  Bose-condensed systems with broken global gauge symmetry are considered. The
description of these systems, as has been shown by Hohenberg and Martin,
possesses an internal inconsistency, resulting in either nonconserving theories
or yielding an unphysical gap in the spectrum. The general notion of
representative statistical ensembles is formulated for arbitrary statistical
systems, equilibrium or not. The principal idea of this notion is the necessity
of taking into account all imposed conditions that uniquely define the given
statistical system. Employing such a representative ensemble for Bose-condensed
systems removes all paradoxes, yielding a completely self-consistent theory,
both conserving and gapless in any approximation. This is illustrated for an
equilibrium uniform Bose system.
