The appearance of these branches is due to the strong
renormalization of the fine structure constant for the massless states.
