Comprehensive theory for reduction of products of spin operators

We present a comprehensive theory that reduces the total power of products of spin operators. This theory improves the previous one P.J. Jensen, F. Aguilera-Granja, Phys. Lett. A 269 (2000) 158] in two aspects. One is that for the set of spin operators S⁺, S⁻, Sz, a new method is suggested where the expansion coefficients in the reduction formula can be solved from linear equations. This new method is of direct physical meaning and is easier to handle. The other is that we show a method to reduce the products of another set of spin operators Sx, Sy, Sz. For this set of operators, the use of permutation regulation of x→y, y→z and z→x can save much time in obtaining some reduction formula. The present comprehensive theory enables one to deal more easily with the decoupling problems in Green' function theory where the set of either S⁺, S⁻, Sz or Sx, Sy, Sz operators is used.