Geometry, the Renormalization Group and Gravity

Summary

Abstract

We discuss the relationship between geometry, the renormalization group (RG) and gravity. We begin by reviewing our recent work on crossover problems in field theory. By crossover we mean the interpolation between different representations of the conformal group by the action of relevant operators. At the level of the RG this crossover is manifest in the flow between different fixed points induced by these operators. The description of such flows requires a RG which is capable of interpolating between qualitatively different degrees of freedom. Using the conceptual notion of course graining we construct some simple examples of such a group introducing the concept of a “floating” fixed point around which one constructs a perturbation theory. Our consideration of crossovers indicates that one should consider classes of field theories, described by a set of parameters, rather than focus on a particular one. The space of parameters has a natural metric structure. We examine the geometry of this space in some simple models and draw some analogies between this space, superspace and minisuperspace.

Introduction

The cosmopolitan nature of Charlie Misner's work is one of its chief features. It is with this in mind that we dedicate this article on the occasion of his 60th birthday. There are several recurring leitmotifs throughout theoretical physics; prominent amongst these would be geometry, symmetry, and fluctuations. Geometry clarifies and systematizes the relations between the quantities entering into a theory, e.g. Riemannian geometry in the theory of gravity and symplectic geometry in the case of classical mechanics. Symmetry performs a similar role, and in the case of continuous symmetries is often intimately tied to geometrical notions.