Perturbation theory for bound states and resonances where potentials and propagators have arbitrary energy dependence

TL;DR

This paper develops a general perturbation theory framework for quantum systems with energy-dependent potentials and propagators, applicable to relativistic quantum field theory, providing practical formulas for energy and wave function corrections.

Contribution

It introduces a new, comprehensive formulation of perturbation theory that handles energy dependence in potentials and Green functions, extending applicability beyond traditional quantum mechanics.

Findings

Provides compact formulas for energy corrections in energy-dependent systems

Enables covariant calculations of bound and resonance states in relativistic quantum field theory

Valid for non-linear energy dependencies in Green functions

Abstract

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations cannot be used, for example, in the context of relativistic quantum field theory. Here we solve this problem by providing a new, general formulation of perturbation theory for calculating the changes in the energy spectrum and wave function of bound states and resonances induced by perturbations to the Hamiltonian. Although our derivation is valid for energy-dependent potentials and is not restricted to inverse free Green functions that are linear in the energy, the expressions obtained for the energy and wave function corrections are compact, practical, and maximally similar to the ones of quantum mechanics. For the case of relativistic quantum field theory, our approach provides a direct covariant way of obtaining corrections to bound and resonance state masses, as well as to wave functions that are not in the centre of mass frame.