The model previously proposed is extended to include multiquark trajectories. Once any trajectories with more than a single quark and antiquark are included, it is necessary to include trajectories where the number of quarks plus the number of antiquarks, which we call the total quark number, is arbitrarily large. The necessary factorization properties of the multiparticle Veneziano amplitudes will hold provided the intercept of the leading trajectory is a polynomial function of the total quark number, and the degeneracy of the levels on all but the leading trajectory will increase with the order of the polynomial. It is possible to construct two different models depending on whether one allows nonplanar duality diagrams. The model with nonplanar diagrams resembles more closely the nonrelativistic harmonic-oscillator quark model, and the nonplanar duality diagrams must be associated with the nonplanar Veneziano amplitudes discussed in a previous paper. One can introduce $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry-breaking by making the intercept depend on the number of strange and nonstrange quarks separately, and one then obtains a modified Gell-Mann-Okubo mass formula.

A new class of multiparticle Veneziano formulas is constructed. The amplitudes correspond to a subclass of nonplanar Feynman diagrams without internal vertices which we call minimal diagrams, just as the current multiparticle Veneziano formulas correspond to planar diagrams. For simplicity, we have not examined diagrams with more than one crossed line. The $r$-point function is represented by an ($r-3\mathrm{}$)-dimensional integral, as it is for planar diagrams. While the new formula has many properties in common with the old one, the three-particle channels of the Feynman diagram have a more complicated spectrum which suggests a system of three quarks.

The unitarity condition is combined with the assumption of infinitely rising Regge trajectories to obtain constraints on the coupling of the Pomeranchuk singularity to different Regge recurrences of particles which can be produced by diffraction dissociation. The results obtained indicate the possibility that the Pomeranchuk singularity "bootstraps" itself.

We have constructed a chiral-symmetry model in which the vector mesons (which are not introduced by means of the Yang-Mills technique) are described in terms of an antisymmetric second-rank tensor $T_{\mu \nu }$. As a result, the interaction Hamiltonian in the interaction representation contains additional contact-type terms. Besides the pseudoscalar and vector-meson nonets, the model contains two nonets with $J^{\mathrm{PC}}=0^{--} \mathrm{and} 1^{-+}$. Their fields appear in even numbers in all terms in the expansion of our Lagrangians. A mass Lagrangian is introduced such that the mesons get their physical masses. This gives generalized partially conserved axial-vector current if higher-order contributions are neglected. A $U{\mathrm{}\left(3\right)\mathrm{}}_{\mathrm{left}}\otimes U{\mathrm{}\left(3\right)\mathrm{}}_{\mathrm{right}}$-invariant Lagrangian is found which gives the well-known $\mathrm{PPV}$ coupling and also a $\mathrm{PVV}$ coupling. A nonet of baryons is introduced, with the ${Y_0}^{*}\mathrm{}\left(1405\right)\mathrm{}$ as ninth baryon. Several chiral-invariant meson-baryon Lagrangians are considered, such that the nonderivative and derivative couplings of the pseudoscalar mesons to baryons, and the vector and tensor couplings of the vector mesons to baryons, are obtained. Also, two symmetry-breaking meson-baryon Lagrangians are needed to give the baryons their physical masses. The coupling constant of the tensor coupling of the $\rho$ mesons to nucleons is correctly predicted. The width of the ${Y_0}^{*}\mathrm{}\left(1405\right)\mathrm{}$ and the ratio $\frac{\left|G_{\mathrm{NKY}}\right|}{\left|G_{\Sigma \pi Y}\right|}$ are predicted to be 62 MeV and 2, respectively, which are close to the experimental values. The decay widths $\Gamma \left[{\eta }^{\prime }\mathrm{}\right(958)\mathrm{}\to \eta +2\mathrm{}\pi ]$, $\Gamma (K^{*}\to K\pi )$, and $\Gamma (\varphi \to K^{+}{K}^{-})$ are in reasonable agreement with experiment.

In this work we consider the Ward identities within the operator formalism of Fubini, Veneziano, and Gordon. We construct an operator $A$ which, when acting on any state of the Hilbert space of oscillators, gives a spurious state. We then show how one can use $A$ to remove the contributions of spurious states in unitary-loop calculations. In particular, we obtain the result of Bardakci, Halpern, and Shapiro for the planar closed loop. No effort is made to prove that $A$ generates the complete set of linear dependences.

There are no known internal symmetries of the isospin invariance variety that are exact. In this paper we study the viewpoint that the "perturbing interactions," usually considered the cause of the "breaking" of internal symmetries, are actually also responsible for the existence of any internal symmetry at all. Consequently, the symmetries are "born imperfect." We study simple examples which give concrete expressions to this hypothesis, and investigate in a model the emergence of two fermion fields with opposite charges from one field upon perturbation. The implication of these considerations for electromagnetic-mass-difference calculations is discussed.

Certain algebraic aspects are abstracted from the duality principle and are incorporated in a simple model of pion $n$-point functions. An algorithm for constructing the $n$-point function in the tree-graph approximation is based on the duality assumption and the Adler condition which states that the amplitudes vanishes if any pion four-momentum vanishes, all others remaining on shell. The resulting amplitudes satisfy the constraints of current algebra and partial conservation of axial-vector current for $n=4, 6, \mathrm{and} 8$, and (we conjecture) for all $n$. In addition, duality specifies a definite form for chiral symmetry breaking.

A variational principle is developed for three-body scattering which resembles in form a combination of the Kohn and Schwinger variational principles. Its principal advantage over previous formulations is that for energies above the three-body breakup threshold, it is not necessary to know the complicated correlation terms in the asymptotic part of the wave function. Application of the principle to both separable and local potentials is discussed. In this paper we present results of calculations with separable potentials for elastic and rearrangement processes beneath the breakup threshold.

An inhomogeneous Lorentz group (different from the Poincaré group) is studied in relation with the relativistic covariance of the Dirac-Weyl equation. A mathematical study of the group structure and of its unitary irreducible representations is presented, and physical consequences concerning neutrino physics (and electromagnetic field) are drawn by the introduction of the Stokes parameters.

Empirical boson assignments on the baryon-antibaryon model are used to analyze exhaustively the masses for 13 out of 16 expected $P$ orbitals; on this basis search areas can be predicted for the missing three states, all having $I=0\mathrm{}$ and $J^P=1^{+}$. The mass analysis alone does not necessarily require much $S{U}_3$ singletoctet mixing. Extrapolation to $D$ states appears to pose a contradiction between this model and exchange degeneracy of normal boson trajectories. The situation can be resolved empirically by determining accurate $J^P$ values for the prominent $I=1\mathrm{}$ bosons in the $R-S$ region of missing-mass spectrometer assignments. Tentative evidence for a first radially excited octet with $J^P=0^{-}$ places it in a mass region consonant with the model.

We find that neutral and $\mathrm{SU}\left(2\right)\mathrm{}$ [and consequently $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$] Sugawara current theories are completely equivalent to canonical Lagrangian field theories of massless scalar fields. A Sugawara current theory is obtained by eliminating all the massive fields in a theory of spontaneous symmetry breakdown and retaining the Goldstone bosons only. The reverse is also true in that all Abelian or $\mathrm{SU}\left(2\right)\mathrm{}$ Sugawara theories are necessarily equivalent to these canonical representations, so that for these groups "Sugawara physics" reduces simply to "Goldstone boson physics."

In the amplitude-phase solution of the Schrödinger equation, it is shown that the bound-state energies are determined by continuing $A^2$ to negative energies and imposing appropriate boundary conditions at its origin. The function $A^2$ can be used to characterize both the regular and irregular solutions of the Schrödinger equation, with the result that in the Green's-function approach to bound-state perturbation theory, the energy shift in any order can be written in the form of an integral over known solutions. Two examples of second-order energy shift are discussed.

The scattering-operator theory developed in a previous publication (I) by one of us (F. R.) is applied here to "${\phi }^n$" models. The fundamental equation for the scattering operator is solved in perturbation expansion. A suitable graphic representation provides diagrams for easier computation. The ${\phi }^3$ model is solved to second and third order as an example, and off-mass-shell unitarity is checked explicitly. General arguments are presented to all orders to show in perturbation expansion, and for the preliminary formulation (I), that (a) for the models which are nonrenormalizable in the conventional theory ($n>\mathrm{}4\mathrm{}$) there is no solution of the $S$-operator equation, and (b) for the renormalizable models ($n=3\mathrm{} \mathrm{and} 4$) the solution of the $S$-operator equation is identical to the renormalized conventional theory if the interaction is suitably chosen. The theory is finite throughout and involves no renormalizations. There are no cutoffs except a technical one for the space-time volume which is removed at the end of the calculation.

This is the third 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-positron annihilation into a nucleon (or hadron) plus "anything else," i.e., $e^{-}+e^{+}\to p+\mathrm{anything},\text{'}\text{'}$ where "anything" refers to all possible hadrons. In particular, we show that the structure functions satisfy a scaling behavior analogous to the Bjorken limit for deep-inelastic scattering. The precise relation of the structure functions for deep-inelastic annihilation processes to those for the deep-inelastic scattering is discussed along with specific experimental implications and tests.

The interaction of a quantized charged vector boson in the presence of an external electromagnetic field is considered. A nonsingular electromagnetic current operator is obtained by separating the points of the field operators by a small spacelike distance ${\epsilon }^{\mu }$ in a gauge-invariant fashion (analogous to the construction of the current operator of spinor electrodynamics). It is shown that the matrix element of the current is gauge-invariant, divergenceless in the limit ${\epsilon }^{\mu }\to 0$, and no more than logarithmically singular, to all orders of the external electromagnetic field. The resultant theory is seen consequently to be gauge-invariant and to be renormalizable.

General and essentially exact formulas for the energy-loss process $A+\gamma \to A+e^{+}+e^{-}$ are given for high-energy cosmic-ray nuclei traversing an isotropic radiation field of arbitrary energy spectrum. These formulas are obtained by integrating the appropriate differential cross section in the rest frame of the cosmic-ray particle. Results are then given for the case of a Planckian photon distribution at arbitrary temperature. The effect of this energy loss on a universal flux of cosmic rays is then investigated along with the possibility that the observed "knee" at high energies is due to pair production. Finally, the energy-loss formula is applied to quasistellar objects, and it is shown that if these objects are at cosmological distances, they are opaque to particles above a certain energy which depends upon the size of the object.

A "pion-links method" is developed to solve the three-dimensional nucleon-pion cascade equations of cosmic rays, and results are computed for showers developing in the atmosphere and in the ionization calorimeter at several values of the primary cosmic-ray energy.

In the present investigation the semiclassical approximation using the time-dependent formulation of the collision theory is extended. The previously used narrow-wave-packet approximation is improved by using, instead of expanding wave packets, wave packets that first contract and then expand. In addition, an integral equation is derived that allows one to obtain corrections to the narrow-wave-packet approximation. Several of these corrections are presented.

It is shown that, for a classical point charge, the "bound" electromagnetic four-momentum contains, besides the generally accepted "Coulomb mass"×four-velocity term, the extra term $-\frac{2}{3}{e}^2{a}^{\mu }$. This is accomplished by exploiting some interesting properties of the usual separation of the retarded field of a moving charge into a velocity and an acceleration part (both retarded). In this way a new derivation of the Lorentz-Dirac equation of motion emerges. In particular, in the light of such a derivation, the physical meaning of the "Schott term" is fully elucidated. The asymptotic condition of uniform motion in the remote past is seen to be essential for the establishment of the differential equation of motion. In contrast with previous discussions, no advanced field need be introduced in any step of the work. Electromagnetic radiation is treated with no single use of asymptotic procedures. Further physical insight is obtained into Rohrlich's criterion for radiation.

Gravitational radiation is calculated for the situation of a small mass passing a large mass in an unbound trajectory, where the velocity of the small mass can be relativistic. This allows one to study gravitational radiation for cases in which the slow-motion approximation is not valid. The gravitational potentials, or perturbations in the metric, arising from the small mass, are determined explicitly by solving the perturbed field equations of general relativity, which are obtained by expanding the metric about a metric representing the geometry of the large mass. From these the energy flux of the emitted waves is calculated. In the nonrelativistic limit, the results agree with those of the slow-motion approximation. The qualitative behavior of the radiation at extreme relativistic velocities is discussed, and is found to disagree with what one would expect from the fast-motion approximation in that same limit. Numerical results are presented for the total energy, power, and angular distribution of energy radiated for a range of velocities from $0.01c$ to $0.9999c$. Significant features in the extreme relativistic limit are the peaking of the radiation in the forward direction and the peaking also in time, which both occur in electromagnetic radiation, and the fact that the total energy radiated in one transit is proportional to ${(1-\frac{v^2}{c^2})}^{-\frac{3}{2}}$.