= Solution
The <Holstein–Primakoff transformation> replaces each finite-dimensional <spin> algebra by a bosonic <creation operator> and <annihilation operator>, while restricting the local <occupation number> to $0\le n_j\le2S$. The number $n_j=a_j^\dagger a_j$ counts deviations from the fully polarized reference <quantum state>. Its zero-boson <Fock vacuum> has $\widetilde S^z=S$.
The representation is exact on this physical subspace: the ordered square root gives $\widetilde S^+|n\rangle=\sqrt{n(2S-n+1)}|n-1\rangle$ and $\widetilde S^-|n\rangle=\sqrt{(n+1)(2S-n)}|n+1\rangle$, reproducing the <spin commutation relations>. For $n_j\ll S$,
$$
\widetilde S^+_j=\sqrt{2S}\left(1-\frac{n_j}{4S}+\cdots\right)a_j,
$$
so the leading <Hamiltonian> is quadratic in bosons and describes <spin waves>.
The square root generates boson interactions beyond leading order and its operator ordering matters. Truncation also enlarges the <Hilbert space> to unphysical occupations unless the low-density approximation remains self-consistent. Large fluctuations, particularly near a gap closing or in low dimensions, can invalidate the <linear spin-wave approximation> even when its algebraic manipulation is straightforward.
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