Similarity renormalization, Hamiltonian flow equations, and Dyson’s intermediate representation

Abstract

A general framework is presented for the renormalization of Hamiltonians via a similarity transformation. Divergences in the similarity flow equations may be handled with dimensional regularization in this approach, and the resulting effective Hamiltonian is finite since states well separated in energy are uncoupled. Specific schemes developed several years ago by G\l{}azek and Wilson and by Wegner are shown to correspond to particular choices within this framework. A modification of Wegner's scheme is introduced with the idea of improving convergence. It is shown that a scheme for the transformation of Hamiltonians developed by Dyson in the early 1950s also arises from a particular choice within the similarity renormalization framework and is particularly suited to analytic computations since the usual counterterm structure used in Feynman perturbation theory is sufficient for this scheme. A logarithmically confining potential is shown to arise at second order in light-front QCD within Dyson's scheme, a result found previously for other similarity renormalization schemes that used sharp cutoffs in momentum space to regularize the Hamiltonian. Steps toward higher order and nonperturbative calculations are outlined. In particular, a set of equations analogous to Dyson-Schwinger equations is developed for both the Dyson scheme and the modified Wegner scheme.