Order and Chaos in Classical and Quantum Mechanics

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Quantum Electrodynamics (QED) has been a very active field in recent years. Twotheoretical techniques, S-matrix theory and effective field theory, allow the systematictreatment of many-electron systems, with the former being most useful for highlycharged ions and the latter for neutral, few-electon atoms.Given the success of the theory, another important direction the field is going in is theuse of the high precision of both experiment and QED theory to make atoms testinggrounds for non-QED physics. The most striking use has been parity nonconservation(PNC) in heavy atoms, and another important use is in nuclear physics.A first major focus of this research will be in the fine structure of helium as a way ofdetermining the fine structure constant. The extreme complexity of the electron g-2approach, recently shown by an error found in Kinoshita?s calculations, indicates thevalue of an alternative approach. Independent checks of a recent calculation ofPachucki are proposed.A second focus will involve research on highly charged ions, continuing a long standingcollaboration with K.T. Cheng of Lawrence Livermore National Laboratory. Completionof calculations of the spectra of sodiumlike ions along with extension to copperlike ionsis proposed. Hyperfine splitting in highly charged ions will also be studied.A third research area is recoil corrections. The treatment of recoil in a highly relativisticsystem has been treated only by one other group. G. Adkins and I have been workingon an independent approach to the problem, and continuation of this research isproposed. Another use of recoil techniques is in the treatment of photon-Z bosonexchange diagrams, which have a binding singularity that has not yet been evaluated ina Bethe-Salpeter framework.A final goal, with broader impact, is writing a textbook on bound state quantum fieldtheory. Most of the techniques used in the research described above have not yet beenwell described in textbook form, and doing this, along with giving a much needed updateof Bethe and Salpeter's monograph, is the intention of this work.