Minimization of an Energy Functional for an Electron in the quantized Radiation Field over quasifree States

TL;DR

This paper analyzes the Bogolubov-Hartree-Fock energy of the Pauli-Fierz Hamiltonian, deriving bounds and conditions for minimizers within the context of quantum electrodynamics.

Contribution

It extends previous work by providing bounds and necessary conditions for the minimizer of the BHF energy in the quantized radiation field.

Findings

Derived upper and lower bounds on the BHF energy.

Suggested positivity of the energy minimizer.

Established a necessary condition for the minimizer under positivity assumption.

Abstract

Building upon the works of Bach, Breteaux, and Tzaneteas (2013) and of Bach and Hach (2022), the Bogolubov-Hartree-Fock (BHF) energy of the Pauli-Fierz Hamiltonian is investigated. Upper and lower bounds on the BHF energy are derived, which suggest positivity of the minimizer, see Theorem IV.6. Under the assumption that the minimizer is indeed positive, a necessary condition on the minimizer is determined by introducing a parameterization, which simplifies the functions of operators appearing in the energy functional.