  We apply the flow equation method for studying the fermion systems where
pairing interactions can either trigger the BCS instability with the symmetry
breaking manifested by the off-diagonal order parameter or lead to the gaped
single particle spectrum without any symmetry breaking. We construct the
continuous Bogoliubov transformation in a scheme resembling the renormalization
group procedure. We further extend this continuous transformation to a case
where fermion pairs interact with the boson field. Due to temporal quantum
fluctuations the single particle excitation spectrum develops a gap which is
centered around the renormalized boson energy. When bosons undergo the Bose
Einstein condensation this structure evolves into the BCS spectrum.
