Gapless Hartree-Fock-Bogoliubov Approximation for Bose Gases

A dilute Bose system with Bose-Einstein condensate is considered. It is shown that the Hartree-Fock-Bogolubov approximation can be made both conserving as well as gapless. This is achieved by taking into account all physical normalization conditions, that is, the normalization condition for the condensed particles and that for the total number of particles. Two Lagrange multipliers, introduced for preserving these normalization conditions, make the consideration completely self-consistent.