= Holstein–Primakoff transformation
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{title2=$S^z=S-a^\dagger a$}
{wiki}
The Holstein–Primakoff transformation represents a spin-$S$ <operator algebra> with a bosonic <creation operator> and <annihilation operator>:
$$
S^+=\sqrt{2S}\sqrt{1-\frac{a^\dagger a}{2S}}\,a,\qquad S^-=\sqrt{2S}\,a^\dagger\sqrt{1-\frac{a^\dagger a}{2S}},\qquad S^z=S-a^\dagger a.
$$
The order of the square root and oscillator operator matters. On the physical <Fock states> $|n\rangle$, $0\le n\le2S$, the raising and lowering matrix elements are $\sqrt{n(2S-n+1)}$ and $\sqrt{(n+1)(2S-n)}$. Their squared difference gives $[S^+,S^-]|n\rangle=2(S-n)|n\rangle$, proving the <spin commutation relations>. Expanding the square root yields the <linear spin-wave approximation> and its interaction corrections.
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