Quantum Vacuum Energy of Self-Similar Configurations

TL;DR

This paper reviews the calculation of quantum vacuum energy in self-similar systems, emphasizing the scaling properties and differences when embedded in fractal spaces, with a focus on Dirichlet boundary conditions.

Contribution

It introduces a systematic method to compute Casimir energy for self-similar structures based on known single-element vacuum energies and explores the effects of fractal geometries.

Findings

Method for calculating Casimir energy using geometrical scaling

Examples of self-similar configurations and their vacuum energies

Discussion of vacuum behavior in fractal versus smooth spaces

Abstract

We offer in this review a description of the vacuum energy of self-similar systems. We describe two views of setting self-similar structures and point out the main differences. A review of the authors' work on the subject is presented, where they treat the self-similar system as a many-object problem embedded in a regular smooth manifold. Focused on Dirichlet boundary conditions, we report a systematic way of calculating the Casimir energy of self-similar bodies where the knowledge of the quantum vacuum energy of the single building block element is assumed and in fact already known. A fundamental property that allows us to proceed with our method is the dependence of the energy on a geometrical parameter that makes it possible to establish the scaling property of self-similar systems. Several examples are given. We also describe the situation, shown by other authors, where the embedded space is a fractal space itself, having fractal dimension. A fractal space does not hold properties that are rather common in regular spaces like the tangent space. We refer to other authors who explain how some self-similar configurations "do not have any smooth structures and one cannot define differential operators on them directly". This gives rise to important differences in the behavior of the vacuum.