Two Quantum Liquids

We describe a theory of the two dimensional Heisenberg-Hubbard model working in the spin and hole fermion representation based on the use of a singular gauge transformation on the spins. We are motivated by a new mean field theory for a system of strongly coupled fermions and bosons which is also described. In the latter case we obtain the bose condensed—fermi liquid phase; in the former we obtain equations governing generalized flux phases. We do not find superconducting solutions to these equations in the atomic limit, J=0. If solutions exist for finite J case then it is argued that an energy gap is present in that case.