  We present the results of numerical studies of superconductivity and
antiferromagnetism in a strongly correlated electron system. To do this we
construct a Hubbard model on a lattice of self-consistently embedded multi-site
clusters by means of a dynamical mean-field theory in which intra-cluster
dynamics is treated essentially exactly. We show that a class of characteristic
features which have been seen in the excitation spectra of high-$T_{c}$
cuprates (e.g., pseudogap and the spin-flip resonance), as well as their
interplay with the onset of a pairing correlations, can be captured within a
dynamical mean-field theory in which short-wavelength dynamics are rigorously
treated. Thus we infer that the observation of the neutron scattering resonance
in the superconducting state of the cuprate superconductors does not appear to
be directly tied to their quasi-2D character. Although our approach is defined
strictly in terms of fermion degrees of freedom, we show that we can readily
identify the emergence of effective low energy bosonic degrees of freedom in
the presence of a well-defined broken symmetry phase as long as their dynamics
are dominated by short-range, short-wavelength fluctuations. Our results reveal
that the dynamics of staggered spin degrees of freedom builds up coherence and
a resonance-like sharp feature emerges as pairing correlations set in. Under
conditions of superconducting broken symmetry our approach thus extends static
BCS mean field theory to provide an exact treatment of quantum fluctuations of
the BCS order parameter.
