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A theory of transformation is presented for the diagonalization of aHamiltonian that is quadratic in creation and annihilation operators or incoordinates and momenta. It is the systemization and theorization of Dirac andBogoliubov-Valatin transformations, and thus provides us an operationalprocedure to answer, in a direct manner, the questions as to whether aquadratic 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 quadraticHamiltonian has a dynamic matrix of its own. The eigenvalue problem of thedynamic matrix determines the diagonalizability of the quadratic Hamiltoniancompletely. In brief, the theory ascribes the diagonalization of a quadraticHamiltonian to the eigenvalue problem of its dynamic matrix, which is familiarto all of us. That makes it much easy to use. Applications to various physicalsystems are discussed, with especial emphasis on the quantum fields, such asKlein-Gordon field, phonon field, etc..

Theory of transformation for the diagonalization of quadratic Hamiltonians