Changes of Variables and the Renormalization Group

TL;DR

This paper introduces a class of exact variable change-based renormalization group transformations that improve approximation accuracy and unify existing methods, with applications to scalar field theories.

Contribution

It proposes a novel class of pure change-of-variables RG transformations that can be exact or asymptotically exact, simplifying and unifying various RG approaches.

Findings

Transformations improve saddle point approximations.

Constructed an exact gauge covariant RG transformation.

Obtained solutions for scalar field theory in epsilon expansion and coupling expansion.

Abstract

A class of exact infinitesimal renormalization group transformations is proposed and studied. These transformations are pure changes of variables (i.e., no integration or elimination of some degrees of freedom is required) such that a saddle point approximation is more accurate, becoming, in some cases asymptotically exact as the transformations are iterated. The formalism provides a simplified and unified approach to several known renormalization groups. It also suggests some new ways in which renormalization group methods might successfully be applied. In particular, an exact gauge covariant renormalization group transformation is constructed. Solutions for a scalar field theory are obtained both as an expansion in {\epsilon}=4-d and as an expansion in a single coupling constant.