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Geometry and Material Effects in Casimir Physics-Scattering Theory

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Casimir Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 834))

Abstract

We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, to nonzero temperatures, and to spatial arrangements in which one object is enclosed in another. Our method combines each object’s classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. This approach, which combines methods of statistical physics and scattering theory, is well suited to analyze many diverse phenomena. We illustrate its power and versatility by a number of examples, which show how the interplay of geometry and material properties helps to understand and control Casimir forces. We also examine whether electrodynamic Casimir forces can lead to stable levitation. Neglecting permeabilities, we prove that any equilibrium position of objects subject to such forces is unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum.

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Notes

  1. 1.

    Because of this relationship, these scattering amplitudes are also referred to as elements of the T-matrix. In standard conventions, however, the T-matrix differs from the matrix elements of the \({\mathbb{T}}\)-operator by a basis-dependent constant, so we will use the term “scattering amplitude” to avoid confusion.

  2. 2.

    Alternatively, we can set up asymptotically incoming and outgoing waves on the outside and regular waves inside. The amplitudes of the outgoing waves are then given by the S-matrix, which is related to the scattering amplitude \({\fancyscript{F}}\) by \({\fancyscript{F}}=( S- I)/2.\) Although these two matrices carry equivalent information, the scattering amplitude will be more convenient for our calculation.

  3. 3.

    The sequence of two changes of variables is known as Hubbard-Stratonovich transformation in condensed matter physics.

  4. 4.

    \({\mathbb{G}}_M\) satisfies \(\left({\varvec\nabla} \times\mu_M^{-1}(ic\kappa){\varvec\nabla} \times+\varepsilon_M(ic\kappa)\kappa^2\right){\mathbb{G}}_M(ic\kappa,{\mathbf x},{\mathbf x}^{\prime})=\delta({\mathbf x}-{\mathbf x}^{\prime}){\mathbb{I}},\) and is related to \(G_M,\) the Green’s function of the imaginary frequency Helmholtz equation, by \({\mathbb{G}}_M(ic\kappa,{\mathbf x},{\mathbf x}^{\prime}) = \mu_M(ic\kappa)\left({\mathbb{I}} + (n_M \kappa)^{-2} {\varvec\nabla}\otimes{\varvec\nabla}^{\prime}\right) G_M(icn_M\kappa,{\mathbf x},{\mathbf x}^{\prime})\).~Here, \(n_M(ic\kappa)=\sqrt{\varepsilon_M(ic\kappa) \mu_M(ic\kappa)}\) is the index of refraction of the medium, whose argument is suppressed to simplify the presentation. Thus \({\mathbb{G}}_M,\) in contrast to \({\mathbb{G}}_0,\) takes into account the permittivity and permeability of the medium when they are different from one.

  5. 5.

    To obtain the free energy at finite temperature, in place of the ground state energy \({\fancyscript{E}},\) \(\int {\frac{d\kappa}{2\pi}}\) is replaced by the sum \({\frac{kT}{\hbar c}}\sum^{\prime}_{\kappa_n\geq 0}\) over Matsubara ‘wavenumbers’ \(\kappa_n = 2\pi n k T/\hbar c\) with the \(\kappa_0=0\) mode weighted by \(1/2\).

  6. 6.

    In practice, \({\mathbb{T}}_A\) and \({\mathbb{T}}_R\) suffice to have the same sign over the frequencies, which contribute most to the integral (or the sum) in (5.43)

  7. 7.

    The first curl in the operator \({\mathbb{V}}_J\) results from an integration by parts. It is understood that it acts on the wave function multiplying \({\mathbb{V}}_J\) from the left.

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Acknowledgements

The research presented here was conducted together with Noah Graham, Steven G. Johnson, Mehran Kardar, Alejandro W. Rodriguez, Pablo Rodriguez-Lopez, Alexander Shpunt, and Saad Zaheer, whom we thank for their collaboration. This work was supported by the National Science Foundation (NSF) through grant DMR-08-03315 (SJR), by the DFG through grant EM70/3 (TE) and by the U. S. Department of Energy (DOE) under cooperative research agreement #DF-FC02-94ER40818 (RLJ).

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Authors and Affiliations

  1. Department of Physics, MIT, 02139, Cambridge, MA, USA

    Sahand Jamal Rahi

  2. Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, CNRS UMR 8626, 91405, Orsay, France

    Thorsten Emig

  3. Center for Theoretical Physics, MIT, Cambridge, MA, 02139, USA

    Robert L. Jaffe

  4. Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Ave, 10065, New York, USA

    Sahand Jamal Rahi

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  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    Diego Dalvit

  2. , Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    Peter Milonni

  3. Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    David Roberts

  4. , Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    Felipe da Rosa

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Rahi, S.J., Emig, T., Jaffe, R.L. (2011). Geometry and Material Effects in Casimir Physics-Scattering Theory. In: Dalvit, D., Milonni, P., Roberts, D., da Rosa, F. (eds) Casimir Physics. Lecture Notes in Physics, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20288-9_5

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