Theory for the reduction of products of spin operators

In this study we show that the sum of the powers of arbitrary products of quantum spin operators such as (S+)l(S−)m(Sz)n can be reduced by one unit, if this sum is equal to 2S+1, S being the spin quantum number. We emphasize that by a repeated application of this procedure all arbitrary spin operator products with a sum of powers larger than 2S can be replaced by a combination of spin operators with a maximum sum of powers not larger than 2S. This transformation is exact. All spin operators must belong to the same lattice site. By use of this procedure the consideration of single-ion anisotropies and the investigation of the magnetic reorientation within a Green's function theory are facilitated. Furthermore, it may be useful for the study of time dependent magnetic properties within the ultrashort (fsec) time domain.