In quantum mechanics, the Holstein–Primakoff transformation is a mapping from boson creation and annihilation operators to the spin operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces.
One important aspect of quantum mechanics is the occurrence of—in general—non-commuting operators which represent observables, quantities that can be measured.A standard example of a set of such operators are the three components of the angular momentum operators, which are crucial in many quantum systems.These operators are complicated, and one would like to find a simpler representation, which can be used to generate approximate calculational schemes.
The transformation was developed[1] in 1940 by Theodore Holstein, a graduate student at the time,[2] and Henry Primakoff. This method has found widespread applicability and has been extended in many different directions.
There is a close link to other methods of boson mapping of operator algebras: in particular, the (non-Hermitian) Dyson–Maleev[3] [4] technique, and to a lesser extent the Jordan - Schwinger map.[5] There is, furthermore, a close link to the theory of (generalized) coherent states in Lie algebras.
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
Sx
Sy
Sz
\left[Sx,Sy\right]=i\hbarSz
S2
Sz
\left|s,ms\right\rangle
| 2 | |
| S | |
| s\right\rangle=\hbar |
s(s+1)\left|s,ms\right\rangle,
Sz\left|s,ms\right\rangle=\hbarms\left|s,ms\right\rangle.
ms
(-s,-s+1,\ldots,s-1,s)
Consider a single particle of spin (i.e., look at a single irreducible representation of SU(2)). Now take the state with maximal projection
\left|s,ms=+s\right\rangle
\left|s,s-n\right\rangle\mapsto
| 1 | |
| \sqrt{n! |
Each additional boson then corresponds to a decrease of in the spin projection. Thus, the spin raising and lowering operators
S+=Sx+iSy
S-=Sx-iSy
[S+,S-]=2\hbarSz
The resulting Holstein–Primakoff transformation can be written asThe transformation is particularly useful in the case where is large, when the square roots can be expanded as Taylor series, to give an expansion in decreasing powers of .
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 spin
s=1/2
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
2s+1
S+=
| \dagger | |
| S | |
| - |
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,
J+=\hbara~, J-=S-~\sqrt{2s-a\daggera}=\hbara\dagger(2s-a\daggera)~, Jz=Sz=\hbar(s-a\daggera)~,
The technique can be further extended to the Witt algebra,[8] which is the centerless Virasoro algebra.