The Reggeization scheme of Carlitz and Kislinger is extended to daughter trajectories.

A class of models describing the collision of Regge poles and their associated Mandelstam branch cuts is described, from which one concludes that Regge trajectories generally develop left-hand branch cuts as a result of such collisions, and in terms of which the singularities in the $l$ plane of the partial-wave amplitude and the asymptotic behavior of the scattering amplitude can be discussed.

The $S$-matrix derivation of broken-symmetry mass formulas previously given for the $J^P=0^{-}$ octet of pseudoscalar mesons is extended to the octet of $J^P={\frac{1}{2}}^{+}$ baryons. We also discuss the relation of crossing to the discrete symmetries of $C$ and $T$ (and of Bose statistics).

We propose a model of the weak interactions for leptons, including scalar intermediate bosons $C^{\pm }$ with a negative metric in addition to the usual weak vector bosons $B^{\pm }$. With this modification, problems of divergence and high-energy behavior are greatly reduced. If the logarithmic weak and electromagnetic self-mass divergences are assumed to cancel each other, the coupling constant $g$ and the mass $m_B$$(=m_C)$ of the weak boson are predicted to be $g^2=\frac{3}{2}{e}^2$ and $m_B=137.7\mathrm{}{m}_p\left(\simeq \frac{m_p}{\alpha }\right)$ (where $m_p$ is the proton mass), respectively. The finite self-masses are $\delta m_l\simeq \left(\frac{3e^2}{16{\pi }^2}\right)m_l$ $\ln {\left(\frac{m_B}{m_l}\right)}^2$ ($\simeq 0.044m_e$ for an electron). It is shown that the renormalization of both the electromagnetic and weak interactions can be consistently accomplished in spite of the existence of the parity-violating interaction. The contributions of the weak interaction to the anomalous magnetic moments of leptons are calculated to be $\frac{1}{2}(g_l-2)=-\frac{G{m_l}^2}{12\sqrt{2}{\pi }^2}$ (≃-7.7 × ${10}^{-10\mathrm{}}$ for $\mu$). Various weak reactions of leptons are discussed. For example, $\sigma ({\nu }_{\mu }+e\to \mu +{\nu }_e)$ approaches a constant value $\frac{G^2{m_B}^2}{\pi }\simeq 2.8\times {10}^{-34\mathrm{}}$ ${\mathrm{cm}}^2$ at high energies.

The electromagnetic properties of the neutrino in the presence of weak interactions have been studied using the two-scalar-boson-exchange model due to Kummer and Segrè. The logarithmically divergent amplitudes of the lowest-order Feynman diagrams are regularized by means of the usual renormalization procedure. The numerical results, such as the charge radius of the neutrino, turn out to be of the same order as those of the previous calculations by Bernstein and Lee based on the intermediate-vector-boson theory.

An infinite-component field theory is proposed to describe the $\rho$ Regge family of particles. Using the generators of the group $O(3, 2)$, a class of first-order wave equations is obtained. The simplest of this class of equations is solved to yield a rising mass spectrum with a hydrogenlike accumulation point. Because only finite-dimensional representations of the homogeneous Lorentz group appear in the theory, it is free from many of the difficulties, including noncausality, which have plagued other infinite-component field theories with nontrivial mass spectra.

We investigate the possible subsidiary conditions compatible with the equations of motion of dual-resonance models for the unrealistic choice of $\alpha \mathrm{}\left(0\right)=1\mathrm{}$. In the language of four-dimensional harmonic oscillators, we find one subsidiary condition for each mode of oscillation. All time components (in the c.m. system) can therefore be eliminated. We discuss the possibility of relaxing the condition $\alpha \mathrm{}\left(0\right)=1\mathrm{}$.

An attempt has been made to extend the minimal current algebra of Bjorken and Brandt starting from a gauge-field Lagrangian and including in it nonets of scalar and pseudoscalar fields and making use of canonical communtation relations both for spin-zero and spin-one fields. To apply it to the problem of weak-interaction divergences, we identify suitably normalized fields with weak currents and scalar and pseudoscalar densities introduced by Gell-Mann. As in the case of Bjorken and Brandt, we go to the limit $m_0\to 0$, $g_0\to 0$ such that $\frac{g_0}{{m_0}^2}=\mathrm{const}\ne 0$, where $m_0$ and $g_0$ are masses and coupling constants of the Yang-Mills field. In the extended minimal algebra, the nonleptonic weak processes are free of all divergences to lowest order and of a class of leading divergences to all orders in the weak-coupling constant.

We extend the results of a group-theoretical analysis of the $t<\mathrm{}0\mathrm{}$ multiperipheral equation to the case $t=0\mathrm{}$ for pairwise equal masses. Using variables discussed in a previous paper, we diagonalize the equation in the Bali-Chew-Pignotti (BCP) model with respect to the $O(2, 1)$ group and relate the solutions to the equation so obtained with the solutions obtained after diagonalization with respect to the $O(3, 1)$ group. Poles in the $O(3, 1)$ partial-wave amplitude give rise to the expected sequence of daughter poles in the $O(2, 1)$ partial-wave amplitude. At general momentum transfer, we establish factorization at the $O(1, 1)$ poles in the decomposition of the BCP amplitude, and present further simplifications to the diagonalized equations based upon this model.

Model amplitudes are discussed which can accommodate nonlinear trajectories, satisfy duality, and have a nondegenerate spectrum of daughter resonances.

We examine the formal foundations of quantum electrodynamics in the infinite-momentum frame. We interpret the infinite-momentum limit as the change of variables $\tau =2^{-\frac{1}{1}}\mathrm{}(t+z)\mathrm{}$, $\mathcal{Z}=2^{-\frac{1}{1}}\mathrm{}(t-z)\mathrm{}$, thus avoiding limiting procedures. Starting from the Feynman rules, we derive a $\tau$-ordered perturbation expansion for the $S$ matrix. We then show how this expansion arises from a canonical formulation of the field theory in the infinite-momentum frame. We feel that this approach should lead to convenient approximation schemes for electrodynamics at high energy.

Perturbation-theory techniques are used to generate an approximate transcendental equation for the leading Regge pole produced by an infinite sum of Feynman ladder diagrams. This equation is not restricted to the weak-coupling domain and can be used to produce trajectories with $\alpha \mathrm{}\left(0\right)>0\mathrm{}$. The equation is solved both below and above the elastic and inelastic thresholds to provide the first complete description of a relativistic Regge trajectory in the ladder-diagram model. The same technique is used to generate potential-theory trajectories which agree with the exact solutions obtained by more complicated methods.

A set of model amplitudes, characterized by two functions $h\left(x\right)\mathrm{}$ and $\omega \mathrm{}\left(x\right)\mathrm{}$, is given which satisfies the conditions of duality, crossing symmetry, and Regge behavior. These model amplitudes can be written as $G\left(\alpha \mathrm{}\right(s), \alpha \mathrm{}(t\left)\right)=\int 0^1\mathrm{dx}h\left(x\right)\mathrm{}{x}^{-\alpha \mathrm{}\left(t\right)\mathrm{}}{\left[\omega \mathrm{}\right(x\left)\right]}^{-\alpha \mathrm{}\left(s\right)\mathrm{}}$ and are generalizations of the beta function. The model has resonances of zero width, like Veneziano's model. We prove that, under certain conditions, the model amplitude has Regge behavior as $s\to \infty$ with $\arg \mathrm{}(-s)<\pi$ and $t$ fixed, and that it dies exponentially as $s\to \infty$ with $0<\left|\arg \mathrm{}\right(s\left)\right|<\mathrm{}\pi$ and $u$ fixed, as does the amplitude in Veneziano's model. We give specific examples of $\omega \mathrm{}\left(x\right)\mathrm{}$ and $h\left(x\right)\mathrm{}$ which satisfy all the required conditions, and show that the models may be generalized to models which bear the same relation to the model $\frac{\Gamma (1-\alpha \mathrm{}(s\left)\right)\Gamma (1-\alpha \mathrm{}(t\left)\right)}{\Gamma (1-\alpha \mathrm{}(s)-\alpha \mathrm{}(t\left)\right)}$ as the above models bear to the beta function.

Given data that are plausibly described by the multi-Regge model, it is shown that a distribution in a certain angle related to a triple subenergy (which in turn is related to $\omega$) is flat if (1) the amplitude is independent of that subenergy and (2) a certain selection of data is made. The test can be applied to up to four internal vertices, thus allowing a complete analysis of $\omega$ dependence in reactions with six, seven, or eight particles in the final state.

The chiral transformation of asymptotic fields in the presence of symmetry breaking is discussed in a model of interaction between a quantized Dirac field and an external electromagnetic field. We conclude from this model that the asymptotic fields will generally transform nonlocally and nonlinearly. Only if the symmetry breaking is small, or if the interaction is sufficiently weak, will the asymptotic fields transform locally and linearly as a first approximation.

The dependence of an eigenvalue hadron mass $m$ on a constituent quark or antiquark mass $M_{\mathrm{}\left(s\right)\mathrm{}}$ must be such that $\left|\frac{\partial m}{\partial M_{\mathrm{}\left(s\right)\mathrm{}}}\right|<\mathrm{}1\mathrm{}$ if the dynamics of a system composed of $n^{\prime }$ quarks and ($n-n^{\prime }$) antiquarks is described by an $n$-particle Dirac Hamiltonian. By evoking the quasi-independent quark-model approximation, we obtain $\frac{\partial m_N}{\partial M}\simeq 2.378$ for the rate at which the nucleon mass would change theoretically subject to a simultaneous increase in the three constituent strangeness-zero quark masses $M$. In the case of a potential with Coulomb-Schwinger homogeneity under scale transformations, we find a linear mass relation without appeal to the quasi-independent quark-model approximation. The latter mass relation yields correct mass values for the parallel-spin hadrons with $L=0\mathrm{}$ (the $J^P=1^{-}$ mesons and $J^P={\frac{3}{2}}^{+}$ baryons) and exhibits consistency in the $\mathrm{}\left|S\right|=1\mathrm{}$ quark-antiquark mass $M^{\prime }$ to $S=0\mathrm{}$ quark-antiquark mass $M$ ratio: $\frac{M^{\prime }}{M}=1.345\mathrm{}\pm \mathrm{}\left(0.005\right)\mathrm{}$.

The stability of the Schwarzschild exterior metric against small perturbations is investigated. The perturbations superposed on the Schwarzschild background metric are the same as those given by Regge and Wheeler, consisting of odd- and even-parity classes, and with the time dependence $\exp \mathrm{}(-ikt)\mathrm{}$, where $k$ is the frequency. An analysis of the Einstein field equations computed to first order in the perturbations away from the Schwarzschild background metric shows that when the frequency is made purely imaginary, the solutions that vanish at large values of $r$, conforming to the requirement of asymptotic flatness, will diverge near the Schwarzschild surface in the Kruskal coordinates. Since the background metric itself is finite at this surface, the above behavior of the perturbation clearly contradicts the basic assumption that the perturbations are small compared to the background metric. Thus perturbations with imaginary frequencies that grow exponentially with time are physically unacceptable, and hence the metric is stable. Perturbations with real values of $k$ representing gravitational waves are also examined. It is shown that the only nontrivial stationary perturbation that can exist is one that is due to the rotation of the source, which is given by the odd perturbation with the angular momentum $l=1\mathrm{}$. The significance of solutions with complex frequencies is pointed out, as is the lack of a theorem (completeness of the eigenfunction) for the even-parity case to parallel the Sturm-Liouville theory, which is applicable to the odd-parity case. Such a theorem would be required to convert the computations indicating stability as given here into a fully rigorous stability theorem.

We show how one can calculate the Veneziano amplitude for an arbitrary nonplanar diagram with one closed loop. In particular, we calculate explicitly the nonplanar diagram with four external particles and one "twisted vertex." If we write the amplitude $M=\int d^4\Pi$ $\mathfrak{M}\left(\Pi \right)$, we find that our result for $\mathfrak{M}\left(\Pi \right)$ is periodic in the loop momentum $\Pi$, so that the expression for $M$ does not converge. We show that, if we integrate over just one period, we obtain the correct result for the imaginary part of the amplitude in the $s$ channel. Consequently, we adopt this as our definition of the unitary nonplanar amplitude. Finally, we show that with this redefinition, $M$ satisfies the duality requirement that it be completely symmetric in the "untwisted" momenta.

We present detailed calculations illustrating the breakdown of the Bjorken limit in perturbation theory, in the "gluon" model of strong interactions. To second order in the gluon-ferrmion coupling constant in the scalar, pseudoscalar, and vector coupling models, we calculate the Bjorken-limit commutator of a pair of currents of arbitrary (vector, axial-vector, scalar, pseudoscalar, tensor) type. To fourth order in the coupling, in the scalar- and pseudoscalar-gluon models, we determine the leading logarithmic behavior of the $S{U}_3$-antisymmetric part of the vector-vector commutator. In the body of the paper we present the main results and discuss their various features and implications. The computational details are relegated to two appendices.

An $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ effective Lagrangian, generated by a massive vector field, has been formulated with the aid of the group-geometrical method employed in our earlier papers. This formalism is manifestly invariant under nonlinear transformations, and permits an exact and compact treatment of the non-linearities. Our $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ treatment incorporates only those minimal departures from $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ that are required to secure a reasonable model, and the new Lagrangian contains no free parameters not already present in the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ model. These parameters are determined by fitting the linearized theory to the pion-nucleon system. Results for other members of the octets, such as the Callan-Treiman formula, are then in satisfactory agreement with experiment or with the current algebra. It is hoped that the formalism will permit the treatment of the nonlinearities to be extended beyond the discussion of the tree diagrams.