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 collision-less regime, which is both conserving and gap-less, 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 the exchange interaction 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 form of the density autocorrelation function and the Green's function is exhibited and discussed.