Holstein–Primakoff transformation

The basic idea can be illustrated for the basic example of spin operators of quantum mechanics.

For any set of right-handed orthogonal axes, define the components of this vector operator as ${\displaystyle S_{x}}$, ${\displaystyle S_{y}}$ and ${\displaystyle S_{z}}$, which are mutually noncommuting, i.e., ${\displaystyle \left[S_{x},S_{y}\right]=i\hbar S_{z}}$ and its cyclic permutations.

In order to uniquely specify the states of a spin, one may diagonalise any set of commuting operators. Normally one uses the SU(2) Casimir operators ${\displaystyle S^{2}}$ and ${\displaystyle S_{z}}$, which leads to states with the quantum numbers ${\displaystyle \left|s,m_{s}\right\rangle }$,

${\displaystyle S^{2}\left|s,m_{s}\right\rangle =\hbar ^{2}s(s+1)\left|s,m_{s}\right\rangle ,}$

${\displaystyle S_{z}\left|s,m_{s}\right\rangle =\hbar m_{s}\left|s,m_{s}\right\rangle .}$

The projection quantum number ${\displaystyle m_{s}}$ takes on all the values ${\displaystyle (-s,-s+1,\ldots ,s-1,s)}$.

Consider a single particle of spin s (i.e., look at a single irreducible representation of SU(2)). Now take the state with maximal projection ${\displaystyle \left|s,m_{s}=+s\right\rangle }$, the extremal weight state as a vacuum for a set of boson operators, and each subsequent state with lower projection quantum number as a boson excitation of the previous one,

${\displaystyle \left|s,s-n\right\rangle \mapsto {\frac {1}{\sqrt {n!}}}\left(a^{\dagger }\right)^{n}|0\rangle _{B}~.}$

Each additional boson then corresponds to a decrease of ħ in the spin projection. Thus, the spin raising and lowering operators ${\displaystyle S_{+}=S_{x}+iS_{y}}$ and ${\displaystyle S_{-}=S_{x}-iS_{y}}$, so that ${\displaystyle [S_{+},S_{-}]=2\hbar S_{z}}$, correspond (in the sense detailed below) to the bosonic annihilation and creation operators, respectively. The precise relations between the operators must be chosen to ensure the correct commutation relations for the spin operators, such that they act on a finite-dimensional space, unlike the original Fock space.

The resulting Holstein–Primakoff transformation can be written as

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.

Alternatively to a Taylor expansion there has been recent progress [6][7] with a resummation of the series that made expressions possible that are polynomial in bosonic operators but still mathematically exact (on the physical subspace). The first method develops a resummation method [6] that is exact for the for spin ${\displaystyle s=1/2}$ while the latter[7] employs a Newton series (a finite difference) expansion with an identical result, as shown below

While the expression above is not exact for spins higher than 1/2 it is an improvement over the Taylor series. Exact expressions also exist for higher spins and include ${\displaystyle 2s+1}$ terms. Much like the result above also for the expressions of higher spins ${\displaystyle S_{+}=S_{-}^{\dagger }}$ and therefore the resummation is hermitian.

There also exists a non-Hermitian Dyson–Maleev (by Freeman Dyson and S.V. Maleev) variant realization J is related to the above and valid for all spins,

${\displaystyle J_{+}=\hbar \,a~,\qquad J_{-}=S_{-}~{\sqrt {2s-a^{\dagger }a}}=\hbar a^{\dagger }\,(2s-a^{\dagger }a)~,\qquad J_{z}=S_{z}=\hbar (s-a^{\dagger }a)~,}$

satisfying the same commutation relations and characterized by the same Casimir invariant.

The technique can be further extended to the Witt algebra,[8] which is the centerless Virasoro algebra.

• Spin wave
• Jordan–Wigner transformation
• Jordan–Schwinger transformation
• Bogoliubov–Valatin transformation
• Klein transformation