Casimir energy of Sierpinski triangles K. V. Shajesh,1,2, ∗ Prachi Parashar,2, † In´es Cavero-Pel´aez,3, ‡ Jerzy Kocik,4, § and Iver Brevik2, ¶ 1Department of Physics, Southern Illinois University–Carbondale, Carbondale, Illinois 62901, USA 2Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 3Centro Universitario de la Defensa (CUD), Zaragoza 50090, Spain 4Department of Mathematics, Southern Illinois University–Carbondale, Carbondale, Illinois 62901, USA (Dated: June 19, 2021) Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sier- pinski 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. I. INTRODUCTION Weyl’s law [1], which was originally discussed for the spectral distribution of the modes allowed inside a Dirich- a a a let cavity, when extended for the Casimir energy per unit length (a) of a polygonal cylindrical cavity with a sin- gle characteristicE scale a, in natural units of ~ = c = 1, 4 s = states that E − 11 E∆ bc 1 a a a (a)= + lim b2A + b1P + b0C , E a2 τ→0 τ 2 τ τ τ h i(1) a a where the coefficients of the divergent terms, A(x), P (x), and C(x), scale like the area of the cavity, the perimeter of the cavity, and the corner angles of the cavity, respec- b 9 tively. That is, s = E − 71 E A(x) x2, P (x) x1, C(x) x0. (2) ∝ ∝ ∝ FIG. 1. Gallery of Sierpinski cylinders with Casimir energies Parameters bc, b2, b1, and b0 in Eq. (1) are dimensionless per unit length Es for the five integrable cylinders studied in constants. The parameter τ is a temporal point-splitting Ref. [3]. Top row: An isosceles right triangle with the equal cutoff parameter introduced in the calculation to regulate sides of length a, an equilateral triangle of side length a, and a the divergences. hemiequilateral triangle with length of hypotenuse a. Bottom Berry conjectured [2], again in the context of spectral row: A square of side length a, and a rectangle of side lengths distribution, that for fractal cavities the Weyl law main- a and b. The Casimir energy per unit length of a Sierpinski triangle is −4/11 times the Casimir energy per unit length of tains the form of Eq. (1) with the only difference that the the respective triangle, E∆, and the Casimir energy per unit coefficients of the divergent terms, A(x), P (x), and C(x), arXiv:1709.06284v2 [hep-th] 30 Nov 2017 length of a Sierpinski rectangle or square is −9/71 times the scale like the Hausdorff-Besicovitch dimension (fractal di- Casimir energy per unit length of the respective rectangle or mension) of the area of the cavity, the perimeter of the square, E. The Casimir energy per unit length, E∆ and E, cavity, and the corner angles of the cavity, respectively. for the five integrable cylinders are summarized in Table I. That is, δ δ δ A(x) x 2 , P (x) x 1 , C(x) x 0 , (3) ∝ ∝ ∝ and δ0 is the fractal dimension of the corner angles of the cavity. where δ2 is the fractal dimension of the area of the cavity, It is, then, not a long shot to envision that the Casimir δ1 is the fractal dimension of the perimeter of the cavity, energy per unit length of a fractal cavity need not scale like the inverse square of length. Thus, presuming that the energy scales like aδc , we can generalize Weyl’s law ∗ [email protected]; http://www.physics.siu.edu/˜shajesh in Eq. (1) as † [email protected]; https://www.ntnu.edu/employees/prachi.parashar ‡ [email protected]; http://cud.unizar.es/cavero δc δc a a a § (a)= bc a + lim τ b2A + b1P + b0C , [email protected]; http://lagrange.math.siu.edu/Kocik/jkocik.htm τ→0 ¶ E τ τ τ [email protected]; http://folk.ntnu.no/iverhb h (4)i 2 Cross section Dirichlet Neumann EM with a numerical value bc 0.0237188, ∼ 0.0237 0.0613 0.0375 Equilateral Tr. − − a2 a2 a2 3√3 3 1 b = , b = , b = , (6) 0.0756 0.0944 0.0187 2 2 1 0 Hemiequilateral Tr. − − 8π −8π 6π a2 a2 a2 m 0.0263 − 0.0454 − 0.0190 of Eq.(1), where ψ( )(z) is the polygamma function of Isosceles Tr. 2 2 2 a a a order m and ζ(z) is the Riemann zeta function. The 0.00483 − 0.0429 − 0.0381 above evaluation was achieved using the mode summa- Square 2 2 2 a a a tion method, which presumes that the Casimir energy of a closed Dirichlet cavity is completely determined by Rectangle Refer Ref. [3]. the modes in the interior of the cavity alone [3]. This should be contrasted with field theoretic methods of Lif- TABLE I. Casimir energy per unit length for cylinders of five shitz et al. [6] and Schwinger et al. [7] that incorporate cross sections from Ref. [3], referred to as E∆ and E in this both the interior and exterior modes in the evaluation. paper. The cutoff independent finite part is presented. The The Casimir energies for the five geometries of Ref. [3], numbers correspond to the constant bc in Eq. (1) for the re- for empty triangles and rectangles or squares, due to in- spective cross sections, presented here to three significant dig- terior modes only, referred to as and in this paper, its without rounding. The second, third, and fourth, columns E∆ E correspond to the boundary conditions imposed on the fields. are summarized in Table I. Before we proceed with our discussion, we present the results for the finite part of the Casimir energy for each of the five geometries presented in Fig. 1, which are ex- where δc is the fractal dimension of the Casimir energy pressed in terms of and of Table I. The expression per unit length of the fractal cavity. E∆ E s = 4 /11 is universal for all Sierpinski triangles, The central theme of this paper is to use the scaling E − E∆ and s = 9 /71 is universal for all Sierpinski rect- arguments and the property of self-similarity introduced angles,E and− areE not restricted to the five geometries of in Ref. [4] to derive the Casimir energies of Sierpinski Ref. [3]. However, we often confine the analysis to the triangles and Sierpinski rectangles. We also introduce a five geometries of Ref. [3], because exact expressions were class of fractals for which the energy does not scale as in- derived for the Casimir energy per unit length for these verse length square, which leads us to introduce a fractal geometries there. dimension for the Casimir energy. One usually associates fractal dimensions to geometrical quantities like perime- ter and area, but being able to introduce fractal dimen- sions to Casimir energy for a class of fractals directly re- lates to the conventional wisdom that Casimir energy of II. SIERPINSKI TRIANGLE cavities, satisfying perfectly conducting boundary condi- tions (or Dirichlet boundary conditions for scalar fields), The Sierpinski triangle is self-similar. That is, it con- is purely geometrical. It should not be very surprising, sists of copies of the scaled-down versions of itself. Fig- because energy has been shown to exhibit fractal nature ure 2 shows the Sierpinski triangle of side length a, which before. For example, the Hofstadter butterfly is a fractal may be viewed as comprised of three Sierpinski triangles that represents the energy of Bloch electrons in a mag- of side length a/2. netic field [5]. We shall consider an equilateral triangle even though most of our discussion holds true for an arbitrary triangle. The Casimir energy of an equilateral triangular cylinder on which a scalar field satisfies Dirichlet boundary con- ditions was calculated exactly, in closed form, in Ref. [3]. This involves the Casimir energy of five cylindrical cross sections, namely an equilateral triangle, a hemiequilat- eral triangle, an isosceles right triangle, a square, and a rectangle, see Fig. 1. For all five geometries the authors of Ref. [3] have shown that the Casimir energy per unit length obeys the Weyl law in Eq. (1). Using Ref. [3], the FIG. 2. Sierpinski triangle. The white regions in the inte- Casimir energy per unit length of an equilateral triangle rior are triangular cavities, each of which contributes to the is described by the parameters Casimir energy per unit length of the Sierpinski triangle. The matter bounding each of the triangles (in blue) are perfectly conducting for the case of electromagnetic fields. 1 √3 (1) 1 (1) 2 8 c b = ψ 3 ψ 3 ζ(3) , (5) −72 " 9 − − π # n o 3 A. Area and δ2 C. Corner angle and δ0 Using the self-similarity of a Sierpinski triangle we can The interior corner angles of a Sierpinski triangle sat- write the following recursion relation for the area As(a) isfy of the cavities inside a Sierpinski triangle: a a a a Cs(a)= C∆ +3Cs , (16) As(a)= A +3As , (7) 2 2 ∆ 2 2 which leads to the series where A∆(a) is the area of an equilateral triangle of side a a 2 a length a. Using Eq.