Theoretical studies of atom-atom collision processes which may be relevant to the interpretation of astrophysical phenomena are reviewed. Work on excitation, ionization, and electron capture in atomic hydrogen by fast protons and ${\mathrm{He}}^{+}$ ions is summarized. Processes involving slow collisions that are listed include radiative charge transfer and association, associative detachment, Penning ionization, spin-change, collision-induced deactivation, mutual neutralization, and rotational excitation.

The method of self-adjoint ladder operators introduced in Part I has been used to study the isotropic harmonic oscillator and both the nonrelativistic and relativistic (Dirac) Kepler problems in a space of $n$ dimensions. In addition, a brief discussion of the dynamical (noninvariance) group $0\left(\right(n+1\mathrm{}), 1)$ of the $n$-dimensional nonrelativistic Kepler problem has been included.

The theory of self-adjoint ladder operators is outlined, a comparison being made with the more usual type of ladder operator. It is shown how the method applies to the $n$-dimensional orbital angular momentum problem, for which it is found that the ladder operator may be expressed as a linear combination of the components of angular momentum, the expansion coefficients being the elements of a generalized spin algebra. Its relationship to the Infeld-Hull factorization method is discussed, the factorization appearing as part of a more general scheme. The physical significance of the ladder operators is emphasized through their application to the relativistic quantum theory of Dirac.

The Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms $\Delta E$ are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless). Starting again with a degenerate version of BW perturbation theory, these methods are used to obtain the Bloch-Horowitz energy expansion, as well as the corresponding wave function, expectation-value, and transition-amplitude expansions. A "reduced" form of the Block-Horowitz expansion is described, and also a "completely linked" version. The latter is suggested as a tool for investigating superfiuid phenomena in nuclear matter, and for establishing contact with the Landau theory of Fermi liquids. The physical interpretation of these expansions is carefully studied, especially with regard to nuclear applications, to determine how they handle such "physical" features as antisymmetry, self-energy effects, wave-function renormalizations, and the distinction between "true" and "model" single-particle occupation probabilities. The problem of a correct theoretical definition for the shell-model potential is carefully examined, and a specific theory is presented. These expansions are seen to form a convenient and very powerful set of tools for studying the structure of actual nuclei.

This work reviews some recent developments in the theory of nuclear matter. Assuming familiarity with the basic Brueckner-Goldstone theory described in the preceding article by B. Day, it is first shown that the Brueckner-Goldstone series does not converge in powers of the reaction matrix, and that the perturbation series for the binding energy has to be rearranged in powers of the density $\rho$. Physical reasons and actual estimates are provided for expecting convergence in powers of $\rho$.

A detailed theory is outlined for the evaluation of the three-body energy, which gives the ${\rho }^2$ term. Attention is paid to both the momentum dependence of the reaction matrix and the tensor nature of nuclear forces. Finally, the last section is devoted to the choice of the single-particle potential energies, suitably designed so as to absorb most of the four-body and higher cluster terms.

The basic ideas of the Brueckner-Goldstone theory of nuclear matter are presented in a simple way. The treatment is aimed at beginners and nonspecialists. It is supposed to provide the necessary background for the review article by Bethe and Rajaraman which follows this paper. Therefore, the discussion is limited to a few important topics, and these are considered in some detail.

The Goldstone expansion is presented (but not derived) and the construction and evaluation of the Goldstone diagrams are explained. The reaction matrix and the correlated two-body wave function are defined, and their properties are discussed. The reference-spectrum method for calculating the reaction matrix is derived, and its use is illustrated. Finally, the related topics of convergence and the definition of single-particle energies are considered. The choice of the single-particle potential energy for occupied states is treated in detail. (Intermediate-state energies will be discussed by Bethe and Rajaraman.) The reason for the divergence of the perturbation series for the binding energy is exhibited; and this series is rearranged into a convergent expansion, for which the density plays the role of small parameter.

We discuss a number of questions relating to phase-shift representations of nucleon-nucleon scattering in an energy range where inelastic channels are open and may contribute significantly to such representations. While attention will be directed specifically toward nucleon-nucleon scattering above 400 MeV, we see the relevance of such questions being applied to similar strongly interacting systems.

Historically it has become a habit in our Laboratory that the phase-shift analysis is carried out by the experimenters. Thus, each experimental investigation is completed by improving the earlier results of the phase-shift analysis and thus, the phase-shift analysis may be considered as an original method for the treatment of experimental data. The phase-shift analysis turns out to be a means first, to test how well new experimental results agree with the whole collection of earlier known data and, second, to more accurately plan further experiments. In our conditions a phase-shift analysis program is used at one energy. The program implies the following assumptions: (1) There is a charge independence of nuclear forces. (2) Nucleon interaction in states with high orbital momenta is correctly described with a one-pion Feynman diagram. (3) Orbital momenta, beginning from those for which a one-pion approximation can be justified, are determined keeping in mind that in the one-pion approximation, polarization in nucleon-nucleon scattering is equal to 0. (4) Pions are produced, mainly, from initial states with total isotopic spin 1. Results of our analyses will be given.

We present a survey of recent works concerning the phase-shift analysis of $p-p$ scattering in the energy region from 400 MeV to a few GeV. The method of analysis and results are described. Physical implications of them are also discussed. A new scheme of analysis is proposed. It modifies the previous method in fixing the imaginary phase shifts to the values from the one-pion model with unitarity. Reanalysis of the $p-p$ data at 970 MeV shows the validity of the new scheme.

In the past several years several bits of evidence have been collected for the presence of a $0^{+}$ isoscalar $S$-wave $\pi -\pi$ resonance. In this paper we review the previous evidence in the light of recent work. For low $\pi -\pi$ masses the decay of the $K^{+}$ meson, $K^{+}\to {\pi }^{+}+{\pi }^{-}+e^{+}+\nu$ gives useful information on the $I=0\mathrm{}$ phase shift. To gain information at higher mass values, a large collection of data from many experiments on the reactions involving pion production in $\pi -p$ collision has been made. Both the data on $K$ decay and $\pi -\pi$ collisions indicate a positive $I=0\mathrm{}$ phase shift at low energies followed by a gradual rise in the phase shift. Although one cannot rule out a resonant behavior near the $\rho$ mass, it seems unlikely. The $I=2\mathrm{}$ phase shift seems to be negative, small (∼20°) and slowly varying. Our results seem to differ from predictions based on current algebra.

A possible application of polarized proton targets and polarized proton and antiproton beams to the direct testing of $T$, $P$, $C$ invariance and the $\mathrm{CPT}$ theorem is considered. Procedure for direct determination of the $T$-violated amplitude $T$ from experimental data is given. Deviations from symmetry properties for polarization tensors in elastic scattering due to the bremsstralung effect are considered. The $P$ invaraiance can be tested both in the up-down asymmetry measurement and in the total cross-section measurements using the polarized beam and the polarized target. The measurement of the total polarization cross section ${\sigma }_{p1p2\mathrm{}}$ is of interest for testing higher isotopic symmetries.

Theoretical models of the nucleon-antinucleon interaction are surveyed in the few-hundred MeV region. Their relation to the nucleon-nucleon interaction, and any light they may shed on it, are discussed.

Knowledge of the two-nucleon interaction can be used as a proving ground for general theories of elementary particles. Within this context three aspects of the nucleon-nucleon problem are examined: (1) the tests of some group theoretical schemes through the nucleon-nucleon interaction; (2) a summary of the status of test of general conservation laws; and (3) an attempt to investigate the general analyticity structure of the $S$ matrix through the two-nucleon interaction.

The present situation in our understanding of high-energy physics is reviewed. In high-energy physics we have a plethora of theories, which often look quite incompatible, but which share one grand claim. For each of them it has been possible to find at least one set of experimental data with which they agree. We summarize the outstanding experimental features which are discernible in the nucleon-nucleon system at energies above about 3-GeV/c laboratory momentum. We consider some of the main theoretical ideas and examine to what extent they are compatible with the experimental properties described.

This paper summarizes very briefly the results of some nuclear structure calculations using realistic two-body interactions which fit nucleon-nucleon scattering data. Using realistic interactions, and many-body techniques such as the reference spectrum and an approximate treatment of three-body correlations the binding energy and density of nuclear matter are quantitatively accounted for. Realistic interactions have also been used in Hartree-Fock calculations and results for the ground state of ${\mathrm{O}}^{16}$ are briefly discussed. Finally there are some remarks concerning the effective interaction to be used in shell-model calculations and in particular, the need to include core excitation in order to fit quantitatively the spectrum of two-particle nuclei such as ${\mathrm{O}}^{18}$.

The present knowledge of the partial-wave proton-proton absorptions due to pion production below 1.5 GeV of laboratory kinetic energy is summarized. The three models which have been used for obtaining this kind of information from inelastic data are discussed: the Mandelstam model, the one-pion-exchange model, and the $K$ matrix, or damping, model. Although there is some disagreement, in the near future one can hope to obtain reasonably trustworthy values for partial wave absorptions as a function of energy from the threshold of pion production to about 1.5 GeV. This information will be useful not only to reduce the number of parameters in high-energy phase-shift analysis, but also to be introduced as known inelasticities in partial wave dispersion relations.

Coupling constants of various scalar and vector mesons to the nucleon are determined from empirical proton-proton partial wave amplitudes of high angular momenta. Various fits are described and compared with predictions of unitary symmetry schemes.