Casimir energies of self-similar plate configurations

TL;DR

This paper introduces a novel method to calculate Casimir energies for self-similar plate configurations, including fractal arrangements like the Cantor set, revealing new insights into vacuum forces in complex geometries.

Contribution

It presents a self-similarity based approach to evaluate Casimir energies, extending to fractal configurations, and connects quantum field regularization with number theory.

Findings

Casimir energy can be positive, zero, or negative depending on configuration.

The method applies to scalar fields and can be extended to electromagnetic fields.

First report of Casimir energy in a fractal configuration.

Abstract

We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory.