Casimir energy of Sierpinski triangles

TL;DR

This paper derives the Casimir energies for Sierpinski fractals using self-similarity and scaling, introduces a fractal dimension for Casimir energy, and discusses the Berry-Weyl conjecture for these geometries.

Contribution

It presents a novel method to calculate Casimir energies for fractal geometries and introduces the concept of fractal dimension of Casimir energy.

Findings

Casimir energies for Sierpinski triangles and rectangles derived

Fractal dimension of Casimir energy introduced

Finite Casimir energy possible for certain fractals despite divergence in individual cavities

Abstract

Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising of compartmentalized cavities, it is possible to establish a finite value to the Casimir energy even while the Casimir energy of the individual cavities consists of divergent terms.