Casimir energies of self-similar plate configurations

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.

arXiv comment13 pages, 5 figures, Minor changes to match the journal article

References in corpus (14)

• Opposites Attract - A Theorem About The Casimir Force
• The Casimir Effect: Physical Manifestations of Zero Point Energy
• Scalar Casimir effect between Dirichlet spheres or a plate and a sphere
• Casimir Forces between Compact Objects: I. The Scalar Case
• Multiple Scattering Methods in Casimir Calculations
• The Reality of Casimir Friction
• Casimir Energy for a Dielectric Cylinder
• Casimir Interaction among Objects Immersed in a Fermionic Environment
• Casimir energy of a non-uniform string
• Many-Body Contributions to Green's Functions and Casimir Energies
• Irreducible Many-Body Casimir Energies of Intersecting Objects
• Problems with the Casimir Vacuum Energy
• Thermodynamic Properties of the Relativistic Composite String - Expository Remarks
• Significance of many-body contributions to Casimir energies

Cited by in corpus (7)

• Quantum Vacuum Energy of Self-Similar Configurations
• Casimir energy of Sierpinski triangles
• Casimir energy of $N$ magnetodielectric $\delta$-function plates
• The closed piecewise uniform string revisited
• Quasiperiodic arrangement of magnetodielectric $\delta$-plates: Green's functions and Casimir energies for $N$ bodies
• Quantum field theory at finite temperature for 3D periodic backgrounds
• Casimir energy for $N$ constant conductivity $\delta$-plates with a neural network perception