Motivated by the difficulty in using the splitting matrix method to obtain parabolic approximations to complicated wave equations, we have developed an alternative method. It is three dimensional, does not a priori assume a preferred direction or path of propagation in the horizontal, determines spreading factors, and results in equations that are energy conserving. It is an extension of previous work by several authors relating parabolic equations to the horizontal ray acoustics approximation. Unlike previous work it applies the horizontal ray acoustics approximation to the propagator rather than to the Green's function or the homogenous field. The propagator is related to the Green's function by an integral over the famous "fifth parameter" of Fock and Feynman. Methods for evaluating this integral are equivalent to narrow-angle approximations and their wide-angle improvements. When this new method is applied to simple problems it gives the standard results. In this paper it is described by applying it to a problem of current interest-the development of a parabolic approximation for modeling global underwater and atmospheric acoustic propagation. The oceanic or atmospheric waveguide is on an Earth that is modeled as an arbitrary convex solid of revolution. The method results in a parabolic equation that is energy conserving and has a spreading factor that describes field intensification for antipodal propagation. Significantly, it does not have the singularities in its range-sliced version possessed by many parabolic equations developed for global propagation. We then discuss two extensions of the method; first to propagation along refracted geodesics and second to a description involving discrete, local, normal modes.

Finite-contour dispersive inequalities are derived for a function f(ξ) which is analytic in the ξ plane except for a right-hand cut. Inequalities are also derived for the derivative of the function. These inequalities are rigorous and the sharpest ones that can be derived using only analyticity.

Using the most general form for the null-plane commutator of the electromagnetic current, sum rules are derived for electroproduction in the deep-inelastic region. The structure of these sum rules is discussed and a free, massless, fermion model is considered as an illustration.

Okubo's recently strengthened inequality is applied to the pion's mechanical form factor, bounding the derivative of that form factor at zero momentum transfer by the magnitude of the same function in the timelike region. In the framework of a model of broken scale and chiral invariance these results are converted into bounds on the dimensions of scale-invariance breaking and on the derivative of the mechanical form factor. Supplementary conditions on the dimensions and on the mechanical form factor are provided by a generalized version of well-known low-energy theorems. A comparison of the information furnished by the inequalities and by the low-energy theorems leads to the following conclusions: In the standard sort of model, in which the entire SU(2)\ensuremath{\bigotimes}SU(2)-violating term and the entire SU(3)\ensuremath{\bigotimes}SU(3)-violating term have the same dimension, the mechanical form factor will be "nonsmooth" and probably will be subtracted. On the other hand, there is a second type of model, in which the entire SU(2)\ensuremath{\bigotimes}SU(2)-violating term and the entire SU(3)\ensuremath{\bigotimes}SU(3)-violating term have different dimensions (zero and two, respectively). In such a theory the mechanical form factor can be "smooth" and unsubtracted.

Assuming the hadronic energy density has the decomposition ${\ensuremath{\Theta}}_{00}={\overline{\ensuremath{\Theta}}}_{00}+\ensuremath{\delta}+u$ (where ${\overline{\ensuremath{\Theta}}}_{00}$ is chiral- and scale-invariant, $\ensuremath{\delta}$ is chiral-invariant but violates scale invariance, and $u$ violates both chiral and scale invariance), we find rigorous bounds on the dimensions of $\ensuremath{\delta}$ and $u$ in the (actual) SU(2)-symmetric limit and in the SU(3)-symmetric limit. If we further assume that $\ensuremath{\delta}$ is a $c$ number, we derive a new sum rule which leads to satisfactory bounds on the dimension of $u$ in the SU(2)-symmetric limit but not in the SU(3)-symmetric limit. Saturation of this sum rule with a scalar meson having a mass of 700 MeV leads to a value of 1.9 for the dimension of $u$.

A prescription is presented for modifying the stress tensor of the Sugawara model in order that it become consistent with general postulates of field theory. The prescription involves redefining the stress tensor as the limit of a spatially nonlocal operator. Within the context of the modified theory, sum rules are derived by considering the vacuum expectation value of equal-time stress-tensor commutators. Variations of our limiting procedure are also considered. It is shown that convergent Weinberg second sum rule in the frame-work of the Sugawara model leads to a null theory. The existence of a nontrivial Sugawara model leads to a specific behavior of the cross section for positron-electron annihilation into hadrons at high energies.