This thesis introduces a new non-perturbative technique into quantum field theory. To illustrate the method, I analyze the much-studied phi/sup 4/ theory in two dimensions. As a prelude, I first show that the Hartree approximation is easy to obtain from the calculation of the one-loop effective potential by a simple modification of the propagator that does not affect the perturbative renormalization procedure. A further modification then susggests itself, which has the same nice property, and which automatically yields a convex effective potential. I then show that both of these modifications extend naturally to higher orders in the derivative expansion of the effective action and to higher orders in the loop-expansion. The net effect is to re-sum the perturbation series for the effective action as a systematic ''accelerated'' non-perturbative expansion. Each term in the accelerated expansion corresponds to an infinite number of terms in the original series. Each term can be computed explicitly, albeit numerically. Many numerical graphs of the various approximations to the first two terms in the derivative expansion are given. I discuss the reliability of the results and the problem of spontaneous symmetry-breaking, as well as some potential applications to more interesting field theories. 40 refs.

The Advent of laser-ion-guiding in the Advanced test Accelerator along with the development of accelerator cavities optimized with respect to beam breakup coupling impedence now make it possible to consider a new class of high current, high emergy linear induction accelerators. The control of the beam breakup and other instabilities by laser guiding and by various magnetic focusing schemes will be discussed along with the scaling laws for the design of such machines to minimize the growth of the beam breakup instability. Many linacs, particularly induction linacs are limited in performance by the beam breakup (BBU) instability. The instability is found in two forms. In the first form the accelerating cavities communicate with one another through interaction with the beam and through propagation of cavity fields through the accelerator structure. In the second form which is the more virulent of the two, the cavities couple to each other only through their interactions with the beam. It is this second form of PPU that will be discussed in this paper.

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