Effective lattice actions for correlated electrons

We present an exact, unconstrained representation of the electron operators in terms of operators of opposite statistics. We propose a path--integral representation for the $t$-$J$ model and introduce a parameter controlling the semiclassical behaviour. We extend the functional approach to the Hubbard model and show that the mean--field theory is equivalent to considering, at Hamiltonian level, the Falikov--Kimball model. Connections with a bond-charge model are also discussed.