Flow Equations for Hamiltonians from Continuous Unitary Transformations

Abstract

This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework of the flow in the manifold of unitarily equivalent matrices, as discovered in the mathematical literature before Wegner's paper, is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model.