  For general quantum systems the power expansion of the Gibbs potential and
consequently the power expansion of the self energy is derived in terms of the
interaction strength. Employing a generalization of the projector technique a
compact representation of the general terms of the expansion results. The
general aspects of the approach are discussed with special emphasis on the
effects characteristic for quantum systems. The expansion is systematic and
leads directly to contributions beyond mean-field of all thermodynamic
quantities. These features are explicitly demonstrated and illustrated for two
non-trivial systems, the infinite range quantum spin glass and the weakly
interacting Bose gas. The Onsager terms of both systems are calculated, which
represent the first beyond mean-field contributions. For the spin glass new
TAP-like equations are presented and discussed in the paramagnetic region. The
investigation of the Bose gas leads to a beyond mean-field thermodynamic
description. At the Bose-Einstein condensation temperature complete agreement
is found with the results presented recently by alternative techniques.
