Theory of transformation for the diagonalization of quadratic Hamiltonians

TL;DR

This paper develops a systematic theory for diagonalizing quadratic Hamiltonians using the eigenvalue problem of their dynamic matrices, simplifying the process and clarifying conditions for diagonalizability across various quantum systems.

Contribution

It formalizes and generalizes Dirac and Bogoliubov-Valatin transformations by linking diagonalization to the eigenvalue problem of the dynamic matrix.

Findings

Provides an operational procedure for diagonalization

Clarifies conditions for uniqueness of the transformation

Applies theory to quantum fields like Klein-Gordon and phonons

Abstract

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..