Radiative corrections problems with

Contributions to the energy which depend only on the small parameters a. and Za. are called radiative corrections. Powers of a arise only from the quantum electrodynamics loops, and all associated corrections have a quantum field theory nature. Radiative corrections do not depend on the recoil factor m/M and thus may be calculated in the framework of QED for a bound electron in an external field. In respective calculations one deals only with the complications connected with the presence of quantized fields, but the two-particle nature of the bound state and all problems connected with the description of the bound states in relativistic quantum field theory still may be ignored. 

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. 

One of the aims of this chapter, then, is to discuss the problem of calculating a property of a many-electron atom with suflicient precision so that the new physics of radiative corrections can be studied. The challenge to many-body theory is quite specific. As will be discussed below, properties of cesium, the atom in which the most accurate PNC measurement has been made [5] must be calculated to the fraction of a percent level to accurately study PNC and radiative corrections to it can this level in fact be reached by modern many-body methods While great progress has been made, the particular nature of this problem, in which relativity has to be incorporated from the start, and a transition between two open-shell states calculated in the presence of a parity-nonconserving interaction, has not permitted solution of the many-body problem to the desired level. It may well be that a reader of this chapter has developed techniques for some other many-electron problem that are of sufficient power to resolve this issue this chapter is meant to clearly lay out the nature of the calculation so that the reader can apply those techniques to what is, after all, a relatively simple system by the standards of quantum chemistry, an isolated cesium atom. 

While there have been a number of papers written on the basic structure problem since Ref. [40] appeared, [55], none of them go qualitatively further than the calculations of that work. Curiously, an experimental paper, [5], claims to have reduced the theoretical error through comparison with new measurements of transition matrix elements. It is the author s opinion, however, that this essentially semiempirical approach is dangerous, and prefers to leave the 1 percent error estimate unchanged at present. However, there are three places in which considerable activity has taken place that we address in turn, the vector polarizability / , the Breit interaction, and radiative corrections. 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

Calculation of the state-dependent nonlogarithmic contribution of order a(Za) is a difficult task, and has not been done for an arbitrary principal quantum number n. The first estimate of this contribution was made in [63]. Next the problem was attacked from a different angle [64, 65]. Instead of calculating corrections of order a(Za) an exact numerical calculation of all contributions with one radiative photon, without expansion over Za, was performed for comparatively large values of Z (n = 2), and then the result was extrapolated to Z = 1. In this way an estimate of the sum of the contribution of order a(Za) and higher order contributions a(Za) was obtained (for n = 2 and Z = 1). We will postpone discussion of the results obtained in this... 

In the case of the polarization insertions the calculations may be simplified by simultaneous consideration of the insertions of both the electron and muon polarization loops [18, 19]. In such an approach one explicitly takes into account internal symmetry of the problem at hand with respect to both particles. So, let us preserve the factor 1/(1 - - m/M) in (9.9), even in calculation of the nonrecoil polarization operator contribution. Then we will obtain an extra factor m /m on the right hand side in (9.12). To facilitate further recoil calculations we could simply declare that the polarization operator contribution with this extra factor m /m is the result of the nonrecoil calculation but there exists a better choice. Insertion in the external photon lines of the polarization loop of a heavy particle with mass M generates correction to HFS suppressed by an extra recoil factor m/M in comparison with the electron loop contribution. Corrections induced by such heavy particles polarization loop insertions clearly should be discussed together with other radiative-recoil... 

Our problem was now that when QED corrections were added the agreement was reduced. However, an experimental pressure dependence of the measured transition wave lengths was found[35] which increases the uncertainty of the experimental data. Our theoretical transition energies do thus appear to be within the current experimental errorbars. The calculated radiative decay rates agree well with experimental work[5]. 

Geometric Optics Results with Emission. When the temperature of a semitransparent layer is large, emission of radiation becomes significant, and the problem of radiative transfer becomes more complex. The change in refractive index at each interface causes total internal reflection of radiation in the medium with higher refractive index at the boundary. This effect must be treated in the RTE at the boundary of the medium, and diffuse boundary conditions are no longer correct for the exact solution of this type of problem. Various approaches have been attempted. 

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