Geometry of theory space and RG flows

TL;DR

This paper explores the geometric structure of theory space using a Fisher-like metric, analyzing RG flows through their expansion, rotation, and shear, and discusses implications and open questions in the context of theoretical physics.

Contribution

It introduces a geometric framework for understanding theory space and characterizes RG flows via their isotropic expansion, rotation, and shear.

Findings

Characterization of RG flows through geometric quantities

Discussion of evolution equations for isotropic expansion

Identification of open questions and potential generalizations

Abstract

The space of couplings of a given theory is the arena of interest in this article. Equipped with a metric ansatz akin to the Fisher information matrix in the space of parameters in statistics (similar metrics in physics are the Zamolodchikov metric or the O'Connor--Stephens metric) we investigate the geometry of theory space through a study of specific examples. We then look into renormalisation group flows in theory space and make an attempt to characterise such flows via its isotropic expansion, rotation and shear. Consequences arising from the evolution equation for the isotropic expansion are discussed. We conclude by pointing out generalisations and pose some open questions.