Holstein-Primakoff transformation

The Holstein-Primakoff transformation in quantum mechanics is a mapping from angular momentum operators to boson creation and annihilation operators. A particular axis is chosen ($z$ in the explicit form below), and the state corresponding to maximal projection of spin along this axis (i.e. $S\_z\; =\; hbar\; S$ for a spin of magnitude $S$) is mapped to the vacuum of bosons. Each added boson then corresponds to a decrease of $hbar$ in the spin projection. The spin raising and lowering operators $S^+\; =\; S^x\; +\; i\; S^y$ and $S^-\; =\; S^x\; -\; i\; S^y$ therefore correspond to the bosonic annihilation and creation operators.

The precise relations between the operators must be chosen to ensure the correct commutation relations for the angular momentum operators. The Holstein-Primakoff transformation can be written as::$S^+\; =\; hbar\; sqrt\{2S\}\; sqrt\{1-frac\{a^dagger\; a\}\{2S\; a$:$S^-\; =\; hbar\; sqrt\{2S\}\; a^dagger\; sqrt\{1-frac\{a^dagger\; a\}\{2S$:$S^z\; =\; hbar(S\; -\; a^dagger\; a)$The transformation is particularly useful in the case where $S$ is large, when the square roots can be expanded as Taylor series, to give an expansion in decreasing powers of $S$.