  The possibility of pion--pair formation in a hot pion gas, based on the
bosonic gap equation, is pointed out and discussed in detail. The critical
temperature for condensation of pion pairs (Evans--Rashid transition) is
determined as a function of the pion density. As for fermions, this phase
transition is signaled by the appearance of a pole in the two--particle
propagator. In bose systems there exists a second, lower critical temperature,
associated with the appearance of the single--particle condensate. Between the
two critical temperatures the pion dispersion relation changes from the usual
quasiparticle dispersion to a Bogoliubov--like dispersion relation at low
momenta. This generalizes the non-relativistic result for an attractive bose
gas by Evans et al. Possible consequences for the inclusive pion spectra
measured in heavy--ion collisions at ultra--relativistic energies are
discussed.
