Perturbative quantum field theory and other physical perturbations - Going to higher order with analytical and approximative schemes

Abstract

Scope and Method of Study: The main areas of research and summary include solving integral equations with application to diffusive scattering studies and calculating non-relativistic energy levels of particles subject to a tractable radially dependent potential plus a less manageable, large radially dependent potential term. The method developed in this thesis for calculating spectra of particles bound in a complicated radual potential can be applied to various spectrum calculations by using a good computer (with math co-processor) and the implementation of Mathematica style source code. Returning to the first mentioned area of rendition and application, diffusice scattering studies discussed and considered in this thesis center around the problem of analyzing and solving integral equations involving the diffusice scattering of radiation in biological material media. There are two things that are of common issue to both mentioned sub-topics of research. First of all, the expressions for the essential phenomena and observables can be and are often expressed as an infinite series. The failure of such and infinite series to get a convergent summation is the second item of common issue in this thesis. This problem occurs with integral equations and in the perturbative treatment given to the quantum mechanical spectra of atomic and nuclear systems. A formalism for partial fractions known as Pade approximants are introduced. These Pade approximants are used to make approximations of the sum that a given infinite series is formally representing.

Findings and Conclusions: Analytical as well as efficient methods for calculating the difficult higher order terms in many infinite series were successfully developed and demonstrated in chapters 2 through 5 of this thesis. Pade approximants, in turn have been applied successfully to all of the examples given (except for one in chapter 2) in order do find consistent convergent results of the various perturbative infinite series involved.