Representation of the $t-J$ model via spin-charge variables

We show that the $t-J$ Hamiltonian is not in general reduced to $H_{t-J}=H(S\to ,f)$, where $S\to$ and $f$ stand for independent $\left(\right[S\to ,f]=0)$ SU(2) (spin) generators and spinless fermionic (holon) fields, respectively. The proof is based upon an identification of the Hubbard operators with the generators of the su$\left(2\mathrm{}|1\right)$ superalgebra in the degenerate fundamental representation and ensuing SU$\left(2\mathrm{}|1\right)$ path-integral representation of the partition function $Z_{t-J}.$