Differential cross sections for the quasi-two-body meson process $\sigma \sigma \to \left(\sigma \sigma \right)\sigma$ (where $\sigma$ is a scalar, isoscalar pion) are calculated using the Bardakci-Ruegg formula for $\sigma \sigma \to \sigma \sigma \sigma$. The slope of $\frac{d\sigma }{\mathrm{dt}}$ in the 2 → 2 reaction is found to change with the ($\sigma \sigma$) mass in a manner similar to experiment. The shape of predicted resonances in the ($\sigma \sigma$) mass is, however, distinctly different from the Breit-Wigner form.

An elementary heuristic discussion of relativistic systems of particles in mutual gravitational interaction is given. Both in the boson and fermion cases, the main features of rigorous treatments are very simply recovered. A conceptual dilemma touching the use of various coordinate systems in such considerations is clarified. Finally, it is shown that in any realistic boson system, the relativistic effects of gravitation are probably masked by the other fundamental interactions, in contradistinction with fermion systems.

The analysis of nonoscillating scattering amplitudes carried on in two previous articles is completed by proving that, as conjectured, the asymptotic relations obtained also hold, in the mean, along physical directions.

That part of the structure of particle reactions of arbitrary spins which is independent of dynamics is investigated in the case when identical particles are contained in the reaction. First, a general discussion of the structure of composite reactions reveals that mirror relations arise naturally in such reactions even when the characteristic quantity contains an even number of letters. It is also shown that there is a close analogy between the constraints of time-reversal invariance and those of identical particles, as applied to their respective self-transforming reactions. It is then easy to determine the changes in the tests of other conservation laws, like parity conservation and time-reversal invariance, in the presence of identical particles. It is found that some tests are erased by the presence of identical particles; others are simplified. With one exception, however, the presence of identical particles does not preclude tests of other symmetries for a reaction. Identical particles also widen the range of self-transforming reactions. Relatively little can be said about reactions with only one pair of identical particles. Two examples illustrate the results in detail.

The possibility is examined of extending the $\pi \pi \to \pi \pi$ and $\pi A_1\to \pi \pi$ Veneziano amplitudes to the case of one or two mesons off the mass shell in a fashion consistent with chiral current algebra, partial conservation of axial-vector current, and conserved vector current. Soft-pion formulas for the pion-vector form factor and the $\sigma$-commutator vertex are deduced. The hard-meson analysis shows that if one wishes to maintain Veneziano form with two mesons off the mass shell, some of the $\pi A_1\to \pi \pi$ amplitudes must develop poles in one of the off-shell momenta at $p^2=0\mathrm{}$. The soft-pion vertex function equations are seen to be a consequence of removing the leading part of this singularity at the soft-pion point $p^{\mu }=0\mathrm{}$. However, the resultant amplitudes are still singular at $p^2=0\mathrm{}$, $p^{\mu }\ne 0$ for a continuum of $s$, $t$, and $u$. The fixed poles implied by the presence of form factors (when two mesons are off shell) are seen to occur only in those amplitudes orthogonal to the $A_1$ polarization vector and do not affect the amplitudes that make nonzero contributions for $\pi A_1\to \pi \pi$ scattering on the mass shell.

A simple bootstrap model of vector mesons is constructed in which the origin of octet enhancement is examined. A parametrized-one-pole approximation is used to solve the $\frac{N}{D}$ equations. The pole parameter is determined by the bootstrap condition and the type of asymmetric solution that emerges. It is demonstrated that the emergence of the Gell-Mann-Okubo formula follows from two general dynamical requirements: the bootstrap self-consistency condition and the requirement that the "vector mesons" are real physical particles and not ghost states.

The Yang (or droplet) model and the Regge model for high-energy diffractive processes are contrasted, their complementarity emphasized. The combination of the physical aspect of the former with the mathematical aspect of the latter gives rise to a bootstrap model which has far-reaching consequences. The assumptions of the bootstrap model are: (a) High-energy inelastic processes are dominated by the two-cluster diffractive fragmentations; (b) the $s$ dependences of diffractive scattering and fragmentation are the same; and (c) partial-wave amplitudes can be continued uniquely into the complex $j$ plane. We study the boot-strap of the Pomeranchon in both the $s$ and the $t$ channel using inelastic unitarity without approximation. The Pomeranchuk singularity is found to be a branch point with $\alpha \mathrm{}\left(0\right)=1\mathrm{}$ exactly; discontinuity of the associated cut vanishes at the tip. Both forward and nonforward cases are considered. Various properties of diffractive scattering and fragmentation at high energy are obtained. On the basis of the bootstrap model, we make predictions on (1) the asymptotic behavior of ${\sigma }_{\mathrm{tot}}$, (2) the ratio of real to imaginary parts of scattering amplitude, (3) the absence of physical manifestation of the Pomeranchon, (4) the dependence of fragmentation cross section on the effective masses of the particle clusters, (5) the average multiplicity of hadron production, (6) the diffraction peak of fragmentation process, and (7) the relationship between $\mathrm{pp}$ and $\gamma p$ diffraction peaks. All of these predictions are consistent with whatever relevant data are available at present. If, in addition, a technical assumption is made concerning the Reggeization of production amplitudes, the precise nature of the Pomeranchuk branch point can be determined.

The multiplet of the pseudoscalar mesons $K$, $\pi$, $\eta$, and ${\eta }^{\prime }$ with their scalar partners is analyzed as a reducible representation of chiral $S{U}_2\otimes S{U}_2$. We introduce a spontaneous breaking mechanism of this symmetry, leaving isospin and hypercharge as residual invariance. We consider also in this scheme a Lagrangian model, which exhibits various remarkable group-theoretical properties, and leads to numerical values, mass, and mixing-angle relations, in very good agreement with experimental data.

Using Joos-Weinberg fields, a Lehmann-Symanzik-Zimmermann reduction formalism, and assuming sufficient analyticity, it is shown that the $S$-matrix elements for processes involving any number of particles with arbitrary spins admit a MacDowell symmetry as a consequence of $\mathrm{CT}$ conservation (or maximum $\mathrm{CT}$ violation).

The asymptotic behavior for electroproduction of one hadron in the limit $q^2\to \infty$ (spacelike) with $\frac{\left(\mathrm{laboratory}\mathrm{energy}\right)}{q^2}$ large and fixed is derived by means of the Bethe-Salpeter equation, which takes account of the vector property of the photon.

We derive an infinite number of sum rules for the process $a+b\to c+d$, where $a$, $b$, $c$, and $d$ are particles of arbitrary mass and spin. Each of the sum rules involves a finite number of partial waves. They are implied by the crossing symmetry of the system and are complete. A classification of these relations into an independent set suggests a basis for the two-variable expansion of the scattering amplitude. The basis has the further virtue that it explicitly displays the kinematic singularities of the amplitude and the threshold and pseudothreshold zeros of the partial waves. The formalism is also valid for a decay process. The partial waves in the sum rules then refer to amplitudes where two of the three final particles are in a state of definite angular momentum, while the region of expansion becomes the Dalitz plot.

An integral representation is given for the matrix element, between single-particle states, of the product of currents at the same space-time point. It is applied to obtain sum rules from the equations of motion in the Sugawara model. These sum rules, unlike the conventional Fubini-type sum rules, involve triple integrals over the absorptive parts of off-mass-shell current-hadron scattering amplitudes. Their physical content and utility require further investigation.

A general study of the space-time symmetries of the scattering amplitude is made. The scattering amplitude in the c.m. frame has $O(2, 1)$ symmetry in the physical regions of the crossed channels. By working in this frame, we can use the same definition of the helicity as that used in the usual angular momentum analysis. For the pairwise equal-mass case, the helicity amplitude in the forward scattering region of the crossed channels has $O(2, 2)$ symmetry in the c.m. frame, and $O(3, 1)$ symmetry in the brick-wall frame. We apply an $O(2, 2)$ expansion to a multiparticle system. We also continue the $O\left(4\right)\mathrm{}$ expansion in the brick-wall frame into the $O(3, 1)$ region, and show the equivalence between the $O\left(4\right)\mathrm{}$ expansion and the $O(3, 1)$ expansion. Finally, we point out the difference between the implications of the three-dimensional and four-dimensional symmetries.

Results previously obtained for the meson trajectories in the Veneziano quark model are here derived for the baryon trajectories. The $\mathrm{SU}\left(0\right)\mathrm{}$ spectrum of particles and the $SU(0, 0)$ spectrum of trajectories are analyzed. Vertex functions for three physical particles or for two physical particles and one Reggeon are calculated. We can adjust two mixing parameters so that the ratios between the couping constants will be those predicted by $\mathrm{SU}{\mathrm{}\left(0\right)\mathrm{}}_W$. The model is applied to determine quantitatively the value of the $\rho$ residue used in Regge fits or pion-nucleon scattering.

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.