Differential equations for perturbations on the Schwarzschild metric have appeared incorrectly in the literature. This paper presents the correct set of equations.

The "no-go" theorem of Grodsky and Streater is extended to nonrenomalizable local theories in which the two-point function may increase faster than any polynomial in momentum space. As a result the non-existence of a local irreducible infinite-component field with a nontrivial mass spectrum is demonstrated without "technical" assumptions.

The Padé approximation method of calculating the $S$ matrix in pion-nucleon pseudoscalar coupling field theory is discussed above production threshold. It is shown that the evaluation of the Padé approximant to the $S$ matrix, reduces, for c.m. energies below second production threshold, to the solution of an integral equation, with an energy-momentum-conserving kernel on the nucleon two-pion three-body subspace. It is shown that a reduction of this type cannot be expected to hold above the second threshold unless further approximations are made.

We show that for elastic processes which have absorptive channels below the physical threshold, the absorptive partial-wave amplitude $a_l\mathrm{}\left(s\right)\mathrm{}$ has the sign of ${\mathrm{}(-1\mathrm{})\mathrm{}}^l$ for $s$ below the physical threshold, where $l$ is the orbital angular momentum. The result is valid for any stable particles of arbitrary spins. More generally, for elastic partial wave transitions between states of definite parity $P=\epsilon {\eta }_1{\eta }_2$ (where ${\eta }_1$ and ${\eta }_2$ are the intrinsic parities and $\epsilon =\pm 1$), the sign of the absorptive amplitude below threshold is that of $\epsilon$.

Relativistic corrections are evaluated for a model three-body problem consisting of three identical bosons in $s$-wave pairs through separable potentials. For the potentials, Shirokov's second-order (in $\frac{v}{c}$) correction, which vanishes in the c.m. frame of a two-body system, is considered. No corrections are considered for the kinetic energy, as it is argued that any such correction is intimately linked with the precise structure assumed for a three-particle dynamics and cannot merely be incorporated through the modification $\frac{p^2}{2m}\to {(m^2+p^2)}^{\frac{1}{2}}-m$ for each particle, which affects even the two-particle dynamics on the energy shell (and hence also the parameters of the two-body potential). The reduction in the quantity analogous to the binding energy of the triton is found to be of the order of 5%, while the curve for the counterpart of the doublet $n-d$ scattering length ($a_{\frac{1}{2}}$) versus the two-body strength parameter ($\lambda$) is slightly shifted to the right as a result of the relativistic correction.

It is explicitly shown that the integration parameter in the background Regge integral can be understood as a time-delay variable, its average value being the time delay and distortion caused on a wave packet by a scatterer, as seen in the Breit system of coordinates. By means of the usual exponential Fourier transform, another time-delay variable is defined with a similar meaning for the scattering process as seen in the laboratory system of coordinates. The average over the energy of the scattering amplitude is related to a cutoff in the time-delay variables in both cases. A smooth energy variation of the scattering amplitudes corresponds to a small range of variation of the time-delay variable in the transformed amplitude. A Fourier analysis of the Veneziano-model amplitude results in a simple periodic structure in the time-delay variable. A parametrization of the scattering amplitudes in the time-delay representation is proposed, valid at high and intermediate energies. It is pointed out that the non-spin-flip pion-nucleon scattering amplitudes are consistent with such a parametrization. An interpretation of models with linearly rising trajectories in terms of multiscattering processes seems natural in the time-delay representation.

The nonforward multiperipheral integral equation for the Reggeon-particle absorptive amplitude is generalized to include complete dependence on the Toller-angle variable.

Starting with the $\mathrm{SW}\left(3\right)\mathrm{}$ group as the fundamental symmetry group, we develop a theory for the continuous breaking of this symmetry, in analogy with the theory based on the $W\left(3\right)\mathrm{}$ group discussed earlier. Despite what appears at first sight, there are considerable differences in the two theories, and in the present case one can avoid many difficulties associated with the $W\left(3\right)\mathrm{}$ theory. Several refinements of the arguments given earlier are also presented, which lead to mass formulas for $\pi$ and $K$ as well as the $\eta$ meson. A method to calculate finite-mass corrections to soft-pion theorems is also proposed and applied to a few problems.

A chiral $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ Lagrangian model with Gell-Mann-Oakes-Renner symmetry breaking is derived explicitly in terms of the pseudoscalar $\mathrm{SU}\left(3\right)\mathrm{}$ octet fields, ${\phi }_i$. We show that the imposition of the Adler consistency condition on $\eta$ mesons implies $\eta ={\phi }_8+c({{\mu }_K}^2-{{\mu }_{\pi }}^2)v_0$. Hence, only if $\eta$ is assumed to transform as ${\phi }_8$ do we arrive at a difficulty of the Veneziano model: ${{\mu }_K}^2={{\mu }_{\pi }}^2={{\mu }_{\eta }}^2$.

The deuteron wave function at short distances is investigated by assuming three types of input separable potentials: (i) central+repulsive core, (ii) central+tensor, (iii) central+tensor+repulsive core ($C+T+H$). The wave function is parametrized through a fit to the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ phase shifts and the deuteron parameters. A detailed comparison with experiment of the electromagnetic form factors of the deuteron, calculated with the above three types of potentials, clearly indicates the importance of the repulsive core in the two-nucleon system. The best results, obtained with the $C+T+H$ potential, give very good agreement with experimental values. Inclusion of Shirokov's relativistic corrections in the basic input potential improves the results still further.

It is demonstrated through an explicit model that the weak high-subenergy tail of the multiperipheral kernel, acting in conjunction with the strong low-subenergy component, is capable of producing a high-ranking output Regge doublet with vacuum quantum numbers. We show that association of the upper doublet member with the $P$ (Pomeranchon) and the lower with the $P^{\prime }$ is consistent with experimental total, elastic, and diffractive dissociation cross sections, as well as with multiplicity of produced pions, and predicts a Pomeranchon slope near $t=0\mathrm{}$ that is roughly half normal. As $t$ becomes negative, the Pomeranchon slope decreases to a small value, while for $t$ positive the slope increases to a normal value, the $P$ trajectory containing the particles usually assigned to the $P^{\prime }$. The latter trajectory has a converse behavior, with small slope for positive $t$ and normal slope at negative $t$. The $P$ and $P^{\prime }$ trajectories thus exchange "normal" and "abnormal" roles near $t=0\mathrm{}$.

We consider the fourth-order contribution to the electromagnetic mass difference of zero-mass pions in a model whose high-energy behavior is that given by algebra-of-fields commutation relations. We find this quantity to be logarithmically divergent. Possible sources of this divergence are discussed.

We continue the work of Coleman, Gross, and Jackiw who have argued that the products of currents occurring in Sugawara's model should be interpreted as a singular limit as the coordinate separations approach zero, if the currents are the usual $\overline{\psi }\frac{1}{2}{\lambda }^a{\gamma }_{\mu }\psi$. They found that ${\theta }_{\mu \nu }$ of the free-fermion theory can be expressed in the Sugawara form. We now point out that their method works even in the presence of interactions; consequently, Sugawara's energy-momentum tensor can only describe the massless free part of any fermion theory. This can also be seen from the fact that Sugawara's equations of motion for fermion currents are equivalent to $\gamma ·\partial \psi =0\mathrm{}$. We suggest that the most natural interaction term to be added to Sugawara's ${\theta }_{\mu \nu }$ is the fully normal-ordered $\frac{1}{2}F:{V_{\lambda }}^a{V}^{a\lambda }$: occurring in Heisenberg's nonlinear spinor theory. The resulting equations of motion for the currents still have the Sugawara form (with an additional normal-ordered term). Finally, we attempt to derive the commutators of ${\theta }_{\mu \nu }$ with itself and with the currents, using current algebra without any reference to the underlying fields. Here we do not know how to handle timelike separations, and with spacelike separations alone we do not find a consistent solution.

We continue our study of the contribution of a Regge-pole family to the scattering amplitude. Total-spin states are constructed, which allow us to expand the off-shell scattering amplitude in a series of representation functions of the Lorentz group. The contribution of the family is transformed into a form which strongly reminds one of a Lorentz-pole contribution, even at nonzero momentum transfers $t$. The most general dependence of the Regge-pole residues on the total helicity $\lambda$ and daughter index $\kappa$ is given in terms of two arbitrary functions of composite variables $\mathrm{tj}(j+1\mathrm{})\mathrm{}$ and $\Delta \lambda$, where $\Delta ={\left\{\right[t-{(m_1-m_2)}^2\left]\right[t-{(m_1+m_2)}^2\left]\right\}}^{\frac{1}{2}}$. The contribution of a family to the scattering amplitude constructed from these residue functions and the trajectories given in a previous paper is the most general one satisfying all kinematic constraint relations at $t=0\mathrm{}$, pseudothreshold, and threshold.

We work out an analytic method for the discussion of broken symmetries. After the demonstration of the power of the method on a quantum-mechanical example, we apply it to the question of broken $SL(2, C)$ symmetry and families of Regge trajectories. As a result, we obtain all the constraints on daughter trajectory functions in a closed functional form for arbitrary Lorentz quantum numbers $\sigma$ and $M$.

The scattering amplitude for photon-photon elastic scattering in quantum electrodynamics is explicitly obtained for the case where the energy is very large while the momentum transfer is small compared with the rest energy of the electron (divided by $c$). This amplitude has no infrared divergence and, at high energies, the fourth- and sixth-order contributions are negligible. The explicit answer for the eighth order is remarkably simple. In terms of helicity states, it is found that the amplitudes $\mathrm{RR}\to \mathrm{RR}$ and $\mathrm{RL}\to \mathrm{RL}$ are the same, while the amplitude $\mathrm{RR}\to \mathrm{LL}$ is different but of the same order of magnitude. By the optical theorem, the total cross section for $\gamma +\gamma \to e^{-}+e^{-}+e^{+}+e^{+}$ is found to be 6.5 μb at high energies.

Conservation of the axial-vector current is assumed, together with the soft-$\pi$-meson hypothesis and the pole dominance of the vector and the axial-vector current. It is shown that there is a symmetry between the families of the residue functions of Regge poles in the $t$ channel and those in the $u$ channel for the $s$-channel helicity amplitude of the process ${\rho }^{+}\left(h_1\right)+{A_1}^{+}\left(h_2\right)\to {\rho }^{+}\left(h^{\prime }\right)+{A_1}^{+}\left(h^{\prime }\right)$. Here $h_i$ and $h^{\prime }$ are the helicities of the particles $\rho$ and $A_1$, and they are restricted to 1 or -1. The direct relation between this symmetry and a Veneziano-type formula is discussed. A relation between the residues of the $\rho$ and $A_2$ Regge poles is also given for the general helicity case.

A discussion is given of Lovelace's conjecture that chiral symmetry for soft mesons plus the absence of exotic resonances yields Veneziano formula with no secondary terms. It is shown that the conditions of the above conjecture imply not only the exchange degeneracy of various Regge trajectories, but also the degeneracy of the reduced residue functions of the $\rho$ and $f$ or the $\omega$ and $A_2$ Regge families (including their daughter poles) of amplitudes of the type $a^{\pm }+{A_{\mu }}^{\pm }\left({V_{\mu }}^{\pm }\right)\to a^{\pm }+{A_{\mu }}^{\pm }\left({V_{\mu }}^{\pm }\right)$, where $a^{+}$ is any particle with positive charge, and $A_{\mu }\left(V_{\mu }\right)$ is the axial-vector (vector) current. The latter statement is proved for arbitrary values of $t$ (or $u$) without suffering from the Regge cuts. This appears to be a necessary condition to write a Veneziano-type formula.

The possibility of retaining the axial-vector field-current identity for interacting $\pi$ and $A_1$ fields which are dynamically independent in the sense that $[a_{\mu }, \pi ]=0\mathrm{}$ and $[a_{\mu }, \dot{\pi }]=0\mathrm{}$ at equal times is shown to lead to an inconsistency unless the mass of the pion is zero. A lower bound on the parameter $\frac{G_{\rho }}{\left(\sqrt{2}{m}_{\rho }{F}_{\pi }\right)}$ is established for massless pions.

The problem of converting nonlinear realizations of $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ has been approached in three different ways. Weinberg has introduced a matrix $\Lambda \left(\pi \right)$ with well-defined properties under infinitesimal chiral transformations and has shown that it can be used to linearize nonlinear fields. Coleman, Wess, and Zumino have achieved the same result by treating the pion fields as parameters of finite chiral rotations; and the present author has made use of the fact that linear realizations are eigenstates of the Casimir operators and close upon themselves under the action of chiral operators. Here we show that the three approaches are equivalent to one another. We first calculate the most general expression for $\Lambda \left(\pi \right)$, and show that certain of its matrix elements have the desired properties with respect to the Casimir and chiral operators. We then show that the method of Coleman, Wess, and Zumino is a special case of $\Lambda \left(\pi \right)$. In the course of the analysis, we find that $\Lambda \left(\pi \right)$ is manifestly covariant under redefinitions of the pion field. To illustrate the usefulness of converting nonlinear fields to linear forms, we calculate $\pi -\pi$ scattering lengths and construct weak currents for meson decay.