Many-body-QED perturbation theory: Connection to the two-electron Bethe–Salpeter equation

Abstract

The connection between many-body theory (MBPT)-in perturbative and nonperturbative form-and quantum-electrodynamics (QED) is reviewed for systems of two fermions in an external field. The treatment is mainly based upon the recently developed covariant-evolution-operator method for QED calculations [Lindgren et al. Phys. Rep. 389, 161 (2004)], which has a structure quite akin to that of many-body perturbation theory. At the same time this procedure is closely connected to the S-matrix and the Green'sfunction formalisms and can therefore serve as a bridge between various approaches. It is demonstrated that the MBPT-QED scheme, when carried to all orders, leads to a Schrödinger-like equation, equivalent to the Bethe-Salpeter (BS) equation. A Bloch equation in commutator form that can be used for an "extended" or quasi-degenerate model space is derived. It has the same relation to the BS equation as has the standard Bloch equation to the ordinary Schrödinger equation and can be used to generate a perturbation expansion compatible with the BS equation also for a quasi-degenerate model space. PACS Nos.: 31.10+z, 31.15Md, 31.30Jv Résumé : French version of abstract (supplied by CJP) [Traduit par la rédaction] Can. J. Phys. : 1-33 () NRC Canada 2 Can. J. Phys. Vol. ,pronounced in the scattering of strongly interacting particles but less so for bound-state systems in weak-coupling [13,14,15,16,17] (see ref.[6] for a review). The earliest applications of the BS equation appeared in atomic physics and concerned the proton recoil contribution to the hydrogen fine structure by Salpeter [18] and the positronium energy level structure by Karplus and Klein [4].An important goal for the equation has been the study of strongly interacting particles, which is a fundamental problem in elementary-particle physics. In recent years there have been numerous applications in QCD, dealing mainly with the quark-quark, quark-antiquark interactions, quark confinement and related problems [14,19,20,21]. Here, the problems mentioned above are more serious, as recently summarized by Namyslowski [6].There have also been many applications in surface and solid-state physics, ranging from electronhole interactions in ion crystals [22] and studies of the two-dimensional Hubbard model [23] and Cooper pairs [24] to quantum dots [25].The BS equation has also been applied to three or more particles [26,27,28], although serious problems have been encountered for more than three particles [29].Various approximation schemes for treating the BS equation have been developed over the time. The simplest approximation is the "ladder approximation", where all intermediate states evolve only in the forward (positive) time direction. This is a useful starting point in the strong-coupling case, where the standard perturbative or self-consistent approach may not converge, and this approximation is, for instance, the basis for the Brueckner theory of nuclear matter [30,31, Sect. 41]. Another approach is the "quasi-potential approximation", which implies that the equa...

Combining Many-Body Perturbation and Quantum Electrodynamics

“…(See [10], [6, Section 6.9].) We shall now show that the procedure presented here is compatible with the Bethe-Salpeter equation [24,25].…”

“…If the interaction is energy independent, this becomes identical to the corresponding folded contribution in standard MBPT (Figure 1). Generalizing the procedure above to arbitrary orders, leads to [10,Equation (100)], [11,Equation (64)…”

“…One of the methods for QED calculations; however, namely, the Covariant-Evolution-Operator (CEO) method, recently developed by the Gothenburg group [8,9], has a structure that is similar to that of MBPT, and it can therefore serve as a basis for a unified procedure. Such a procedure has been developed by the group during the last few years and is described in a number of publications [10,11] as well as in a recent book [6]. The procedure is now being implemented [12][13][14][15], and at a later stage interesting comparison can be made with accurate experimental data that presently cannot be explained.…”

A numerical procedure for combined many‐body‐QED calculations

“…The last term is caused by the energy dependence of the potential. When this procedure is carried out to all orders, it can be shown that it leads to [23] …”

Helium fine structure: Unsolved many‐body‐QED Problem

“…Work is ongoing in our group to extend the CEOT to include strong correlation by combining it with the CCA 12. This is essentially a many‐body theory for energy‐dependent perturbations.…”