Many-body quantum electrodynamics

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

Fock-Breit model, Ej fb and define all other energy corrections to this order of relativistic quantum electrodynamics, as a many-body effects, so that... 

The transition from a macroscopic description to the microscopic level is always a complicated mathematical problem (the so-called many-particle problem) having no universal solution. To illustrate this point, we recommend to consider first the motion of a single particle and then the interaction of two particles, etc. The problem is well summarized in the following remark from a book by Mattuck [18] given here in a shortened form. For the Newtonian mechanics of the 18th century the three-body problem was unsolvable. The general theory of relativity and quantum electrodynamics created unsolvable two-body and single-body problems. Finally, for the modem quantum field... 

Theoretical calculations of two-electron ion energy levels have been the topic of much research since the discovery of quantum mechanics. The contribution of relativistic effects via the Dirac equation and QED contributions has been intensely studied in the last three decades [1]. Two-electron systems provide a test-bed for quantum electrodynamics and relativistic effects calculations, and also for many body formalisms [2]. 

The earliest quantitative theory to describe van der Waals forces between two colloidal particles, each containing a statistically large number of atoms, was developed by Hamaker, who used pairwise summation of the atom-atom interactions. This approach neglects the multi-body interactions inherent in the interaction of condensed phases. The modem theory for predicting van der Waals forces for continua was developed by Lifshitz who used quantum electrodynamics [19,20] to account for the many-body molecular interactions and retardation within and between materials. Retardation is a reduction of the interaction because of a phase lag in the induced dipole response that increases with distance. 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

These three examples reflect various aspects of quantum electrodynamics theory. The electron anomalous magnetic moment follows from free-electron QED, the transition frequencies in hydrogen follow from bound-state QED, and, at least in principle, the relevant condensed matter theory follows from the equations of many-body QED. 

Highly-ionized atoms DHF calculations on isoelectronic sequences of few-electron ions serve as the starting point of fundamental studies of physical phenomena, though many-body corrections are now applied routinely using relativistic many-body theory. Relativistic self-consistent field studies are used as the basis of investigations of systematic trends in ionization energies [137-144], radiative transition probabilities [145-148], and quantum electrodynamic corrections [149-151] in few-electron systems. Increased experimental precision in these areas has driven the development of many-body methods to model the electron correlation effects, and the inclusion of Breit interaction in the evaluation of both one-body and many-body corrections. 

Quantum Electrodynamics and Many-body Perturbation Theory. - In his... 

In his work, Goldstone70 introduced the graphical techniques into many-body physics making use of Feynman-like diagrams. However, interactions were taken to be instantaneous and the effects of relativity were ignored. In recent years, the growing interest in the treatment of relativistic and quantum electrodynamic effects in atoms and molecules is necessitating the reintroduc-... 

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. 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

Keywords Covariance Many-body perturbation Quantum-electrodynamical effects Electron correlation Fine structure Heliumlike ions... 

We have presented a relativistically covariant many-body perturbation procedure, based upon the CEO and the GO. This represents a unification of the many-body perturbation theory and quantum electrodynamics. Applied to all orders, the procedure leads in the equal-time approximation to the BSE in the effective-potential form. By relaxing this restriction, the procedure is consistent with the full BSE. The new procedure will be of importance in cases where QED effects beyond first order in combination with high-order electron correlation are significant. 

H. M. Quiney, H. Skaane, I. P. Grant. Relativistic, quantum electrodynamic and many-body effects in the water molecule. Chem. Phys. Lett, 290 (1998) 473-480. 

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