Abstract
Casimir forces arise from vacuum fluctuations. They are fully understood only for simple models, and are important in nano- and microtechnologies. We report our experience of computer algebra calculations toward the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases confidence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Green’s functions is at the boundatry of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that indicate the possibility of alternative regularizations. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and suggest combined numerical and symbolic approaches toward a more general computational framework.
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References
Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51, 793–795 (1948)
Bressi, G., Carugno, G., Onofrio, R., Ruoso, G.: Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88(4), 041804 (2002)
Decca, R.S., LĂłpez, D., Fischbach, E., Krause, D.E.: Measurement of the Casimir force between dissimilar metals. Phys. Rev. Lett. 91(5), 050401 (2003)
Hertlein, C., Helden, L., Gambassi, A., Dietrich, S., Bechinger, C.: Direct measurement of critical Casimir forces. Nature 451(175), 172–175 (2008)
Lamoreaux, S.K.: Demonstration of the Casimir force in the 0.6 to 6 \(\upmu \)m range. Phys. Rev. Lett. 78(1), 5–8 (1997)
Munday, J.N., Capasso, F., Parsegian, A.V.: Measured long-range repulsive Casimir-Lifshitz forces. Nature 457(7226), 170–173 (2009)
Dzyaloshinskii, I.E., Lifshitz, E.M., Pitaevskii, L.P.: The general theory of van der Waals forces. Adv. Phys. 10(38), 165–209 (1961)
Philbin, T.G., Leonhardt, U.: No quantum friction between uniformly moving plates. New J. Phys. 11(3), 033–035 (2009)
Bordag, M., Klimchitskaya, G.L., Mohideen, U., Mostepanenko, V.M.: Advances in the casimir effect. Oxford University Press, Oxford (2009)
Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Statistical physics, part 2. Butterworth-heinemann, Oxford (1980)
Heun, K.: Theorie der Riemann’schen Functionen zweiter Ordnung mit vier Verzweigungspunkten. Mathematische Annalen 3, 161 (1899)
Boyer, T.H.: Quantum electromagnetic zero-point energy of a conducting spherical shell and the Casimir model for a charged particle. Phys. Rev. 174(5), 1764–1776 (1968)
Philbin, T.G., Xiong, C., Leonhardt, U.: Casimir stress in an inhomogeneous medium. arXiv:0909.2998, 2009 (Preprint)
Acknowledgments
The authors are supported by EPSRC grant EP/CS23229/1. UL is also supported by a Royal Society Wolfson Research Merit Award. We would like to acknowledge the critical discussions we have had with Tom Philbin.
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Xiong, C., Kelsey, T.W., Linton, S.A., Leonhardt, U. (2014). Towards the Calculation of Casimir Forces for Inhomogeneous Planar Media. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_15
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DOI: https://doi.org/10.1007/978-3-662-43799-5_15
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