  In the present work it is shown that the family of first-order Lagrangians
for the t-J model and the corresponding correlation generating functional
previously found can be exactly mapped into the slave-fermion decoupled
representation. Next, by means of the Faddeev-Jackiw symplectic method, a
different family of Lagrangians is constructed and it is shown how the
corresponding correlation generating functional can be mapped into the
slave-boson representation. Finally, in order to define the propagation of
fermion modes we discuss two alternative ways to treat the fermionic sector in
the path-integral formalism for the t-J model.
