Quantum electrodynamical corrections to the fine structure of helium

Theory of strong interactions

A boundary condition at t = ± ∞ (t being the “relative” time variable) is obtained for the four-dimensional wave function of a two-body system in a bound state. It is shown that this condition implies that the wave function can be continued analytically to complex values of the “relative time” variable; similarly the wave function in momentum space can be continued analytically to complex values of the “relative energy” variable P 0. In particular one is allowed to consider the wave function for purely imaginary values of t, or respectively p 0, i.e., for real values of x 4 = ict and p 4 = ip 0. A wave equation satisfied by this function is obtained by rotation of the integration path in the complex plane of the variable p 0, and it is further shown that the formulation of the eigenvalue problem in terms of this equation presents several advantages in that many of the ordinary mathematical methods become available.

Properties of Bethe-Salpeter Wave Functions

Determination of eh, Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants

http://cds.cern.ch/record/427076/files/0002158.pdf

Theory of light hydrogenlike atoms

A functional integro-differential equation for the electron-positron Green's function is derived from a consideration of the effect of sources of the Dirac field. This equation contains an electron-positron interaction operator from which functional derivatives may be eliminated by an iteration procedure. The operator is evaluated so as to include the effects of one and two virtual quanta. It contains an interaction resulting from quantum exchange as well as one resulting from virtual annihilation of the pair. The wave functions of the electron-positron system are the solutions of the homogeneous equation related to the Green's function equation. The eigenvalues of the total energy of the system may be found by a four-dimensional perturbation technique. The system bound by the Coulomb interaction is here treated as the unperturbed situation. Numerical values for the spin-dependent change of the energy from the Coulomb value in the ground state are finally obtained accurate to order $\ensuremath{\alpha}$ relative to the hyperfine structure ${\ensuremath{\alpha}}^{2}$ Ry. The result for the singlet-triplet energy difference is $\ensuremath{\Delta}{W}_{\mathrm{ts}}=\frac{1}{2}{\ensuremath{\alpha}}^{2}{\mathrm{Ry}}_{\ensuremath{\infty}}[\frac{7}{3}\ensuremath{-}(\frac{32}{9}+2\mathrm{ln}2)\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}]=2.0337\ifmmode\times\else\texttimes\fi{}{10}^{5}\frac{\mathrm{Mc}}{sec}.$ Theory and experiment are in agreement.

Electrodynamic displacement of atomic energy levels. iii. the hyperfine structure of positronium

A general method is described whereby the low-energy behavior of collision processes can be investigated systematically. The basic procedure is the construction of the renormalized Green's functions for the participating particles in the presence of external sources. For processes involving the collision of a boson with a fermion,---photon scattering, meson scattering, and photoproduction of mesons---one requires the propagation function of a nucleon in the presence of external electromagnetic and meson fields, as well as the propagator for a meson in an electromagnetic field. The rather direct relation of these functions to the various transition matrices is derived. The possible forms of the Green's functions are limited by the principles of Lorentz, gauge, and charge conjugation invariance. Their detailed structure is determined in part by the requirement that they describe particles of known mass, charge, magnetic moment, and mesic coupling constant. These conditions are sufficient to imply a number of theorems. For example, it is shown that the amplitude for photon-proton scattering is given correctly up to terms of first order in the photon energy by a Born approximation calculation, if one assigns to the proton its experimental charge and magnetic moment. If one includes the meson-nucleon coupling constant in this set of parameters, then perturbation theory is also valid for the leading terms in a momentum-energy expansion of the $P$-wave in meson-nucleon scattering, the $S$- and $P$-waves associated with the nucleon current in photomeson production, and the entire meson current for the same process.The methods employed to establish the theorems are extended in order to provide a phenomenological framework for the description of the experiments for energies low enough so that the expansions in boson energy are still valid, but for which the deviations from the theorems are of practical significance. Thus, it is suggested that for the description of photon scattering one should attribute an electric and a magnetic polarizability to the nucleon. The description of the $P$-wave in meson-nucleon scattering also requires the introduction of two additional parameters. From the manner in which these enter the scattering amplitude, it can be concluded that the phase shift in the state of angular momentum and isotopic spin $\frac{3}{2}$ is enhanced compared to its Born value, if only the phase shifts in the other states deviate in the opposite direction. Finally, parameters are introduced to describe the $S$-wave in meson-scattering, and the basis for a phenomenological description of photoproduction is indicated.

Low-Energy Theorems for Renormalizable Field Theories

The vacuum fluctuations of the photon and pair fields modify the interaction of an electron with an electromagnetic field. The effects on the energy levels are conveniently described in terms of the mass operator and the vacuum polarization potential. An operator calculus for handling the mass operator is used to separate the contribution of low energy quanta, for which the external electromagnetic field may not be treated as weak. The remainder is given as a power series in the field. The additional displacement of order $Z\ensuremath{\alpha}$ with respect to the published values for the Lamb shift is calculated by specializing the external field to the Coulomb field of the nucleus and by taking the matrix element of the operators in an $S$-state of a hydrogen-like atom. It is found that the $\mathrm{nS}$-level is raised by $\frac{{Z}^{5}{\ensuremath{\alpha}}^{4}}{{n}^{3}}\mathrm{Ry}\left(1+\frac{11}{128}\ensuremath{-}\frac{1}{2}\mathrm{ln}2+\frac{5}{192}\right).$ The theoretical value for the Lamb shift of an atom with infinitely heavy nucleus becomes 1058.42 Mc/sec. Effects of finite nuclear mass and size change this value to 1057.8 Mc/sec and 1058.9 Mc/sec for hydrogen and deuterium, respectively.

Electrodynamic Displacement of Atomic Energy Levels. II. Lamb Shift

The general method of deducing the Tamm-Dancoff equaltimes formalism, as generalized by L\'evy, from the relativistic two-body equation of Bethe-Salpeter and Schwinger is given. Only processes which are finite ab initio are considered. The essence of the procedure is the relation between a set of conventional matrix elements of the Tamm-Dancoff formalism and the Feynman diagram which summarizes them; this relationship provides a convenient guide for enumerating all matrix elements of a specified type and precludes the possibility of omission of any members of the set. Rules are also given for writing down any matrix element. The method is then applied to the derivation of the fourth-, sixth-, and eighth-order adiabatic potentials on the symmetrical pseudoscalar-pseudoscalar theory. Some discrepancies with the results of L\'evy are noted: In connection with the fourth-order potential these are first, that a more careful treatment of the energy denominators of the leading two-pair terms brings to light contributions that cancel with all other two-pair matrix elements that are of relative order $\frac{\ensuremath{\mu}}{M}$ compared to the leading ones; second, that the one-pair terms do not vanish but yield a repulsive interaction which substantially alters the qualitative picture of the fourth-order potential; third, that for the no-pair terms the result should agree with the previously calculated fourth-order potential for the pseudoscalar-pseudovector theory. The sixth- and eighth-order results are also in disagreement with L\'evy. Finally, an analysis of the problem of many-particle forces is given and explicit results obtained for the leading terms of the three- and four-particle forces as well as for certain smaller contributions to the three-particle interaction.

Abraham Klein

Papers

Electrodynamic displacement of atomic energy levels. iii. the hyperfine structure of positronium

Low-Energy Theorems for Renormalizable Field Theories

Electrodynamic Displacement of Atomic Energy Levels. II. Lamb Shift

The Tamm-Dancoff Formalism and the Symmetric Pseudoscalar Theory of Nuclear Forces

Electrodynamic Displacement of Atomic Energy Levels