Casimir energies of self-similar plate configurations

Abstract

We construct various self-similar configurations using parallel δ-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 the 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 δ-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.

Quantum Vacuum Energy of Self-Similar Configurations

“…A generalization of the two-body interaction of more than two bodies was given in Reference [20,21] in the context of multiple-scattering techniques and in References [22,23] using Green's function method, but explicit solutions for the Green functions were reported only for configurations with three bodies. In [11], the authors find solutions to the Green functions for four bodies, and then they go further and express the solution to the Green function for N bodies as a recursion relation in terms of the Green's functions for (N − 2) bodies. This procedure then lets them extend their solutions for the Green functions for an infinite sequence of objects by taking the limit N → ∞.…”

“…which requires some elaboration because we have used the idea of self-similarity in writing Equation (11). The interaction energy of the complete stack of plates in Figure 1 is on the left side of Equation (11).…”

“…which requires some elaboration because we have used the idea of self-similarity in writing Equation (11). The interaction energy of the complete stack of plates in Figure 1 is on the left side of Equation (11). This is the term of interest for us, and the notation indicates that we consider the interaction energy of a self-similar plate configuration of total width a.…”

“…This approach is explained in Section 2. There, we describe the method developed in [11] and extended in [12], where we calculate the quantum vacuum energy of some self-similar structures. In the former reference, we use the multiple scattering technique to develop a rather simple-looking method that we verify by reproducing our result using a Green's function approach .…”

Casimir energy of Sierpinski triangles

“…The central theme of this paper is to use the scaling arguments and the property of self-similarity introduced in Ref. [4] to derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. We also introduce a class of fractals for which the energy does not scale as inverse length square, which leads us to introduce a fractal dimension for the Casimir energy.…”