Holstein–Primakoff transformation

In quantum mechanics, the Holstein–Primakoff transformation is a mapping between to the spin operators from boson creation and annihilation 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 in 1940 by Theodore Holstein, a graduate student at the time, 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 technique, and to a lesser extent the Jordan–Schwinger map. There is, furthermore, a close link to the theory of (generalized) coherent states in Lie algebras.