What is dimension?

TL;DR

This paper examines the concept of dimension in sets, discussing various mathematical characterizations, their generalizations to multifractals, and the role of renormalization group flow in understanding complex and anomalous dimensions.

Contribution

It introduces a comprehensive framework connecting different notions of dimension, including multifractals and complex dimensions, with applications to physical phenomena like localization.

Findings

Different power law characterizations of Euclidean space dimensions

Extension to multifractals and complex dimensions

Application of RG flow equations to localization phenomena

Abstract

This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the generalization to multifractals, and discrete and continuous scale invariance with the emergence of complex dimensions. The idea of renormalization group flow equations can be introduced in this framework, to show how the power laws determined by dimensional analysis (engineering dimensions) get modified by extra anomalous dimensions. As an example of the RG flow equation, the scaling of conductance by disorder in the context of localization is used. A few technicalities, including the connection between entropy and fractal dimension, can be found in the appendices.