Exact transformation from paulions to bosons

First let us remark that the replacement of Pauli operators by Bose operators applied in this chapter is only approximate since the occupation numbers of paulions can be either 0 or 1, whereas the occupation numbers for bosons can take all nonnegative integer values 0,1,2. .28 Therefore the replacement of the operators Ps and Pj by Bose operators can provoke uncontrolled errors in all 

28The results of this section can also be applied in the theory of magnetism (see, for example, the papers by Goldhirsh et al. (87), by Goldhirsh and Yakhot (88) and the supplement by Yu. Rudoy and Yu. Tserkovnikov in Tyablikov s book (89)). 

Molecular transition Dipole approx. 6-state model II L MO method 9-state model II M  

We can, however, improve the transition from Pauli operators to Bose operators Bs and B] requiring that for an arbitrary number of bosons the number of paulions will equal 0 or 1 (90). To this aim we write the Pauli operators in the form 

So we obtained the exact transition from Pauli to Bose operators in the form 

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