The relationship between charge quantization and the compactness of the gauge group is discussed. Also, remarks are made about charge quantization and the observation of flux quantization in superconductors.

We propose a parameter-free regularization of the one-loop planar unitary dual diagram, very similar to the renormalization in quantum field theory, compatible with duality, Regge behavior, unitarity, and crossing symmetry.

It is shown that an approximate summation of the "crossed-ladder" Feynman diagrams for the scattering of two charged particles leads to the formula $s={m_1}^2+{m_2}^2+2\mathrm{}{m}_1{m}_2{[1+\frac{Z^2{\alpha }^2}{{(n-{\epsilon }_j)}^2}]}^{-\frac{1}{2}}$ for the squared mass of bound states. This formula neglects radiative corrections. It includes recoil effects properly, and reduces in the limit of one infinite mass to the corresponding spectrum of a relativistic particle in a static Coulomb potential. In the particular case of positronium, its expansion in powers of $\alpha$ coincides up to order ${\alpha }^4$ with the singlet energy levels. In an appendix we investigate some properties (gauge invariance, static limit) of this series of graphs.

An exactly soluble model of relativistically invariant and local quantum field theory is described. It satisfies all the usual postulates except the requirement of the positive definiteness of either the energy or the metric in the space of state vectors. The $S$ operator is nontrivial and is given explicitly. The two-particle scattering amplitude is calculated. Various generalizations of the model are possible.

We have discovered a covariant stress tensor for free massless field theories of integer spin. We construct and explicitly give this tensor. Until now, no covariant stress tensor was known in such theories except for the anomalous cases of spins 0 and 1. Indeed, we show that for massless field theories of spin greater than 1, no ordinary second-rank stress tensor can even exist. It is thus significant that the stress tensor we have found belongs to an infinite-dimensional representation of the homogeneous Lorentz group. This infinite-component stress tensor has remarkable properties which we study in detail.

A model is presented for three- and four-point functions of currents which satisfies the Ward identities implied by chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ for arbitrary vector and axial-vector spectral functions. In the limit in which the spectral functions are single-particle dominated, the three- and four-point functions of the usual hard-pion approach are recovered. However, this limit is not forced upon us by the assumption of smoothness of primitive vertices, since the singularity structure is extracted in a manner somewhat different from the hard-pion approach.

We show that the second-order radiative correction changes the electron eikonal function into $[1+F(k^2\left)\right]\times [1+F^{\prime }(k^2\left)\right]E\left(k^2\right)$, where $E\left(k^2\right)$ is the bare eikonal, $k$ is the momentum transfer between the electrons ($e$ and $e^{\prime }$), and $F\left(k^2\right)$ is the second-order electric form factor. This suggests that in high-energy near-forward electrodynamic processes, the eikonals should be regarded as the basic carriers of the interaction.

The phase-contour method is applied to the study of $\pi \pi$ scattering. We include the effect of the $\sigma \left(0^{+}\right)$ resonance, for which there is now very good experimental evidence, and we discuss its relation with the Adler zero. We have obtained a general description of the phase contours for several isospin amplitudes having different types of symmetry, in particular, mixed symmetry and exchange-degenerate trajectories in one channel.

We study in this work the saturation of the charge-current equal-time commutation relation $\left[{F_{\alpha }}^5\right(x_0), {{\mathcal{I}}_{{\beta }^0}}^5\mathrm{}(x\left)\right]=i{f}_{\alpha \beta \gamma }{\mathcal{I}}_{{\gamma }^0}\mathrm{}\left(x\right)\mathrm{}$ in the infinite-momentum frame by employing a pure as well as a mixed configuration of $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W\otimes O\left(3\right)\mathrm{}$. We examine the consistency of such finite saturation schemes for the above commutator in comparison with the corresponding charge-charge commutator. Our study shows, as expected on heuristic grounds, that the consistency requirement cannot be fulfilled over the entire range of the momentum transfer $q$. We also discover that the unitary-symmetric and -antisymmetric combinations of the above commutator display different degrees of contradictions: The former admits only a zero "range of consistency," while the latter yields a family of sum rules near $q=0\mathrm{}$, which offer predictions for the zero-$q^2$ intercepts and slopes of the various form factors involved. Among these we find the disturbing results ${\mu }^{\prime }\mathrm{}\left(p\right)={\mu }^{\prime }\mathrm{}\left(n\right)={\mu }^{*}\left(N^{*}\right)=0\mathrm{}$, whereas others have yet to be tested experimentally. It is shown that even a substantial extension of the saturation scheme fails to induce visible improvement in the consistency problem. While the sum rules appear to show some response to the admixture, many of the results remain unaffected. A closer investigation of these problems suggests the desirability of considering modified transformation properties for the local current operators as an alternative approach to the present problem.

An analysis of typical microphysical systems is presented in the hydrodynamic formulation of quantum mechanics. The emphasis is on the physical peculiarities appearing in the hydrodynamic picture, and on the mathematical treatment of the nonlinear quantum-hydrodynamic field equations. Further, quantum-hydro-dynamic uncertainty relations are derived, which relate the minimum uncertainty products to the interior quantum stresses.

It is shown that many of the satellite terms possible in Veneziano models are linearly dependent, and that, in fact, specification of the residues of the poles is arbitrary, but completely determines all the coefficients of the satellite terms. Implications of this for ghosts, parity doubling, daughters, asymptotic behavior, Adler zero, and finite-energy sum rules are discussed. A specific numerical example for $\pi -\pi$ scattering is given.

The general relativistic equations for a stationary axially symmetric rotating perfect fluid, in a comoving coordinate system, are presented in an elegant general form. The coordinates can then be chosen in any of several ways to further simplify the equations. One choice reduces the number of independent metric functions to three.

A semiquantitative calculation is performed for the specific heat of a plane, rigid, electric-dipole rotator having classical interactions with thermal electromagnetic radiation including electromagnetic zero-point radiation. The calculation is termed semiquantitative because of the introduction of an unevaluated parameter to account for the failure (at low frequencies) of a familiar approximation. The calculation seems of interest for three reasons: (i) It provides a rough classical understanding of qualitative behavior which is usually attributed to quantum interactions. (ii) It again emphasizes that in statistical thermodynamics a consistent classical theory including electromagnetism must include electromagnetic zero-point radiation. (iii) It is another step in a general program attempting to use electromagnetic zero-point radiation as an alternative hypothesis to quanta. From the calculation, the specific heat of the rotator is found to vanish with vanishing slope as the temperature $T\to 0$, and to go smoothly over to the traditional classical value $\frac{1}{2}k$ at high temperatures. Moreover, the rotator probability distribution with frequency, which departs from the traditional classical theory at absolute zero, becomes the usual Boltzmann distribution in the limits of high temperature or large moment of inertia. These results are in contrast to the classical calculation of Fokker in 1914 involving the Planck spectrum without zero-point radiation, which found behavior in complete contradiction with experiment—including infinite slope for the specific heat at $T=0\mathrm{}$ and a failure to approach $\frac{1}{2}k$ at high temperatures. Despite some qualitative features in agreement with experiment, it is suggested that the new results cannot be compared directly with quantum theory or with molecular specific heats, because the calculation involving classical radiation damping does not assure a separation of variables from the internal molecular variables, which would also be influenced by electromagnetic zero-point radiation.

Improved results on the total photo-ionization cross section of atomic helium are given. The total cross section includes contributions from simultaneous ionization and excitation of ${\mathrm{He}}^{+}$ and double ionization; the combined effect of these two processes adds about 10% to the normal photo-ionization process where ${\mathrm{He}}^{+}$ is left in the ground state. A lower abundance (10.92 based on 12.00 for hydrogen) is adopted for helium, based on recent radio determinations. In calculating the opacity due to $K$-shell photo-ionization of heavy elements, a lower abundance (8.00) is also adopted for neon. Brief mention is made of the effects of irregularities in the density distribution of the interstellar gas on the problems of both x-ray and radio-wave absorption.

The effects of a possible magnetic charge of the moon upon the magnetic field in the lunar vicinity have been analyzed. Magnetic field observations obtained by the GSFC magnetometer aboard Explorer 35 have been studied to search for these effects. Using these observations, it is possible to obtain a measure of the net difference between the number of northern and southern monopoles within the moon which are of opposite sign. The search has resulted in negative findings and places an approximate upper limit on the average difference in the number of monopoles within the moon at 1.6×${10}^{-7\mathrm{}}$ ${\mathrm{cm}}^{-3\mathrm{}}$ or 7×${10}^{-32\mathrm{}}$ per nucleon.

Cosmic-ray muons produced in the earth's atmosphere were measured at a depth of 8.71×${10}^5$ g ${\mathrm{cm}}^{-2\mathrm{}}$ with a large-area (170 ${\mathrm{m}}^2$) liquid-scintillation-detector hodoscope. These data taken together with those of other workers lead to an improved muon vertical-depth intensity curve, $I_v\mathrm{}\left(h\right)=a_{\mu }{e}^{-\frac{h}{\lambda }}$, where $I_v\mathrm{}\left(h\right)=\mathrm{vertical}\mathrm{intensity}$, $a_{\mu }=\left({{1.04}_{-0.13\mathrm{}}}^{+0.21\mathrm{}}\right)\times {10}^{-6\mathrm{}}$ ${\mathrm{cm}}^{-2\mathrm{}}$ ${\sec }^{-1\mathrm{}}$ ${\mathrm{sr}}^{-1\mathrm{}}$, $\lambda =\left({{8.04}_{-0.39\mathrm{}}}^{+0.35\mathrm{}}\right)\times {10}^4$ g ${\mathrm{cm}}^{-2\mathrm{}}$, and $h=\mathrm{depth}$ in g ${\mathrm{cm}}^{-2\mathrm{}}$. A comparison of these results with those expected from the sea-level muon spectra via an improved calculation may indicate the need for an increased energy loss, probably via the photonuclear interaction as suggested by Keuffel et al. The results are not inconsistent with the presence of the $X$ process of Keuffel et al.

Hartle and Taylor have shown that the cluster law imposes certain restrictions on the allowed symmetries of first-quantized systems of identical particles. We extend their results to show that all particles which are neither bosons nor fermions and which obey the cluster law can be divided into two classes, those of finite order, and those of infinite order. For every positive integer $p$ there are two types of finite-order particle, which we call parabosons and parafermions of order $p$. Ordinary bosons and fermions can be fitted into this scheme as particles of order 1. We conjecture that the finite-order particles can be identified with the parafermions and parabosons of the second-quantized, parafield theory. Infinite-order particles would seem to have no analog in the second-quantized theory, as presently formulated.

The transverse electrical conductivity is computed for a system of degenerate relativistic electrons in a strong magnetic field $H\approx {10}^{10}-{10}^{13}$ G. As suggested by pulsar models, such fields exist in nature in collapsed bodies, like neutron stars. The present computation is valid in the outer regions of the star where the scatterers are not degenerate, while the electrons are taken to be at zero temperature. The scattering mechanism is assumed to be the electron-ion Coulomb scattering. Numerical values of the transverse conductivity are given in the range ${10}^9\le H\le {10}^{13}$ G, and a comparison is made with the longitudinal and zero-field conductivities. It is found that for densities ${10}^7\le \rho \le 5\times {10}^7$, ${\sigma }_{\mathrm{II}}\approx 20{\sigma }_0$ and ${\sigma }_0\approx 3{\sigma }_{\perp }$. As the density increases, both ${\sigma }_{\perp }$ and ${\sigma }_{\mathrm{II}}$ tend to ${\sigma }_0$.

Forces are found that satisfy the covariance condition of Currie and Hill for Newtonian equations of motion of two particles in four-dimensional space-time. Examples are given that can decrease, at any prescribed rate, for large particle separation. Unfortunately these latter forces become complex and even singular for certain regions of velocity space. The technique used is to simplify the covariance condition by using four-vector variables as much as possible.

Results of a Monte Carlo simulation of the longitudinal development of nuclear-electromagnetic cascades in glass, iron, and tungsten absorbers are presented. The simulation was fitted first to measurements made with an iron-absorber ionization spectrometer that was exposed at an accelerator to protons of known energies. The calculations were then extended to higher primary energies and to the other two absorbers. The results presented are for the energy range 10-1000 GeV. The maximum absorber depths considered are 56, 96, and 192 radiation lengths for glass, iron, and tungsten, respectively. Average cascade development curves and their fluctuations, as well as the ionization energy deposited by the cascades, are given as a function of the absorber depth. The accuracies with which the cascades can be used to determine unknown primary energies are also presented. It is shown that these accuracies increase rapidly with the total absorber depth up to an optimum depth for a given primary energy and given absorber. Over the energy range considered and for moderately large absorber depths, the accuracy also improves with increasing primary energy.