Quantum electrodynamics vacuum polarization

When innermost core shells must be treated explicitly, the four-component versions of the GREGP operator can be used, in principle, together with the all-electron relativistic Hamiltonians. The GRECP can describe here some quantum electrodynamics effects (self-energy, vacuum polarization etc.) thus avoiding their direct treatment. One more remark is that the... 

Heitler spoke on the quantum theory of damping, which is a heuristic attempt to eliminate the infinities of quantum field theory in a relativistic invariant manner, Peierls spoke of the problem of self-energy, and Op-penheimer gave an account of the developments of the last years in electrodynamics in which he discussed the problem of the vacuum polarization and charge renormalization with special reference to the recent work of Schwinger and Tomonaga. 

The vacuum polarization is well known to have an analog in quantum electrodynamics [46], the photon self-energy. The latter has no classical analog on the U(l) level, but one exists on the 0(3) level, thus saving the correspondence principle. The classical vacuum polarization on the 0(3) level is transverse and vanishes when oo = 0. It is pure transverse because, as follows, the hypothetical E0) field is zero on the 0(3) level... 

The dominant interaction within the muonium atom is electromagnetic. This can be treated most accurately within the framework of bound state Quantum Electrodynamics (QED). There are also contributions from weak interaction which arise from Z°-boson exchange and from strong interaction due to vacuum polarization loops with hadronic content. Standard theory, which encompasses all these forces, allows to calculate the level energies of muonium to the required level of accuracy for all modern precision experiments1. 

The very-short-time behavior of micro-cavity quantum electrodynamics for characterizing micro-laser arrays for high speed applications, where highly nonlinear coupling of the vacuum fluctuations and the atomic polarizations exist, requires the full time-dependent quantum treatment discussed here. 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

The theoretical energy levels are determined to high accuracy by the Dirac eigenvalue, quantum electrodynamic effects such as the self energy and vacuum polarization, finite-nuclear-size corrections, and nuclear motion effects. 

The leading quantum electrodynamic effects to be accounted for in electronic structure calculations are the radiative corrections known as electron self-energy interaction and vacuum polarization. For the energy of electronic systems, the latter is usually small compared to the former, but only the latter can be expressed in terms of an effective additive potential to be included in the electronic structure calculations. The total vacuum polarization potential can be expanded into a double power series in the fine structure constant a and the external coupling constant Za. The lowest-order term, the Uehling potential, can be expressed as [110-112] ... 

This statement implies that not only the Coulomb interaction is included in Er and Exc but also the (retarded) Breit interaction. It thus points at the fact that a consistent and complete discussion of many-electron systems and consequently of RDFT must start from quantum electrodynamics (QED). RDFT necessruily has to reflect the various features of QED, both on the formal level and in the derivation of explicit functionals. The most important differences to the noiu-elativistic situation arise from the presence of infinite zero point energies and ultraviolet divergencies. In addition, finite vacuum corrections (vacuum polarization, Casimir energy) show up in both fundamental quantities of RDFT, the four current and the total energy. These issues have to be dealt with by a suitable renormalization procedure which ultimately relies on the renormalization of the vacuum Greens functions of QED. The first attempt to take... 

The Dirac equation did not take all the physical effects into account. For example, the strong electric field of the nucleus polarizes a vacuum so much that electron-positron pairs emetge from the vacuum and screen the electron-nucleus interaction. The quantum electrodynamics (QED) developed by Feynman, Schwinger, and Tomonaga accounts for this and similar effects and brings theory and experiment to an agreement of unprecedented accuracy. 

Table 3.1. Contributions of various physical effects (non-relativistic, Bieit, QED, and beyond QED, distinct physical contributions shown in bold) to the ionization energy and the dipole polarizability a of the helium atom, as well as comparison with the experimental values (all quantities are expressed in atomic units i.e.. e = 1. fi = 1, mo = 1- where iiiq denotes the rest mass of the electron). The first column gives the symbol of the term in the Breit-Pauli Hamiltonian [Eq. (3.72)] as well as of the QED corrections given order by order (first corresponding to the electron-positron vacuum polarization (QED), then, beyond quantum electrodynamics, to other particle-antiparticle pairs (non-QED) li,7T,. ..) split into several separate effects. The second column contains a short description of the effect. The estimated error (third and fourth columns) is given in parentheses in the units of the last figure reported. 

One should note that the relativistic effect on the core IPs can already be seen for Ne and progressively increases for heavier atoms. Thus, to compare theoretical results with the experimental data for the ionization core levels one needs to account for relativistic contributions even for relatively light systems. The still existing disagreement between the experimental data and theoretical lOTC results for the inner Is core IPs can be attributed to the neglection of other higher-order relativistic quantum electrodynamic contributions such as Breit, self energy and vacuum polarization terms [34]. 

The Breit corrections are sometimes classified as nonradiative effects in contrary to the radiative affects which are true quantum-electrodynamical effects due to the electron self energy and vacuum polarization [30-32]. 

As shown in O Fig. 28.3, in the upper region of the X-ray transitions, the particle may interact with and thus probe the electronic shell, while in the lower one it already probes the atomic nucleus as its wave function partially penetrates it. The region between is also interesting as those transitions are weakly influenced both by the remaining or recombined electrons and by the field of the nucleus there one can get information on the properties of the captured particle or test the effects of quantum electrodynamics. As an example, O Fig. 28.7 presents (Horvath and Lambrecht 1984) the various energy terms in the 5g 9/2 4f 7/2 transition in muonic lead here the contribution of the Lamb shift, which is the dominant such term in ordinary atoms, is quite small as compared to the other vacuum polarization effects. 

J. Schwinger. Quantum Electrodynamics. II. Vacuum Polarization and Self-Energy. Phys. Rev., 75(4) (1949) 651-679. 

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. 

Quantum electrodynamic (QED) effects are not included in the present work. A recent calculation of these effects for s electrons [66] gave estimates of about 0.04 eV for the ionization potential of the Tl + 6s electron (self energy 0.05 eV, vacuum polarization —0.01 eV) and 0.06 eV for the El 11 Is electron (self energy 0.09 eV, vacuum polarization —0.03 eV). Since the p electrons responsible for the transitions discussed in this work exhibit much weaker penetration into the nucleus, QED effects here are expected to be considerably smaller, at most 0.01-0.02 eV, within the error limits of the method ( 0.05 eV) estimated by comparing calculated and experimental results for thorium (Tables 2.5 and 2.6). 

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