Intro

There are a few interesting tranformation that appears in quantum theories.

The list

Holstein–Primakoff transformation: maps from bosonic ladder operators to spin operators

Jordon-Wigner transformation: maps from spin operators to fermionic ladder operators

I purposely put the above two transformations together, as it begs the naive question: if we compose the two maps one after the other, does that mean there is a map from bosonic ladder operators to fermionic ladder operators? This smells like some supersymmetry.

Klein transformation: used to fix the (anti-)commutation relation between field operators such that the canonical (anti-)commutation relation is satisfied in accordance with the spin-statistics theorem.

Bogoliubov–Valatin transformation: maps from bosonic to bosonic or from fermionic to fermionic; it is used to diagonalize the BCS Hamiltonian.

Jordan-Schwinger map: according to Wikipedia, it is “a map from matrices Mij to bilinear expressions of quantum oscillators which expedites computation of representations of Lie algebras occurring in physics”.

Schrieffer-Wolff transformation: This transformation is quite different from the previous ones. It is used to obtain the effective low-energy effective Hamiltonian, by diagonalize a Hamiltonian to 1st order in interaction. This is the operator version of the 2nd-order perturbation theory, where virtual-state is used to connect two low-energy states.