Cosmic-ray flux measurements in the energy region ${10}^{10}$-${10}^{14}$ eV obtained by calorimeters on the satellites Proton I and II have shown results that are at variance with previous data. While a single power law provides an approximate fit to the all-particle spectrum, the primary proton flux falls sharply at energies above ∼5×${10}^{11}$ eV, indicating that at high energies protons become progressively scarcer in the primary flux. The cross section for particle production by protons on carbon is found to rise by 20% in the interval between 2×${10}^{10}$ and ${10}^{12}$ eV. Assuming that, in the energy region of interest, (1) the real proton flux is given by a single power law, and (2) the nuclear composition remains constant, we show that the satellite flux measurements can be explained by an energy-loss mechanism in the calorimeter, the loss being a function of the energy per nucleon rather than the total energy. Furthermore, this "$X$" process has a cross section of the right magnitude to account for the $p$-carbon cross-section measurements. The $X$ process could be described in terms of particle production or dissociation of the primary protons.

A formalism is presented giving the development of $\psi \left(\overline{x}\right)$ in time with the interpretation that $\psi \left(\overline{x}\right)$ is the probability amplitude for observing an event at a space-time point $\overline{x}$. No properties other than the four space-time coordinates are associated with an event. A Hilbert space is defined in which $\psi \left(\overline{x}\right)$ is the result of a scalar product. The space-time position operators defined in this Hilbert space have no association with particle properties, such as mass. These operators cannot be defined in the Hilbert space spanned by solutions of a Schrödinger equation, since the operators lead out of the Hilbert subspace belonging to a given mass. It is shown that state vectors in Hilbert space that are eigenvectors of $P_{\mu }{P}^{\mu }$ produce position amplitudes satisfying the Klein-Gordon equation. The relation between this Hilbert space and the one introduced by Dirac is discussed. Spin is not considered.

An approach to the measurement of static gravitational couplings is discussed. A plausible mathematical treatment of the nongravitational interactions of particles in the presence of a small, constant, classical tensor field without the assumption of universal coupling to the field is outlined. Some nonstandard features, principally the breakdown of strict Lorentz covariance, are pointed out. Calculations based on this formalism, or, alternatively, on simple energy-momentum conservation, indicate that some particle processes which are forbidden if the coupling to the tensor field is universal are allowed if it is not. The processes considered in some detail are spontaneous emission of a photon by a free particle, spontaneous photo-production of a particle-antiparticle pair, and spontaneous neutrino decay. These occur under certain conditions involving the energies and the coupling constants to the tensor field. The tensor field is identified with the local gravitational field and rates are estimated. Certain existing data on the absence of such reactions give evidence that the gravitational coupling of the muon in particular is equal to that of the photon to within parts per ten thousand, and that all particles with $m{c}^2$ less than a few GeV, excepting possibly the neutrinos and the graviton itself, have gravitational coupling which is at most the normal amount—to varying degrees of accuracy. The latter result is subject to an assumption that the algebraic sign of the local gravitational field is what one expects on intuitive grounds.

Martin and Newton have proved some sufficient conditions of solubility and uniqueness for the unitarity integral equation. Application to experimentally known differential cross sections justified by appropriate assumptions shows that these conditions are, in general, not satisfied. Similarily, the angular distribution discussed by Crichton does not meet the solubility condition.

We present an extremely simple derivation of the Jin-Martin bound in a slightly weakened form. Explicitly, let $f^{\mathrm{AB}\to \mathrm{AB}}\mathrm{}\left(s\right)\mathrm{}$ be the spin-averaged forward-scattering amplitude for the process $\mathrm{AB}\to \mathrm{AB}$ [normalized by ${\sigma }_{\mathrm{tot}}=4\mathrm{}\pi \frac{\mathrm{Im}f\left(s\right)\mathrm{}}{k\surd s}$]. Then we prove that there is a constant $C$ and a sequence $s_n\to \infty$ such that either $f^{\mathrm{AB}\to \mathrm{AB}}\left(s_n\right)>C{s_n}^{-2\mathrm{}}$ (all $n$) or $f^{A\overline{B}\to A\overline{B}}\left(s_n\right)>C{s_n}^{-2\mathrm{}}$ (all $n$). We also clarify the connection between allowable asymptotic behavior and the sign of the scattering length found by Jin and Martin.

In order to find the effect of spin and isospin on the construction of a Veneziano amplitude, we have analyzed the reaction $A\pi \to A\pi$. This reaction involves the (degenerate) $\rho -f$ trajectory in both channels. In order to construct an amplitude possessing crossing symmetry, proper asymptotic behavior, and pole structure, as well as signature and factorization for the leading trajectory, it is necessary to include three independent beta functions in an expansion of the form (polynomials) × (beta functions). It is found that, to avoid parity doubling on the leading trajectory and to keep the $A\rho \pi$ coupling from vanishing, it is necessary to have a conspiracy among three of the invariant amplitudes for all values of $s$. The conspiracy, signature, and factorization constraints provide a large number of constraints, but do not fully determine the amplitude. The $A\rho \pi$ couplings are undetermined by the foregoing constraints. The amplitude is sufficiently flexible to satisfy the Adler condition and to agree with current-algebra determination of the threshold amplitude. (These conditions only lead to two constraints.) The simpler reaction $A\pi \to \pi \pi$ is analyzed in detail. For this reaction, it is possible to find a minimal amplitude satisfying asymptotic behavior and soft-pion constraints, and only one over-all undetermined constant.

On a dynamical basis which incorporates analyticity, multi-soft-pion production, and multiparticle unitarity, it is concluded that in a world in which the pion mass is zero, the Regge parameters $\alpha$ for nucleon (and nucleon-resonance) trajectories should have the threshold behavior. $\mathrm{Im}{\alpha }_{l=J+\frac{1}{2}}\mathrm{}\left(k\right)\mathrm{}\underset{k\to 0}{\sim }{k}^{2\alpha \mathrm{}\left(k\right)\mathrm{}}, \mathrm{Im}{\alpha }_{l=J-\frac{1}{2}}\mathrm{}\left(k\right)\mathrm{}\underset{k\to 0}{\sim }{k}^{2\alpha \mathrm{}\left(k\right)\mathrm{}}.$

This is to be contrasted with the normal behavior in a world with $m_{\pi }\ne 0:\mathrm{Im}{\alpha }_{l=J+\frac{1}{2}}\sim k^{2\alpha \mathrm{}\left(k\right)+2\mathrm{}}$, $\mathrm{Im}{\alpha }_{l=J-\frac{1}{2}}\sim k^{2\alpha \mathrm{}\left(k\right)\mathrm{}}$. A discussion is given of the relationship of these results to (a) possible parity doubling of baryon trajectories and (b) unitarization schemes for models with initially real trajectories.

The sum of all ladder-type diagrams contributing to the scattering of two scalar nucleons is evaluated in the high-energy approximation. It results in an expression for the eikonal function in closed form which is, however, purely real. Possible sources of this difficulty are discussed.

A set of invariant amplitudes for the reaction $\omega \pi \to \omega \pi$ is constructed within the framework of the Veneziano model by expansions of the form $\Sigma \left(\mathrm{polynomials}\right)\times \left(\mathrm{beta}\mathrm{function}\right)$. The dominant asymptotic behavior corresponds to the $f$ trajectory in the $t$ channel and the $\rho$ and $B$ trajectories in the $s$ and $u$ channels. Separate treatments are given for the $\rho$ and $B$ contributions. In every case the leading trajectory poles have correct spin-parity as a consequence of enforcing proper asymptotic behavior on the invariant amplitudes. These relations are nontrivial and differ for the $\rho$ and $B$ contributions. The detailed construction of the amplitudes for $\omega \pi$ scattering differs from most previous applications because of the absence of an exotic channel. $\omega \pi$ scattering is notable in the number of terms required to represent the amplitude; a 72-parameter amplitude built using $B_{11}$, $B_{12}$, and $B_{21}$ cannot satisfy simple physical requirements for the $\rho$ trajectory. One must add further terms lacking the $\rho$ pole (such as $B_{22}$) in order to avoid decoupling the $\rho$ trajectory. The $B$ contribution is easily treated in direct analogy to the $\rho$ trajectory in $A\pi$ scattering. The implication of the hypothesis of partially conserved axial-vector current (PCAC) that the invariant amplitude $T_1$ (coefficient of $e^{\prime }·e$) should vanish at the Adler point is somewhat delicate because of the near degeneracy of the $\omega$ and $\rho$ mesons. The signature and amplitude conspiracy relations ensure the suppression of the apparent pole.

We analyze in detail the spontaneous breakings of chiral $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ and $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$. We determine the directions along which the two groups may break spontaneously. We discuss also the physical implications of these group-theoretical results, as the appearance of Goldstone particles, the particle mixings, and the consequences of the residual invariance.

The harmonic-oscillator model of hadrons is extended so that it is dual- and crossing-symmetric for any value of the ground-state mass. The new feature of the model system is an extra nonspatial harmonic mode associated with each quark.

Using the harmonic-oscillator model for hadrons, in which any hadron is described by two quarks embedded in a one-dimensional harmonic continuum, and allowing quanta to couple to either quark, we were able to construct the full crossing-symmetric Veneziano type of amplitude in the tree-graph approximation. This generalization also gives us a set of rules to compute the nonplanar single-loop diagrams.

Using the generalized harmonic-oscillator model of hadrons, we construct the amplitude for dual-symmetric Feynman-like diagrams with a single loop. Our result is essentially identical to the expression derived by Veneziano and Virasoro.

The harmonic-oscillator model of hadrons is generalized to include higher harmonics so that the degrees of freedom of the internal state of a hadron are equivalent to those of a violin string or organ pipe. Scattering amplitudes for the scattering of particles by the oscillator are constructed and shown to be equivalent to the dual-symmetric Veneziano model. Our method provides a direct construction of the amplitude in a manifestly factorized form.

It is shown that unitarity allows fixed poles at certain nonsense points of either right or wrong signature. The conditions for the existence of these poles are found. These conditions are then used to locate the poles allowed in hadronic reactions. Possible mechanisms for the poles are considered. It is then argued that fixed poles provide the most natural explanation of the conspiracy phenomenon.

A no-subtraction dispersion relation for the electron self-energy is written in the momentum of the electron. The integrand of the dispersion relation is specified to order $e^2$ by truncating a sum over a complete set of states with a one-electron one-photon state. The result, which is a consequence of a mixture of perturbation theory and dispersion theory, is an expression for the electron self-energy as a sum over the $E_1$ and $M_1$ phase shifts of the multipole expansion of the Compton scattering amplitude.

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.