  A two component model of negative U centers coupled with the Fermi sea of
itinerant fermions is discussed in connection with high-temperature
superconductivity of cuprates, and superfluidity of atomic fermions. We examine
the phase transition and the condensed state of this boson-fermion model (BFM)
beyond the ordinary mean-field approximation in two and three dimensions. No
pairing of fermions and no condensation are found in two-dimensions for any
symmetry of the order parameter. The expansion in the strength of the order
parameter near the transition yields no linear homogeneous term in the
Ginzburg-Landau-Gor'kov equation and a zero upper critical field in
any-dimensional BFM, which indicates that previous mean-field discussions of
the model are flawed. Normal and anomalous Green's functions are obtained
diagrammatically and analytically in the condensed state of a simplest version
of 3D BFM. A pairing of bosons analogous to the Cooper pairing of fermions is
found. There are three coupled condensates in the model, described by the
off-diagonal single-particle boson, pair-fermion and pair-boson fields. These
results negate the common wisdom that the boson-fermion model is adequately
described by the BCS theory at weak coupling.
