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Casimir energies of self-similar plate configurations

K. V. Shajesh1,2,*, Iver Brevik2,†, Inés Cavero-Peláez3,‡, and Prachi Parashar1,2,§

  • 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

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Phys. Rev. D 94, 065003 – Published 6 September, 2016

DOI: https://doi.org/10.1103/PhysRevD.94.065003

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.

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See Also

Casimir energy of Sierpinski triangles

K. V. Shajesh, Prachi Parashar, Inés Cavero-Peláez, Jerzy Kocik, and Iver Brevik
Phys. Rev. D 96, 105010 (2017)

Article Text

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Phys. Rev. D 94, 065003– Published 6 September, 2016
  • Received 5 July 2016

© 2016 American Physical Society

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