Conserving and gapless model of the weakly interacting Bose gas

The dielectric formalism is used to set up an approximate description of a spatially homogeneous weakly interacting Bose gas in the collisionless regime, which is both conserving and gapless, and has coinciding poles of the single-particle Green's function and the density autocorrelation function in the Bose-condensed regime. The approximation takes into account the direct and exchange interactions in a consistent way. The fulfillment of the generalized Ward identities related to the conservation of particle number and the breaking of the gauge symmetry is demonstrated. The dynamics at long wavelengths is considered in detail below and above the phase transition, numerically and in certain limits also in analytical approximations. The explicit forms of the density autocorrelation function and the Green's function are exhibited and discussed.