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The right-hand side of (B-11) contains an expression completely independent of the space S(N) or A(N) in which we defined the action of the operator . Since we defined operator as acting as in each of these subspaces having fixed N, we can simply write:

(B-12)

This is the expression of one-particle symmetric operators we were looking for. Its form is valid for any value of N and the particles are no longer numbered; it contains equal numbers of creation and annihilation operators, which only act on occupation numbers.

Comment:

Choosing the proper basis {|ui〉} it is always possible to diagonalize the Hermitian operator and write:

(B-13)

Equality (B-11) is then simply written as:

(B-14)

where is the occupation number operator in the state |uk〉 defined in (A-28).