We examine the question whether local relativistic quantum field theory can account for the observed asymptotic behavior of the electromagnetic form factor of the proton. Particular attention is devoted to the theory of pions and nucleons interacting via isospin-symmetric pseudoscalar coupling, with minimal coupling to the electromagnetic field. New techniques are developed for calculating and summing all asymptotically dominant contributions to vertex functions and to propagators from all orders of perturbation theory. The results indicate that summing Feynman graphs in local quantum field theory is unlikely to provide an explanation of the observed rapid decrease of the form factor at large momentum transfers.

Using spurion techniques on the five-point function for $\pi \sigma \to \pi \pi \pi$, we obtain an expression for $A^{\mu }\pi \to \pi \pi$ for a conserved axial-vector current which has poles along the $\pi -A_1$ trajectory in the current momentum squared whose residues are Veneziano-type strong-interaction amplitudes satisfying the Adler consistency condition. Using the soft-pion theorem, we obtain a simple expression for the pion form factor, $F\left(q^2\right)={\pi }^{-1\mathrm{}}\left[\frac{\Gamma \left(\frac{1}{2}\right)\Gamma (-q^2+\frac{1}{2})\Gamma (-{\gamma }_2+\frac{1}{2})}{\Gamma (-q^2-{\gamma }_2+\frac{1}{2})}\right]$, where $-{\gamma }_2$ is a constant and is related to the $s$-wave coupling constant in the $A_{1\rho \pi }$ vertex.

We show that two proposals for the incorporation of currents in dual $n$-point functions are very unlikely to be correct because of anomalies in their factorization or level-structure properties. The amplitude proposed by Bander has infinitely degenerate Regge trajectories even at the leading level. Amplitudes constructed according to a prescription by Sugawara and Ohba have a richer spectrum than that of hadronic $n$-point functions and, even worse, are internally inconsistent in their factorization properties. Finally we construct a set of dual functions for $n$ external bosons, one vector current, and one scalar current which satisfy the Ward identities of a local vector-scalar current algebra. Only the scalar current has structure (pole-dominated) in these amplitudes. The vector current is structureless. Because of this simplification, the factorization inconsistency mentioned above does not apply, and it is very possible that a resonance-saturated current algebra which realizes essentially the Dashen-Gell-Mann program can be constructed from these amplitudes.

The energy levels of the hadron are described by a Lorentz-invariant wave equation in the form of supermultiplets. The mass levels belong to the finite-dimensional reducible representations of the group $G=U(3, 1)\otimes SO(3, 2)$ whose generators form the algebraic basis of the wave equation. All the states are occupied, and therefore, for baryons, there are no supplementary conditions imposed on the wave function. The level spacings are mostly regulated by a constant $\rho$ of the theory, which for $\rho >1\mathrm{}$ leads to a hadron type of spectrum for various multiplets, in which case the deviation from the $G$ symmetry is of the order $\frac{1}{\rho }$. The dimension numbers $N$ and $N_0$ of the $U(3, 1)$ and $SO(3, 2)$, respectively, assume the roles of principal quantum numbers to differentiate between the various spin supermultiplets. The $N_0$ distinguishes also between Fermi-Dirac ($N_0=4\mathrm{}$) and Bose-Einstein systems ($N_0=5, 10$). The spin multiplicities for $N_0=4\mathrm{}$ and $N=1, 4, 6, 10, 15, 20$ are obtained, and the corresponding eight mass levels of positive parity are derived as functions of the three parameters and of the spin and other quantum numbers for $N_0=4\mathrm{}$ and $N=1, 4, \mathrm{and} 6$ only. Agreement (3 parts in 1200) with the observed masses of ${\Xi }^0$, ${\Xi }^{-}$, ${\Sigma }^0$, ${\Sigma }^{+}$, $Y^{+}$, $p$, ${\Delta }^{-}$, ${\Delta }^{+}$ is obtained. The transitions between principal levels [$N_0, N$] as well as within the spin multiplets of a given level are briefly described. The Lorentz-invariant generators for the group $\mathrm{SU}\left(2\right)\mathrm{}$ constructed in terms of the generators of the $U(3, 1)$ and the Poincaré group are used as a space-time isospin group. Finally, in the limit $\rho =\infty$ all masses, for any supermultiplet [$N_0, N$], coalesce into the same mass $m$ (one of the parameters of the theory), and the wave equation reduces to a Dirac-type equation for half-integral spin ($N_0=4\mathrm{}$), or to a Kemmer-type equation for integral spin ($N_0=5, 10$), depending on the representation of $G$.

The properties with respect to the crossing transformation of previously suggested relativistic two-variable expansions are considered. Signature amplitudes with definite symmetries are introduced, and it is shown that if the total amplitude is an analytic function satisfying a subtractionless Mandelstam representation, and if the signature amplitudes are square-integrable functions (with a suitable measure) in the physical region of each channel, then two-variable expansions of one type can be continued into expansions of another type, from one channel into the other. The expansion coefficients, called Lorentz amplitudes, in both physical regions, carrying all the dynamics, are shown to be two separate pieces of a general analytic function of the complex angular momentum $l$. It is suggested that these analyticity properties should be made use of to relate low-energy and high-energy scattering data.

The purpose of this paper is to find to what extent the notion of position in the relativistic quantum mechanics of free elementary systems of nonzero mass is defined (or precluded) by the requirement that the descriptions of a localized state as seen from two observers in different frames of reference should be physically consistent. Neither the existence of a time component of the position operator nor the compatibility of the space components of position is assumed. Hermiticity with respect to the quantum-relativistic scalar product in state-vector space is neither assumed nor rejected. There are three possibilities: (i) The position is represented as usual by a Hermitian operator; this is the point-type position. (ii) Non-Hermitian operators are accepted, and the $k$ component of position can represent segments of the $k$ axis with length of the order of $\frac{1}{m}$; when the operator is not only non-Hermitian, but also non-normal, this is the extended-type position. (iii) Non-Hermitian operators are accepted, but the segments are reduced to points; this is the limiting case of the extended-type position. Case (ii) is not studied in this paper. General conditions are found for any spin as regards cases (i) and (iii), and their consequences are obtained for spin 0 and spin ½. The results are the following: There is no position operator in case (i), but in case (iii) the position operator is unique for spin 0 and unique up to a parameter for spin ½. For both spins, in spite of the lack of relativistic Hermiticity, the eigenvalues of position are real, and the velocity is the expected one. The eigenstates of one component of the position are found. The components of the position are compatible with each other for spin 0, and incompatible for spin ½. In the former case, the simultaneous eigenstates of all components are Philip's class-I localized states. Related work is discussed.

In this paper we extend the treatment in Paper I of this series by studying high-energy scattering processes from the viewpoint of the projectile system, i.e., the system in which the incident particle is at rest. In particular, we give two equivalent sets of rules for obtaining the high-energy scattering amplitudes from this standpoint, one for the momentum space and the other for the position space. It is found that both sets of rules give directly the exact expression of the impact factor, and no more approximations need to be made even for the numerators of the integrals involved. We conclude that the impact factor is best studied in the projectile system, while the potential is most conveniently taken care of in the laboratory system, and that they can be separately treated. In the case of high-energy collision of two particles, each impact factor is best studied in its own projectile system. These two sets of rules are shown to be equivalent to those in the laboratory system. In the Appendix we also study high-energy scattering processes in the position space from the viewpoint of the laboratory system, and explicit rules are given.

Recent theoretical results on high-energy scattering are extended. It is found that the form of the matrix elements in the high-energy limit can be readily understood in terms of energy denominators in pre-Feynman perturbation theory. On this basis, we give explicit rules to obtain the high-energy behavior of the sums of large classes of Feynman diagrams, for both the scattering of two extremely relativistic particles and that of one such particle by an external static potential. Application of these rules yields immediately all the previous results on high-energy elastic scattering, and also leads to a deeper understanding of exponentiation. These considerations suggest a simple and natural physical picture for high-energy elastic scattering and, by trivial extensions, also for diffraction and some inelastic scattering processes. We emphasize that this physical picture has the virtue of correctly yielding all the high-energy results of quantum electrodynamics.

This is the second in a series of four papers devoted to a theoretical study based on canonical quantum field theory of the deep-inelastic lepton processes. In the present paper we present the detailed calculations leading to the limiting behavior—or the "parton model"—for deep-inelastic electron scattering. It follows from this work that the structure functions depend only on the ratio of energy to momentum transfer $\frac{2M\nu }{q^2}$ as conjectured by Bjorken on general grounds. To accomplish this derivation, it is necessary to introduce a transverse momentum cutoff so that there exists an asymptotic region in which $q^2$ and $M_{\nu }$ can be made larger than the transverse momenta of all the virtual constituents or "partons" of the proton that are involved. We also derive the ladder approximation for the leading contribution, order by order in the strong interaction and to all orders in the coupling, to the asymptotic behavior of these structure functions with increasing ratio of energy to momentum transfer. Finally, we draw and discuss the experimental implications.

We investigate the headlight effect for nonelectromagnetic radiation and find that, depending upon the wave and source velocities, the effect may be normal (radiation concentrated in the forward direction) or anomalous (radiation concentrated in some direction other than forward). Expressions are derived for the distribution of the radiation as a function of angle for both an infinite line source and a point source, and the critical wave velocity at which the headlight effect makes a transition from anomalous to normal is determined for both cases as a function of source velocity. A possible application to the decay of radioactive particles is suggested.

Final results on the first satellite experiment on abundances of cosmic-ray nuclei having a mean energy of several GeV per nucleon are presented. A nuclear-emulsion detector, exposed on Gemini XI in a near-equatorial orbit ranging between geographic latitudes ±29°, collected 619 high-quality tracks above the earth's atmosphere. Time resolution (within about 5 min) was provided by movement of a lower emulsion stack relative to an upper one. The detector was covered by only 0.07 g/${\mathrm{cm}}^2$ of aluminum and was favorably oriented for 18 h. The results on abundances, requiring no correction for secondary production in the atmosphere, are characterized by (a) a pronounced odd-even effect, with low abundances for elements of atomic number 7, 9, 11, and 13, compared to those of neighboring elements with even $Z$; (b) approximately equal fluxes of neon, magnesium, and silicon, each being about one-fourth that of oxygen; and (c) an abundance gap in the region $15\le Z\le 19$. The observed ratios of Be+B, $10\le Z\le 19$, and $Z\ge 20$ to the medium group, $6\le Z\le 9$, provide no evidence for significant variation of composition with rigidity between 3.5 and 30 GV. A primordial composition—prior to interactions of the cosmic rays with the interstellar medium—is calculated. This source composition is compared with "universal" and solar abundances.

The Kohn-type variational principle for the elements of the scattering matrix is derived in a form which displays the second-order error term explicitly. With the scattering energy taken to be a complex parameter, a calculable bound on the magnitude of this error term is obtained. This allows for a systematic improvement in the accuracy of the variational calculation. The motivation for this study lies in recent demonstrations that accurate analytic continuation to physical energies can in fact be performed. The applicability of this technique to the determination of the level-shift operator and the effective potential is pointed out.

An elasticity theory in general relativity is formulated and includes a measure of the strain which is identical to the classical concept. The theory is developed essentially by generalizing the classical elasticity theory. The physical interpretation of the work is simplified by retaining the three-dimensional form of the classical quantities. As a further aid to understanding the theory, some of the thermodynamics and the weak-field limit are studied. As part of the latter investigation the classical theory is reformulated so that there is no explicit dependence on the displacement u. The resulting equations near a strong similarity to the general-relativistic field equations when they are cast in a manner called the (3+1) form.

Criticism by Brody et al. of the CERN-I $\pi p$ phase-shift analysis is answered.

The calculations of a previous paper on Reggeized $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}\otimes O\left(3\right)\mathrm{}$ and absorptive corrections are extended to $K^{+}n\to K^{0}p$ at 5.5 GeV/c with the addition of no new parameters. Good agreement with experiment is obtained.

Some recent results of Abarbanel, Drell, and Gilman on high-energy proton-proton scattering are shown to be the consequence of a general property of the Baker-Blankenbecler approximation for the unitarity condition they used. The latter approximation is shown to give results generally consistent with earlier suggestions of Wu and Yang.

A formalism is developed to treat the effects of final-state interactions on the properties of unstable particles produced in high-energy collisions. The formalism is applied to the photoproduction of $\rho$ mesons.