  We propose a new boson expansion method using a norm operator. The small
parameter expansion, in which the boson approximation becomes the zeroth-order
approximation, requires the double commutation relations between phonon
operators that are not closed between the phonon excitation modes adopted as
boson excitations. This results in an infinite expansion regardless of whether
the type of the boson expansion is Hermitian or non-Hermitian. The small
parameter expansion does not hold when the commutation relations are closed.
The norm operator is expressed as a function of the number operator in the
physical subspace, which enables us to obtain substantially a finite boson
expansion regardless of the Hermitian or non-Hermitian type. We also point out
the problems of the conventional boson expansion methods. The normal-ordered
linked-cluster expansion theory has failed to refute Marshalek's claim that
KT-1 and KT-2 are of chimerical boson expansion. The Dyson boson expansion
theory does not have exceptional superiority over other types. Previous studies
using the boson expansion methods should be re-examined.
