Furry Representation

For the bound electrons (more exactly for the electrons moving in external fields) it is convenient to develop PT based on the bound electron wave functions. This representation is cadled Furry representation, or the Furry picture... 

There are two particular aspects of radiative corrections in atomic physics that will be emphasized here. One has to do with the correct implementation of QED to many-electron atoms and ions, a subject also discussed by Labzowsky and Goidenko in Chapter 8 of this book. While QED has been tested quite stringently over the years, and is unlikely to be fundamentally incorrect, to actually carry out bound state calculations is a highly nontrivial task. Even the introduction of relativity has raised serious questions about the stability of atoms, referred to as the Brown-Ravenhall wasting disease [2] or continuum dissolution [3]. While the problem is, in a practical sense, still open for neutral systems, we will show that a particular way of applying QED to atoms, use of the Furry representation [4], allows a consistent and accurate treatment of these questions for highly charged ions. 

The plan of this chapter is as follows. In the next section an overview of the history of the weak interactions in general, and atomic PNC in particular, is given. In section 3, Furry representation is introduced and applied to a calculation of a transition energy of a highly charged ion, Bi ". Section 4 describes the theory of cesium PNC, starting with low-order many-body perturbation theory (MBPT) methods, and then generalizing to all-orders methods based on coupled cluster theory. Section 5 closes the chapter with a brief description of the closely related field of atomic electric dipole moments. 

Rather than review MBPT, which will be dealt with in considerable detail when cesium PNC is treated later, it is useful at this point to instead directly introduce this same breakup in the framework of QED, which is done with a particular kind of interaction representation known as Furry representation [4]. The basic idea is to divide the complete Hamiltonian of QED into a noninteracting part Hq and an interaction term Hj that... 

The purpose of the preceding discussion of this particular transition in lithiumlike Bismuth was to show that use of the Furry representation firstly allows a consistent implementation of QED for the many-electron problem, with both correlation and radiative effects treated as Feynman diagrams, and secondly to show that when the extra expansion parameter 1/Z is present that extremely precise predictions result that agree well with experiment. There is no reason in principle, therefore, that QED cannot be applied to all atoms and molecules. In practice, however, without the rapid convergence provided by factors of 1/Z, the utility of this approach for neutral systems can be questioned. The best way to combine many-body methods and QED in this case is one of the forefront problems of the field. We now turn to a neutral system, the cesium atom, and describe the progress that has been made in the search for new physics in this much more challenging case. 

Investigation of the connection between the Many-Body Perturbation Theory (MBPT) approach and the Furry representation of Quantum Electrodynamics (QED) has been shown to allow a precise definition of QED effects.82 Every MBPT diagram has a corresponding Feynman diagram, but there are Feynman diagrams that have no MBPT counterpart. 

Feynman rules for the Green function. In the Furry picture, in addition to the standard Feynman rules in the energy representation (see [24,13]), the following vertices and lines appear (we assume that the Coulomb gauge is used)... 

Unless carefully implemented the representation of the Dirac spectrum obtained within the algebraic approximation may exhibit undesirable properties which are not encountered in non-relativistic studies. In particular, an inappropriate choice of basis set may obliterate the separation of the spectrum into positive and negative energy branches. So-called intruder states may arise, which are impossible to classify as being of either positive or negative energy character. The Furry bound state interaction picture of quantum electrodynamics is thereby undermined. 

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