Intensive boson operators

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become... 

We also note that had we not used the method of intensive boson operators but rather evaluated Eq. (7.18) exactly, we would have obtained (van Roosmalen, 1982)... 

This is a function of the complex variables E,j, ,2- Once more, by making use of intensive boson operators, one can easily obtain Hd(pi,q, p2, q2). The potential functions can then be defined as... 

The expectation value of H can again be simply evaluated if one introduces intensive boson operators such that... 

In the language of intensive boson operators, we simply have to double the definitions (5.6) and (5.7) to account for two families of boson operators and (f and i ), k =, 2. This method, when applied to Eq. (5.24), leads to the following potential surface ... 

As a final example of application of the intensive boson operator technique to the one-dimensional algebraic model, we consider the case of the n m Fermi operator introduced in Section III.E [Eq. (3.128)]. A straightforward use of the aforementioned method leads to the classical potential surface... 

Three-dimensional problems can also be addressed within the framework of the intensive boson operators method, by simply introducing, in place of the scalar quantities a and jS of Eq. (5.6), two (complex) scalars and two vectors associated with the scalar and vector boson operators of U(4), respectively. In this way it is possible to obtain the classical limit of... 

This expression can be obtained by applying the intensive boson method to the invariant operator of the coupled algebra Ui2(2). 

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