Holstein–Primakoff transformation

Holstein - Primakoff bosons or the Holstein - Primakoff transformation is a method to describe spins.

Here, the spin operators are replaced by the following bosonic creation and annihilation operators:

How to show easily that satisfy the so-defined operators of the angular momentum algebra.

The Holstein - Primakoff representation is used, among others, in the theory of spin waves, here is often considered the limiting case of small Bosonenzahl, where the roots can be developed.

In addition, there are other methods to map angular momentum operators on boson creation and annihilation operators ( Freeman Dyson, Julian Schwinger ).