Holstein–Primakoff transformation

The Holstein-Primakoff transformation in quantum mechanics is a mapping 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[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 technique

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 ${\displaystyle S_{x}}$, ${\displaystyle S_{y}}$ and ${\displaystyle S_{z}}$, which are mutually noncommuting, i.e., ${\displaystyle \left[S_{x},S_{y}\right]=i\hbar S_{z}}$ and its cyclic permutations.

In order to uniquely specify the states of a spin, one may diagonalise any set of commuting operators. Normally one uses the SU(2) Casimir operators ${\displaystyle S^{2}}$ and ${\displaystyle S_{z}}$, which leads to states with the quantum numbers ${\displaystyle \left|s,m_{s}\right\rangle }$,

${\displaystyle S^{2}\left|s,m_{s}\right\rangle =\hbar ^{2}s(s+1)\left|s,m_{s}\right\rangle ,}$

${\displaystyle S_{z}\left|s,m_{s}\right\rangle =\hbar m_{s}\left|s,m_{s}\right\rangle .}$

The projection quantum number ${\displaystyle m_{s}}$ takes on all the values ${\displaystyle (-s,-s+1,\ldots ,s-1,s)}$.

Consider a single particle of spin s (i.e., look at a single irreducible representation of SU(2)). Now take the state with maximal projection ${\displaystyle \left|s,m_{s}=+s\right\rangle }$, the extremal weight state as a vacuum for a set of boson operators, and each subsequent state with lower projection quantum number as a boson excitation of the previous one,

${\displaystyle \left|s,s-n\right\rangle \mapsto {\frac {1}{\sqrt {n!}}}\left(a^{\dagger }\right)^{n}|0\rangle _{B}~.}$

Each additional boson then corresponds to a decrease of ħ in the spin projection. Thus, the spin raising and lowering operators ${\displaystyle S_{+}=S_{x}+iS_{y}}$ and ${\displaystyle S_{-}=S_{x}-iS_{y}}$, so that ${\displaystyle [S_{+},S_{-}]=2\hbar S_{z}}$, correspond (in the sense detailed below) to the bosonic annihilation and creation operators, respectively. The precise relations between the operators must be chosen to ensure the correct commutation relations for the spin operators, such that they act on a finite-dimensional space, unlike the original Fock space.

The resulting Holstein–Primakoff transformation can be written as

${\displaystyle S_{+}=\hbar {\sqrt {2s}}{\sqrt {1-{\frac {a^{\dagger }a}{2s}}}}\,a~,\qquad S_{-}=\hbar {\sqrt {2s}}a^{\dagger }\,{\sqrt {1-{\frac {a^{\dagger }a}{2s}}}}~,\qquad S_{z}=\hbar (s-a^{\dagger }a)~.}$

The transformation is particularly useful in the case where s is large, when the square roots can be expanded as Taylor series, to give an expansion in decreasing powers of s.

The nonhermitean Dyson-Maleev variant realization J is related to the above,

${\displaystyle J_{+}=\hbar \,a~,\qquad J_{-}=S_{-}~{\sqrt {2s-a^{\dagger }a}}=\hbar a^{\dagger }\,(2s-a^{\dagger }a)~,\qquad J_{z}=S_{z}=\hbar (s-a^{\dagger }a)~,}$

satisfying the same commutation relations and characterized by the same Casimir invariant.

The technique can be further extended to the Witt algebra,[6] which is the centerless Virasoro algebra.

References

1. T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 - 1113 (1940) doi:10.1103/PhysRev.58.1098
2. "Theodore D. Holstein, Physics: Los Angeles". University of California. Retrieved 23 December 2015.
3. A. Klein and E. R. Marshalek, Boson realizations of Lie algebras with applications to nuclear physic,s http://link.aps.org/doi/10.1103/RevModPhys.63.375 doi:10.1103/RevModPhys.63.375
4. "This Week's Citation Classic by F. J. Dyson, August 4, 1986" (PDF). Current Contents (36): 16. 8 Sep 1986.
5. Schwinger, J. (1952). "On Angular Momentum", Unpublished Report, Harvard University, Nuclear Development Associates, Inc., United States Department of Energy (through predecessor agency the Atomic Energy Commission), Report Number NYO-3071 (January 26, 1952).
6. D Fairlie, J Nuyts, and C Zachos (1988). Phys Lett B202 320-324. doi:10.1016/0370-2693(88)90478-9