The Casimir effect for parallel plates in the spacetime with a fractal extra compactified dimension

TL;DR

This paper investigates how a fractal extra compactified dimension influences the Casimir effect between parallel plates, revealing that the fractal nature alters the force and energy compared to integer-dimensional models.

Contribution

It introduces a method to analyze the Casimir effect with a fractal extra dimension and identifies how the fractal scale dimension affects the Casimir energy and force.

Findings

Negative Casimir energy requires scale dimension δ > 1/2.

Larger fractal scale dimension increases the Casimir force correction.

Casimir force behavior differs between integer and fractal extra dimensions.

Abstract

The Casimir effect for massless scalar fields satisfying Dirichlet boundary conditions on the parallel plates in the presence of one fractal extra compactified dimension is analyzed. We obtain the Casimir energy density by means of the regularization of multiple zeta function with one arbitrary exponent. We find a limit on the scale dimension like $\delta>1/2$ to keep the negative sign of the renormalized Casimir energy which is the difference between the regularized energy for two parallel plates and the one with no plates. We derive and calculate the Casimir force relating to the influence from the fractal additional compactified dimension between the parallel plates. The larger scale dimension leads to the greater revision on the original Casimir force. The two kinds of curves of Casimir force in the case of integer-numbered extra compactified dimension or fractal one are not superposition, which means that the Casimir force show whether the dimensionality of additional compactified space is integer or fraction.