A relativistic quasipotential equation is derived from the conventional Hamiltonian formalism and old-fashioned "noncovariant" off-energy-shell perturbation theory in a similar way to that by which the four-dimensional Bathe-Salpeter equation is obtained from the off-mass-shell Feynman rules. The three-dimensional equation for the (off-energy-shell) scattering amplitude appears as a straightforward generalization of the nonrelativistic Lippmann-Schwinger equation. The corresponding homogeneous equation for the bound-state wave function and the normalization condition for its solutions are derived from the equation for the complete four-point Green's function. In order to obtain a solvable model, we consider a simplified version of the quasipotential equation which still reproduces correctly the on-shell scattering amplitude and is consistent with the elastic unitarity condition. It involves a "local" approximation to the potential $V(p-q)\mathrm{}$ which defines the kernel of our integral equation (the integration being carried over a two-sheeted hyperboloid in the energy-momentum space). It is shown that for the scalar Coulomb potential $V(p-q)=\frac{\alpha }{{\mathrm{}(p-q)\mathrm{}}^2}$, our model equation is equivalent to a simple infinite-component wave equation of the type considered by Nambu, Barut, and Fronsdal. The energy eigenvalues for the bound-state problem are calculated explicitly in this case and are found to be $O\left(4\right)\mathrm{}$ degenerate (just as in the nonrelativistic Coulomb problem and in Wick and Cutkosky's treatment of the Bethe-Salpeter equation in the same approximation).

The electroproduction inelastic structure functions $W_i(\kappa , \nu )$ ($i=1, 2$; $\kappa =q^2=\mathrm{momentum}\mathrm{transfer}\mathrm{squared}$; $\nu =q·p=\mathrm{energy}\mathrm{transfer}$; $p^2=1\mathrm{}$) are studied in the Bjorken ($A$) limit ($\nu \to \infty$, $\rho \equiv -\frac{\nu }{\kappa }$ fixed) and in the Regge ($R$) limit ($\nu \to \infty$, $\kappa$ fixed). Finite $A$ limits [$F_2\left(\rho \right)=\lim \nu W_2(\kappa , \nu )$, $F_1\left(\rho \right)=\lim W_1(\kappa , \nu )$] and Pomeranchuk-dominated $R$ limits [$w^2\left(\kappa \right){\nu }^{-1\mathrm{}}=\lim W_2$, $w_1\left(\kappa \right)\nu =\lim W_1$] are assumed. These two limits are first related by use of Deser-Gilbert-Sudarshan (DGS) representations for causal functions $V_i(\kappa , \nu )$ related to the $W_i$ by $W_2=\kappa V_2$ and $W_1=\kappa V_1-{\nu }^2{V}_2$. The above $A$- and $R$-limit assumptions, together with a smoothness assumption on the DGS spectral functions motivated by the existence of some equal-time commutators, are shown to imply that $F_2\left(\infty \right)=w_2(-\infty )\equiv w_2=\mathrm{const}$ and ${\lim }_{\rho \to \infty }\frac{F_1\left(\rho \right)}{\rho }=-{\lim }_{\kappa \to -\infty }{w}_2\left(\kappa \right)\kappa \equiv w_1=\mathrm{const}$. These results agree with experiment if $w_i\ne 0$. The properties of the spectral functions that are obtained are used to discuss some equal-time commutators and properties of the photon structure function and the cross section ${\sigma }_{\gamma }\left(\nu \right)$. The empirical value of ${\sigma }_{\gamma }\left(\infty \right)$ is used to roughly calculate $w_2$. It is stressed, however, that the above assumptions do not preclude the possibility that $w_2=0\mathrm{}$. The Fourier transforms ${\hat{W}}_i(x^2, p·x)$ of the $W_i(\kappa , \nu )$ are next studied and used to again relate the $A$ and $R$ limits. Restrictions on the $W_i$ imposed by the requirements of finite $A$ limits and correct $R$ limits are determined and results equivalent to the above ones are obtained. This analysis determines the configuration-space behavior corresponding to the $A$ and $R$ limits for large $\rho$ and $\kappa$, respectively. These limits are shown to determine the behavior of the ${\hat{W}}_i$ near the light cone $x^2=0\mathrm{}$, and the results are that $\pi {\hat{W}}_2\sim \delta \left(x^2\right)\frac{1}{2}\epsilon \left(x_0\right){\hat{f}}_2\left(x_0\right)$ and $\pi W_1\sim {\delta }^{\prime }\left(x^2\right)2\left|x_0\right|{\hat{f}}_1\left(x_0\right)$ for $x^2\sim 0$, where $w_2=\frac{1}{2}i\int d{x}_0{\hat{f}}_2\left(x_0\right)$ and $w_1=2i\left[f_1\right(\infty )-{\hat{f}}_1(-\infty \left)\right]$. Corresponding properties of the $V_i$ are derived, and the equivalence of this approach with the one using integral representations is established. The light-cone behavior of each component of $〈p\left|\right[J_{\mu }\mathrm{}\left(x\right), J_{\nu }\mathrm{}\left(0\right)\left]\right|p〉$ can be determined, and it is shown in particular that $〈p\left|\right[J_0-J_3, J_0-J_3\left]\right|p〉\sim \delta \left(x^2\right)\epsilon \left(x_0\right){\hat{f}}_2\left(x_0\right)$, apart from total derivatives with respect to $x_0-x_3$. The light-cone behavior equivalent to the Fubini-Dashen-Gell-Mann sum rule is then put in a form which can accommodate this result so well that we are led to propose a highly symmetric universal structure for the light-cone commutator of two $\mathrm{SU}\left(3\right)\mathrm{}$ currents. The universality is made precise by use of the ${\mathrm{}\left[SU\right(3\mathrm{})\mathrm{}\otimes \mathrm{SU}(3\left)\right]}_{\beta }$ equal-time commutation relations. The equal-time implications of the proposal are considered and are shown to be consistent with, and suggested by, the gluon mode. In the context of this model, $w_2$ is numerically estimated and found to agree with the experimental value.

A self-contained theory of Dirac fermions interacting with (pseudo-) scalar bosons at infinite momentum has been constructed. The fields are coupled with a $\overline{\psi }\Gamma \psi \phi$ interaction where $\Gamma =1\mathrm{}$ or ${\gamma }^5$, depending on the parity of the boson. The theory is designed to give the correct relativistic dynamics when the space of intermediate states is restricted to those states which can be interpreted as the infinite-momentum limit of sets of particles with finite momentum, that is, those states which correspond to the particle configurations observed in nature. The intermediate states which were eliminated are those containing particles moving away from the infinite-momentum observer. The dynamical effect of these states is incorporated into a counterterm in the Hamiltonian. The effect of this counterterm is to produce a set of four-point vertices in the old-fashioned Hamiltonian perturbation expansion. The perturbation-theory rules for this model were described in an earlier paper. A general formalism for handling interactions at infinite momentum is presented here. It is used to derive the counterterm for our model theory. A complete set of generators for the Poincaré group in a basis appropriate to the infinite-momentum limit is demonstrated. These generators satisfy the commutation algebra of the Poincaré group, proving the relativistic invariance of the theory and its internal consistency.

The perturbation series for electron-electron elastic scattering in quantum electrodynamics is studied in the limit of high energies. For this matrix element, in addition to the previously known terms which are proportional to $s$ (the square of the c.m. energy) and hence lead to a constant total cross section at high energies, there are found terms of the orders of magnitudes $s\ln s$, $s{\left(\ln s\right)}^2$, $s{\left(\ln s\right)}^3$, etc. For $n=1, 2, 3, \dots$, the coefficient of $s{\left(\ln s\right)}^n$ is a power series in the fine-structure constant $\alpha$, where the leading term is proportional to ${\alpha }^{2(n+1\mathrm{})\mathrm{}}$ and is due to Feynman diagrams with $n$ closed electron loops. Physically, through the optical theorem, the presence of these terms is intimately related to the production of low-energy electron-positron pairs in high-energy electron-electron scattering, but is independent of whether the spin-1 particle is a photon with zero mass or massive neutral vector meson. These leading terms of order ${\alpha }^{2(n+1\mathrm{})\mathrm{}}$ are explicitly found for all $n$, and are all imaginary, representing absorption. The procedure of summing the leading term is carried out, and the result demonstrates dramatically the importance of unitarity in the direct, or $s$, channel for high-energy processes. Generalization to two-body diffraction processes $a+b\to a^{\prime }+b^{\prime }$ is immediate.

The derivation of an action form from a time-symmetric formalism for quantum electrodynamics is given. A constraint added to a Fokker-type functional leads directly to an expression given recently by Schwinger; however, no supplementary renormalization condition is required here.

An infinity of four-point functions which are dual, display Regge behavior, and contain no "ancestors" has been proposed independently by Bitar, Fairlie, and Gardiner. We construct the corresponding $N$-point functions in this paper consistent with duality and factorization. They too display Regge behavior and are devoid of "ancestors." The existence of these functions shows that the principles of duality and factorization are more general and flexible than any one of their concrete realizations.

The influence of nonlinear (fourth-order) terms in the Lagrangians of some classical relativistic field theories is investigated, independently of the mass and coupling-constant parameters. In the spinor case, stable localized configurations with a lowest energy state are shown to exist always for positive values of the coupling constant.

Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincaré quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group ${\mathcal{G}}_5$ is suggested as the symmetry group of dynamics. ${\mathcal{G}}_5$ contains both the nonrelativistic Galilei group and the Poincaré group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of ${\mathcal{G}}_5$ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\tilde{\mathcal{G}}}_5$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\tilde{\mathcal{G}}}_5$ correspond to infinite towers of states with increasing spin.

A complete discussion of the classical theory of high-intensity Thomson scattering by free electrons is presented. Neglecting the radiation reaction, the equations of motion for an electron in an arbitrarily intense, elliptically polarized, plane electromagnetic wave can be solved exactly. From the solutions for the electron motion, the radiated power, momentum, and harmonics are calculated in two special Lorentz frames: the laboratory frame and the frame in which the electron is on the average at rest. The difference between the radiated power measured by an observer and that emitted by the electron is discussed for each frame. A sum rule for the radiated harmonics is derived. The limitation due to the neglect of radiation reaction is considered. Finally, the high- and low-intensity behavior of the spectrum and angular distribution of the radiation is analyzed in both frames.

A relativistically accurate table of values is calculated for the Compton energy exchange rate between photons in a Planckian distribution at temperature $T_R$ varying from 5 to 50 keV and electrons in a non-degenerate Maxwell-Boltzmann distribution at temperatures $T_e$ varying from 0 to 100 keV.

The rate of energy transfer due to Compton collisions from a photon gas to a Maxwell electron gas is calculated to fourth order in the expansion parameters $\frac{k{T}_e}{m{c}^2}$ and $\frac{k{T}_R}{m{c}^2}$. The photon distribution is taken to be Planckian at a temperature $T_R$ not equal to the electron temperature $T_e$. It is found that the nonrelativistic expression is accurate only for $k{T}_R\le 2.5$ keV and $k{T}_e\le 20$ keV, while the fourth-order expression is accurate for $k{T}_R\le 10$ keV and $k{T}_e\le 80$ keV.

The "threshold epoch" of classical cosmology is first discussed from a classical point of view. It is shown that if the density fluctuations at this epoch are proportional to $N^{-n}$ (where $N$ is the number of particles in a disturbance), then $n$ must exactly, or very closely, equal the value $\frac{2}{3}$ in order that such fluctuations develop into protogalaxies. This result is independent of the value of the threshold epoch, and of the equation of state for the very early universe. Arguments of a speculative nature are then presented which indicate that the threshold epoch occurs at the Planck density. It is also proposed that the initial conditions of galaxy formation are metric fluctuations at the threshold epoch of classical cosmology. The evidence in favor of this theory is (i) the density fluctuation obeys the $n=\frac{2}{3}$ rule, and (ii) the spin fluctuation gives an eventual angular momentum of the order $N^{\frac{4}{3}}\hslash$, in reasonable agrement with that of the Galaxy.

The generalized Kerr solution is tentatively accepted as the description for the state of a charged rotating mass which has undergone complete gravitational collapse. Mass accretion by such a system is demonstrated to damp out the rotation. Even the most favorable selective capture of matter leaves the angular momentum bounded by $m^2$, precisely the upper limit that the Kerr geometry can accommodate without change of character. Mach's principle, exemplified by the rotation of inertial frames, is employed to obtain approximate expressions for perihelion precession of satellites, deflection of light trajectories, and the rotation of polarization of light. Results are compared with exact expressions, when available.

Recent data on photoproduction from deuterium, which have been proposed as evidence for $t$-channel exchange of exotic meson states, are examined. It is shown that the effects that were associated with exchange of exotic states can result from shadow effects in the deuteron.

It is shown that baryon trajectories can be constructed which are in agreement with the observed $\pi -N$ resonance spectrum as well as backward scattering data and yet have no MacDowell partners. This is accomplished by including appropriate physical branch points in the trajectory function.

The generalization of $W$ spin to orbital extensions of the quark model is discussed. We point out the Lorentz-invariant properties and the predictive power of the assumptions, and discuss a possible manifestation of the breaking of the rest (and collinear) symmetry.

We use a Regge-pole model with absorptive corrections to calculate the differential cross sections and polarizations for the reactions ${K_2}^{0}p\to {K_1}^{0}p$ and ${K_2}^{0}p\to {K_2}^{0}p$.

We have studied the data on ${\rho }^0$ production in ${\pi }^{-}p$ interactions which have been used in tests of the vector-dominance model. We find inconsistencies between the assumptions made in the tests and the detailed characteristics of the data. In particular, the density matrix element ${\rho }_{1-1}$ for ${\rho }^0$-meson decay does not show the ${sin}^2\theta$ dependence expected ($\theta$ is the polar decay angle in the helicity frame), assuming only $s$- and $p$-wave $\pi \pi$ interactions. We conclude that the breakdown of the model has not been established by these tests.