Explicit Diagonalization of Pair Interaction Models

TL;DR

This paper introduces a systematic algebraic method for diagonalizing quadratic bosonic Hamiltonians, enabling explicit computation of ground state energies and broad applicability to quantum field models.

Contribution

It provides a general algebraic construction of Bogoliubov transformations that diagonalize pair interaction models, including explicit formulas for ground state energies.

Findings

Diagonalization of various quantum field models achieved

Explicit formulas for ground state energies derived

Method applicable to harmonic oscillator and Pauli-Fierz 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.