Explicit Diagonalization of Pair Interaction Models

Abstract

We provide a general method for constructing bosonic Bogoliubov transformations that diagonalize a general class of quadratic Hamiltonians. These Hamiltonians describe the pair interaction models. Bogoliubov transformations are constructed algebraically, and the resulting Hamiltonians become the second quantizations of explicit one-particle Hamiltonians. Moreover, an explicit formula for the ground state energies is given. Our method systematically diagonalizes various models of quantum field theory, including a model of a harmonic oscillator coupled to a Bose field and the Pauli-Fierz models in the dipole approximation.