Electron-positron virtual pairing

The anomalous contribution to the magnetic moment of an electron has been explained by the quantum electrodynamic theory. The additional contribution—the radiative correction —arises from the interaction of the electron-positron virtual pair emitted and absorbed by the real electron. A theoretical expression in terms of the fine structure constant a is... 

The magnetic field is oriented perpendicular to the plane inscribed by a completely polarized electron-positron pair [17]. The virtual electron-positron... 

At high values of Lorentz factors the probability of passage of an atom through a layer of matter becomes greater than the one that follows from the usual exponential dependence. This phenomenon was predicted in [8] and was given the name superpenetration . The quantitative theory of superpenetration was developed in [9,10,11]. For ultrarelativistic A2e, the time of formation from the virtual electron-positron pair is strongly dependent on the thickness of the target [12],... 

Taking into account the time of formation A2e from the virtual electron-positron pair for ultrarelativistic A2e changes very strongly the effective value of the thickness of target for A2e production [12],... 

Fig. 1. Feynman diagrams representing various contributions to the Lamb shift. A solid line represents an electron, a wavy line a virtual photon and a cross denotes exchange of a Coulomb photon (a) Leading self-energy term (b) One-loop vacuum polarisation term. The loop represents a virtual electron-positron pair (c) Some diagrams contributing to the two-loop binding correction...

Table 5. Lamb shift contribution for the ground state of 208Pb81+ i0n (in eV). The notations are the same as in Table 4. The finite nuclear size correction is calculated for a Fermi distribution with (r2 1,/2 = 5.505 0.001 fm. The SESE (a) (irred) correction is obtained by an interpolation from the known values for Z = 70, 80,92. The inaccuracy of the Uehling approximation for VPVP (f) and S(VP)E corrections is neglected. The zero value presented for the nuclear polarization is due to the cancellation of the usual nuclear polarization [35] with the mixed nuclear polarization (NP)-vacuum polarization correction [36]. The latter effect arises when the nucleus interacts with a virtual electron-positron pair. For lead, due to the collective monopole vibrations, specific for this nucleus, mixed NP-VP effect becomes rather large. Therefore, the nuclear polarization effects which otherwise limit very precise Lamb shift predictions are almost completely negligible for 208Pb, making this ion especially suitable for the most precise theoretical predictions...

Heat capacity nearly constant Cy - (5/2)R Dissociation begins, Cv rises Dissociation virtually complete. System now consists of 2 moles of H Cv - 3R Ionization of atoms noticeable, Cv rising Ionization virtually complete. The system consists of 2 moles of protons and 2 moles of electrons Cv has leveled off near 6 R Cv remains nearly constant Collisions of particles produce electron-positron pairs, Cv rising Proton-proton collisions produce pions, Cv rising... 

The other principal radiative correction is the vacuum polarization (Fig. 3 b)). It describes the interaction of a fermion with virtual electron-positron pairs which can be thought present in the vacuum for short times without violating the energy-time uncertainty relation. If external fields are present, these virtual pairs are influenced and act like a polarizable medium. Therefore the Coulomb interaction of the nucleus with the electrons is modified which leads to an energy shift compared to the pure Coulomb potential energy eigenvalue. 

A further term, which has no analogue in hydrogen, arises in the fine structure of positronium. This comes from the possibility of virtual annihilation and re-creation of the electron-positron pair. A virtual process is one in which energy is not conserved. Real annihilation limits the lifetimes of the bound states and broadens the energy levels (section 12.6). Virtual annihilation and re-creation shift the levels. It is essentially a quantum-electrodynamic interaction. The energy operator for the double process of annihilation and re-creation is different from zero only if the particles coincide, and have their spins parallel. There exists, therefore, in the triplet states, a term proportional to y 2(0). It is important only in 3S1 states, and is of the same order of magnitude as the Fermi spin-spin interaction. Humbach [65] has given an interpretation of this annihi-... 

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. 

Qvac is the total charge of the vacuum, which vanishes for free electrons, but is finite in the presence of an external field (the phenomenon of vacuum polarization). Note that whilst Q is conserved for all processes, the total number of particles need not be it is always possible to add virtual states incorporating electron-positron pairs without changing Q. The neglect of such terms in the total wavefunction of an n-electron system is called the no-pair approximation. 

The simplest approximation to this Fock space theory consists in projecting the n-particle Hamiltonian to electronic states, i.e. to ignore the creation of virtual electron-positron pair states, whence the name no-pair theory. The next step after a no-pair theory would be a formalism, in which an n-electron state mixes, e.g. with an (n -1- l)-electron-1-positron state etc.. Not the particle number, but the charge is a constant of motion. In this way one takes care of vacuum polarization, which is a real physical effect. It is, however, not recommended to treat it in such a brute-force way, but rather to use the apparatus of QED. 

Although the proper point of departure for relativistic atomic structure calculations is quantum electrodynamics (QED), very few atomic structure calculations have been carried out entirely within the QED framework. Indeed, almost all relativistic calculations of the structure of many-electron atoms are based on some variant of the Hamiltonian introduced a half century ago by Brown and Ravenhall [1] to understand the helium fine structure. By decoupling the electron and radiation fields in QED to order a (the fine-structure constant) using a contact transformation. Brown and Ravenhall obtained a relativistic momentum-space Hamiltonian in which the electron-electron Coulomb interaction was surrounded by positive-energy projection operators. Owing to the fact that contributions from virtual electron-positron pairs are automatically projected out of... 

Later, when making comparisons with nonrelativistic calculations, we subtract the electron rest energy mc firom e. ) The choice of the potential U r) is more or less arbitrary one important choice being the (Dirac) Hartree-Fock potential. Eigenstates of Eq. (1) fall into three classes bound states with —mc < k < rn< , continuum states with > m( , and negative energy (positron) states ej < —me . Since contributions from virtual electron-positron pairs are projected out of the no-pair Hamiltonian, we will be concerned primarily with bound and continuum electron states. 

The first and second terms in the right-hand-side of Ekj. (186) come from virtual electron-electron and positron-positron pairs in the intermediate states, respectively, while the third and fourth terms are from electron-positron pairs. Dirac compared with... 

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