Accelerating the loop expansion

This thesis introduces a new non-perturbative technique into quantum field theory. To illustrate the method, the author analyze the much-studied phi4 theory in two dimensions. As a prelude, the author 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 suggests itself, which has the same nice property, and which automatically yields a convex effective potential. The author 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. The author discuss the reliability of the results and the problem of spontaneous symmetry-breaking, as well as some potential applications to more interesting field theories