Quantum electrodynamics perturbative development

It now remains to expand the operators in (3.235) using the definitions given in (3.230) but before we do so we must draw attention to a difficulty with (3.235). The final term, containing the operator (00)2 is not obtained if a more sophisticated treatment starting from the Bethe Salpeter equation is used. The reader will recall our earlier comment that the interaction term in the Breit Hamiltonian is acceptable provided it is treated by first-order perturbation theory. Rather than launch into quantum electrodynamics at this stage, we shall proceed to develop (3.235) but will omit the (00)2 term without further comment. 

This paper is dedicated to Professor Ingvar Lindgren in connection with his 65th anniversary in view of his many outstanding contributions to physics and particularly to the development of perturbation theory and its applications to the non-relativistc theory of atomic and molecular systems and in some cases also to relativistic corrections by means of quantum electrodynamics. 

Perturbation theory has been one of the most frequently used and most powerful tools of quantum mechanics. The very foundations of relativistic quantum theory—quantum electrodynamics—are perturbative in nature. Many-body perturbation theory has been used for electron correlation treatments since the early days of quantum chemistry, and in more recent times multireference perturbation theories have been developed to provide quantitative or semiquantitative information in very complex systems. 

The procedures for many-body perturbation calculations (MBPT) for atomic and molecular systems are nowadays very well developed, and the dominating electrostatic as well as magnetic perturbations can be taken to essentially all orders of perturbation theory (see, for instance, [1]). Less pronounced, but in many cases still quite significant, are the quantum electrodynamical (QED) perturbations—retardation, virtual pairs, electron self-energy, vacuum polarization and vertex correction. Sophisticated procedures for their evaluation have also been developed, but for practical reasons such calculations are prohibitive beyond second order (two-photon exchange). Pure QED effects beyond that level can be expected to be very small, but the combination of QED and electrostatic perturbations (electron correlation) can be significant. However, none of the previously existing methods for MBPT or QED calculations is suited for this type of calculation. 

---