Relativistic photon-electron interaction

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

The correct frame of description of interacting relativistic electrons is quantum electrodynamics (QED) where the matter field is the four-component operator-valued electron-positron field acting in the Fock space and depending on space-time = (ct, r) (x = (ct, —r)). Electron-electron interaction takes place via a photon field which is described by an operatorvalued four-potential A x ). Additionally, the system is subject to a static external classical (Bose condensed, c-number) field F , given by the four-potential (distinguished by the missing hat)... 

The precise form of this correction depends upon the gauge condition used to describe the electromagnetic field. In the Coulomb gauge, which has been employed more often in relativistic atomic structure, the electron-electron interactions come from one-photon exchange process and is sum of instantaneous Coulomb interaction and the transverse photon interaction. 

The retarded electron-electron interaction presented above arises from the first of the one-photon Feynman diagrams in figure 5.1. In terms of an expansion of the relativistic interactions in powers of 1 /c, this interaction contains the lowest-order terms. As pointed out above, the Breit interaction contains all terms of order c . After the Breit interaction, the lowest-order interactions come from the other two one-photon diagrams, the vacuum polarization and self-energy terms, which are 0 c ). The energy contribution from these two terms is called the Lamb shift, after its discoverer W. E. Lamb Jr. (1952), and its calculation has been an important testing ground for QED theories. 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

A last possibility to produce relativistic electrons in the cluster atmospheres is due to the interaction of high-E gamma-rays with low energy (CMB/IR) photons, with the subsequent production of e pairs (Timokhin et al. 2003). The main uncertainty of this last model is the unknown origin of the very high-E gamma-rays in clusters. 

This constant explains far more than the appearance of the hydrogen atom s spectrum, however. The fine-structure constant is recognized as one of the most important constants in physics. We know, for example, that the fine-strucmre constant is a measure of the strength of the interaction between photons and electrons. Thus, this constant will appear in all simations that reveal quantum and relativistic properties of electrically charged particles. If electrons and light did not interact, the fine-structure constant would be zero. 

Following a leptonic model, gamma rays may arise from inverse Compton (IC) processes when the relativistic electrons of the jet interact with external photons from the massive companion star. If the jets are hadronic, the gamma rays may come from the decay of neutral pions in the interaction between protons from the jet and ions from the wind of the companion star. 

In view of its field-theoretical basis, this energy functional not only accounts for the relativistic kinematics of both electrons and photons, but, in principle, also for all radiative corrections. With the Ritz principle,1 avoiding the question of interacting u-representability (Dreizler and Gross 1990), we may then formulate the basic... 

To not leave the reader with the impression that these extensions are trivial, let us recall that a relativistic reformulation caimot ignore the virtual creation of electron-positron pairs nor the fact that the Breit interaction involves the exchange of transverse photons. 

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