Casimir energy of Sierpinski triangles

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.

Quantum Vacuum Energy of Self-Similar Configurations

“…For an equilateral triangle C = 8. The same behavior is supposed for the calculation in [12]. This expansion seems to have a universal behavior, meaning that the coefficients of the divergences associated have an specific meaning, reminiscent of the geometry under study.…”

“…We have calculated the vacuum energy of structures that cover the whole space and systems that mimic quasi-periodic configurations. In future work, we plan to further develop the system that in [12] we call "Inverse Sierpinski triangle".…”

“…Using the scaling argument of Equation (12) in Equation ( 11), we identify the relation involving the Casimir interaction energy of the infinite sequence of plates in Figure 1,…”

“…We study the Casimir effect of the self-similar structure that emerges from this set up, where the oscillating modes are calculated in the vacuum space inside the triangular cavity. The arguments given here are developed in Reference [12].…”

“…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 .…”