Bound-state quantum electrodynamics

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 spectrum of hydrogen and one-electron ions provides a direct test of bound-state quantum electrodynamics. Except for finite nuclear-size and mass (recoil)... 

For all these reasons Ps is an ideal candidate for precision tests of bound state quantum electrodynamics (QED). 

Positronium and muonium are the simplest atoms composed of leptons and are of great importance for testing quantum electrodynamics and, more generally, the modem standard theory. In particular, these atoms are useful for studying bound state quantum electrodynamics and for determining the properties of the positron as antiparticle to the electron and of the muon as a "heavy" electron. 

The precise measurement of the g-factor of the electron bound in a hydrogen-like ion is a sensitive test of bound-state Quantum Electrodynamics (QED) at high fields. Highly charged ions are ideal for such studies, because the electromagnetic... 

Nonrelativistic quantum electrodynamics (NRQED) [11] is an attempt to combine the simplicity of the quantum mechanical description with the power and rigor of field theory. The idea is to write ordinary relativistic quantum electrodynamics in the form of a nonrelativistic expansion with a Lagrangian containing vertices with arbitrary powers of fields. This is useful if we want to consider essentially nonrelativistic processes, like nonrelativistic bound states and threshold phenomena. In such a physical situation the dominant dynamics is nonrelativistic, and the calculations could be in principle simplified if... 

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. 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

Abstract. Muonium is a hydrogen-like system which in many respects may be viewed as an ideal atom. Due to the close confinement of the bound state of the two pointlike leptons it can serve as a test object for Quantum Electrodynamics. The nature of the muon as a heavy copy of the electron can be verified. Furthermore, searches for additional, yet unknown interactions between leptons can be carried out. Recently completed experimental projects cover the ground state hyperfine structure, the ls-2s energy interval, a search for spontaneous conversion of muonium into antimuonium and a test of CPT and Lorentz invariance. Precision experiments allow the extraction of accurate values for the electromagnetic fine structure constant, the muon magnetic moment and the muon mass. Most stringent limits on speculative models beyond the standard theory have been set. 

We will review here experimental tests of quantum electrodynamics (QED) and relativistic bound-state formalism in the positron-electron (e+,e ) system, positronium (Ps). Ps is an attractive atom for such tests because it is purely leptonic (i.e. without the complicating effects of nuclear structure as in normal atoms), and because the e and e+ are antiparticles, and thus the unique effects of annihilation (decay into photons) on the real and imaginary (related to decay) energy levels of Ps can be tested to high precision. In addition, positronium constitutes an equal-mass, two-body system in which recoil effects are very important. 

J.R. Sapirstein and D.R.. Yennie Theory of Hydrogenic Bound States . In Quantum Electrodynamics, ed. by T. Kinoshita (Singapore, World Scientific 1990) pp. 

Spectroscopy of positronium provides a sensitive test of the bound state theory in Quantum Electrodynamics (QED). Because of the small value of the electron mass, the effects of strong interactions are negligible compared to the accuracy achieved in present experiments. For this reason positronium represents a unique system which can, in principle, be described with very high accuracy by means of QED only. Tests of QED predictions are made possible thanks to a high precision of positronium spectroscopic measurements. 

The difference in the energy of the 2 Sand 2 Pjy2 levels in hydrogenic atoms is a purely electrodynamic effect due to the interaction of the bound electron with the quantized electromagnetic field. The measurement of this splitting was a major stimulus for the development of renormalization theory and still provides an important test of Quantum Electrodynamics. The precise measurement of this split-ting is difficult because of the short radiative lifetime of the 2 P 2 state. 

Such comparisons promise interesting tests of QED. Unfortunately, however, the theory of hydrogen is no longer simple, once we try to predict its energy levels with adequate precision [36]. The quantum electrodynamic corrections to the Dirac energy of the IS state, for instance, have an uncertainty of about 35 kHz, caused by numerical approximations in the calculation of the one-photon self-energy of a bound electron, and 50 kHz due to uncalculated higher order QED corrections. 

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. 

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-... 

Since the spectrum of the DCB Hamiltonian is not bounded from below it is not possible to optimize the wave function by minimization of the energy. The unphysical unboundedness is due to the fact that not all possible normalizible antisymmetric wave functions of N coordinates are states of an N-electron system. The set of possible solutions also contains wavefunctions in which one or more negative energy levels are occupied and it is the mixing with such states that gives rise to unphysical arbitrarily low energies. One needs the second quantization formalism of quantum electrodynamics (QED) for a proper treatment of these states. As this is discussed in more depth elsewhere in this... 

An overview of relativistic density functional theory (RDFT) is presented with special emphasis on its field theoretical foundations and the construction of relativistic density functionals. A summary of quantum electrodynamics (QED) for bound states provides the background for the discussion of the relativistic generalization of the Hohenberg-Kohn theorem and the effective single-particle equations of RDFT. In particular, the renormalization procedure of bound state QED is reviewed in some detail. Knowledge of this renormalization scheme is pertinent for a careful derivation of the RDFT concept which necessarily has to reflect all the features of QED, such as transverse and vacuum corrections. This aspect not only shows up in the existence proof of RDFT, but also leads to an extended form of the single-particle equations which includes radiative corrections. The need for renormalization is also evident in the construction of explicit functionals. 

Unless carefully implemented the representation of the Dirac spectrum obtained within the algebraic approximation may exhibit undesirable properties which are not encountered in non-relativistic studies. In particular, an inappropriate choice of basis set may obliterate the separation of the spectrum into positive and negative energy branches. So-called intruder states may arise, which are impossible to classify as being of either positive or negative energy character. The Furry bound state interaction picture of quantum electrodynamics is thereby undermined. 

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