Self-consistent theory of Bose-condensed systems

In the theory of Bose-condensed systems, there exists the well known problem, the Hohenberg–Martin dilemma of conserving versus gapless approximations. This dilemma is analysed and it is shown that it arises because of the internal inconsistency of the standard grand ensemble, as applied to Bose systems with broken global gauge symmetry. A solution of the problem is proposed, based on the notion of representative statistical ensembles, taking into account all constraints imposed on the system. A general approach for constructing representative ensembles is formulated. Applying a representative ensemble to Bose-condensed systems results in a completely self-consistent theory, both conserving and gapless in any approximation.