  A theory of transformation is presented for the diagonalization of a
Hamiltonian that is quadratic in creation and annihilation operators or in
coordinates and momenta. It is the systemization and theorization of Dirac and
Bogoliubov-Valatin transformations, and thus provides us an operational
procedure to answer, in a direct manner, the questions as to whether a
quadratic Hamiltonian is diagonalizable, whether the diagonalization is unique,
and how the transformation can be constructed if the diagonalization exists.
The underlying idea is to consider the dynamic matrix. Each quadratic
Hamiltonian has a dynamic matrix of its own. The eigenvalue problem of the
dynamic matrix determines the diagonalizability of the quadratic Hamiltonian
completely. In brief, the theory ascribes the diagonalization of a quadratic
Hamiltonian to the eigenvalue problem of its dynamic matrix, which is familiar
to all of us. That makes it much easy to use. Applications to various physical
systems are discussed, with especial emphasis on the quantum fields, such as
Klein-Gordon field, phonon field, etc..
