Similarity Renormalization, Hamiltonian Flow Equations, and Dyson's Intermediate Representation

TL;DR

This paper introduces a comprehensive framework for Hamiltonian renormalization using similarity transformations, compares different schemes including Dyson's, and demonstrates how confinement potentials emerge in light-front QCD.

Contribution

It unifies various Hamiltonian renormalization schemes within a single framework and advocates for Dyson's scheme due to its computational simplicity.

Findings

Dyson's scheme is preferable for ease of computation.

Logarithmic confinement arises at second order in light-front QCD.

Framework facilitates higher order and nonperturbative calculations.

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 Glazek and Wilson and contemporaneously by Wegner correspond to particular choices within this framework, and the relative merits of such choices are discussed from this vantage point. It is shown that a scheme for the transformation of Hamiltonians introduced by Dyson in the early 1950's also corresponds to a particular choice within the similarity renormalization framework, and it is argued that Dyson's scheme is preferable to the others for ease of computation. As an example, it is shown how a logarithmically confining potential arises simply at second order in light-front QCD within Dyson's scheme, a result found previously for other similarity renormalization schemes. Steps toward higher order and nonperturbative calculations are outlined. In particular, a set of equations analogous to Dyson-Schwinger equations is developed.