Theory of transformation for the diagonalization of quadratic Hamiltonians

Understand

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.