Holstein–Primakoff transformation
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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.
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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.