Boson-like operator

The boson creation operators are Ypi where p = iv,v indicates proton or neutron and i = L,M the angular momentum of the basis bosons. We consider even L, even parity for the bosons, which are supposed to represent pairs of like particles (or holes). The numbers Np = Ejnpi are assumed to equal 1/2 the number of proton or neutron particles or holes from the nearest closed shell, and are thus well defined functions of N and Z, to the... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

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