Daniel Hedendahl

Coupled Clusters and Quantum Electrodynamics

We start by reviewing, as a background, the standard non-relativistic and relativistic many-body perturbative and coupled-cluster approaches in the way we have implemented them. The covariant-evolution-operator method that we introduced more recently for quantum-electrodynamical (QED) calculations is then described, and it is demonstrated how this method can be extended to combine electron-correlation and QED effects in a covariant manner. This can be included in a many-body calculation of coupled-cluster type, and it is demonstrated that this leads for two-particle systems eventually to the full Bethe–Salpeter equation. It is indicated how this procedure can also be applied to systems with more than two electrons. Preliminary numerical results are given for the ground state of some heliumlike ions.

ENERGY-DEPENDENT MANY-BODY PERTURBATION THEORY: A ROAD TOWARDS A MANY-BODY-QED PROCEDURE

A rigorous procedure for energy-dependent many-body perturbation theory (MBPT) is presented. This can be applied for numerical evaluation of many-body-QED effects by combining QED with electron correlation to arbitrary order. So far, it has been used only for the exchange of a single retarded photon together with an arbitrary number of instantaneous Coulomb interactions. For heliumlike neon this represents more than 99% of the nonradiative effect on the energy beyond standard MBPT.

Combining Quantum Electrodynamics and Electron Correlation

There is presently a large interest in studying highly charged ions in order to investigate the effects of quantum-electrodynamics (QED) at very strong fields. Such experiments can be performed at large accelerators, like that at GSI in Darmstadt, where the big FAIR facility is under construction. Accurate experiments on light and medium-heavy ions can also be performed by means of laser and X-ray spectroscopy. To obtain valuable information, accurate theoretical results are required to compare with. The most accurate procedures presently used for calculations on simple atomic systems are (i) all-order many-body perturbative expansion with added first-order analytical QED energy corrections, and (ii) two-photon QED calculations. These methods have the shortcoming that the combination of QED and correlational effects (beyond lowest order) is completely missing. We have developed a third procedure, which can remedy this shortcoming. Here, the energy-dependent QED effects are included directly into the atomic wave function, which is possible with the procedure that we have recently developed. The calculations are performed using the Coulomb gauge, which is most appropriate for the combined effect. Since QED effects, like the Lamb shift, have never been calculated in that gauge, this has required some development. This is now being implemented in our computational procedure, and some numerical results are presented.

QED Calculations on Highly Charged Ions, Using a Unified MBPT-QED Approach

There is presently a great interest in studying QED effects in highly charged ions by means of large accelerators, and the best information is usually gained from the study of multi-electron ions. It might then be possible to detect the combined effect of QED and electron correlation, which has so far never been observed. That could be possible also from accurate laser or X-ray data. For the corresponding theoretical analysis it will then be necessary to treat the effects of QED and electron correlation simultaneously in a coherent manner. This is not possible with presently available techniques but will require the new generation of atomic calculations that is now being developed at our laboratory. The calculations have to be performed in the Coulomb gauge, and a procedure for renormalization in that gauge has very recently been tested for hydrogen-like ions. Work is now in progress to perform unified MBPT-QED calculations on multi-electron systems.

Combining Many-Body Perturbation and Quantum Electrodynamics

It has been a long-sought problem to be able to combine many-body perturbation theory and quantum electrodynamics into a unified, covariant model. Such a model has recently been developed at our laboratory and is outlined in the present paper. The model has potential applications in many areas and opens up the possibility of studying the interplay between various interactions in different system. The model has so far been applied to highly ionized helium-like ions, and some numerical results are given. It is expected that the combined effect—that has never been calculated before—could have a significant effect on certain experimental data. The radiative effects are being regularized using the dimensional regularization in Coulomb gauge, and the first numerical results have been obtained.

Relativistically Covariant Many-Body Perturbation Procedure

A covariant evolution operator (CEO) can be constructed, representing the time evolution of the relativistic wave unction or state vector. Like the nonrelativistic version, it contains (quasi-)singularities. The regular part is referred to as the Green’s operator (GO), which is the operator analogue of the Green’s function (GF). This operator, which is a field-theoretical concept, is closely related to the many-body wave operator and effective Hamiltonian, and it is the basic tool for our unified theory. The GO leads, when the perturbation is carried to all orders, to the Bethe–Salpeter equation (BSE) in the equal-time or effective-potential approximation. When relaxing the equal-time restriction, the procedure is fully compatible with the exact BSE. The calculations are performed in the photonic Fock space, where the number of photons is no longer constant. The procedure has been applied to helium-like ions, and the results agree well with S-matrix results in cases when comparison can be performed. In addition, evaluation of higher-order quantum-electrodynamical (QED) correlational effects has been performed, and the effects are found to be quite significant for light and medium-heavy ions.

Field‐Theoretical Approach to Many‐Body Perturbation Theory: Combining MBPT and QED

Many‐Body Perturbation Theory (MBPT) is today highly developed. The electron correlation of atomic and molecular systems can be evaluated to essentially all orders of perturbation theory—also relativistically (RMBPT)—by means of techniques of Coupled‐Cluster type. When high accuracy is needed, effects beyond RMBPT will enter, i.e., effects of retarded Breit interaction and of radiative effects (Lamb shift), effects normally referred to as QED effects. These effects can be evaluated by means of special techniques, like S‐matrix formulation, which cannot simultaneously treat electron correlation. It would for many applications be desirable to have access to a numerical technique, where effects of electron correlation and of QED could be treated on the same footing. Such a technique is presently being developed and gradually implemented at our laboratory. Some numerical results will be given.