Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2014 / Paper 1 / 1 / Solution

Use the Holstein–Primakoff transformation only on the physical occupation sector $0\le n\le 2S$, with $n=a^{†}a$ and $S^{-}=a^{†}\sqrt{2S-n}$. Acting on number states givesConsequently $S^{+}{S}^{-}=(n+1)(2S-n)$ and $S^{-}{S}^{+}=n(2S-n+1)$, soThe endpoint factors keep the spin operators inside the physical sector. The square-root ordering matters; replacing it by an unrestricted linear oscillator would not give the exact finite-spin algebra.