  The operators of localized spins within a magnetic material commute at
different sites of its lattice and anticommute on the same site, so they are
neither fermionic nor bosonic operators. Thus, to construct diagrammatic
many-body perturbation theory, requiring the Wick theorem, the spin operators
are usually mapped to the bosonic ones -- the most popular in magnonics and
spintronics literature has been the Holstein-Primakoff (HP) transformation.
However, the square root of operators in the HP transformation has to be
expanded into an infinite series, which is then truncated to some low order to
make calculations tractable. This poses a question on the {\em range of
validity of truncated HP transformation} when describing nonequilibrium
dynamics of localized spins interacting with conduction electron spins -- a
problem frequently encountered in numerous transport phenomena in spintronics.
Here we apply exact diagonalization techniques to Hamiltonian of fermions
(i.e., electrons) interacting with HP bosons vs. Hamiltonian of fermions
interacting with the original localized spin operators in order to compare
their many-body states and one-particle Green functions. For this purpose, we
consider a one-dimensional quantum Heisenberg ferromagnetic metallic spin-$S$
XXX chain of $N \le 7$ sites, where $S=1$ or $S=5/2$ and electrons can hop
between the sites while interacting with localized spin via $sd$ exchange
interaction. For two different versions of the Hamiltonian for this model, we
compare the structure of their ground states; time-evolution of excited states;
spectral functions computed from the retarded Green function in equilibrium;
and the lesser nonequilibrium Green function depending on two times.
