Gapless kinetic theory beyond the Popov approximation
  We present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body $T$ matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. 
J. Wachter, R. Walser, J. Cooper, and M. Holland