Casimir forces between compact objects: The scalar case

T. Emig, N. Graham, R. L. Jaffe, and M. Kardar

Phys. Rev. D 77, 025005 – Published 9 January 2008

Abstract

We have developed an exact, general method to compute Casimir interactions between a finite number of compact objects of arbitrary shape and separation. Here, we present details of the method for a scalar field to illustrate our approach in its most simple form; the generalization to electromagnetic fields is outlined in Ref. [T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Phys. Rev. Lett. 99, 170403 (2007).]. The interaction between the objects is attributed to quantum fluctuations of source distributions on their surfaces, which we decompose in terms of multipoles. A functional integral over the effective action of multipoles gives the resulting interaction. Each object’s shape and boundary conditions enter the effective action only through its scattering matrix. Their relative positions enter through universal translation matrices that depend only on field type and spatial dimension. The distinction of our method from the pairwise summation of two-body potentials is elucidated in terms of the scattering processes between three objects. To illustrate the power of the technique, we consider Robin boundary conditions $ϕ - λ \partial_{n} ϕ = 0$ , which interpolate between Dirichlet and Neumann cases as $λ$ is varied. We obtain the interaction between two such spheres analytically in a large separation expansion, and numerically for all separations. The cases of unequal radii and unequal $λ$ are studied. We find sign changes in the force as a function of separation in certain ranges of $λ$ and see deviations from the proximity force approximation even at short separations, most notably for Neumann boundary conditions.

Received 16 October 2007

DOI:https://doi.org/10.1103/PhysRevD.77.025005

Authors & Affiliations

T. Emig^1,2, N. Graham^3,4, R. L. Jaffe⁴, and M. Kardar⁵

¹Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
²Laboratoire de Physique Théorique et Modèles Statistiques, CNRS UMR 8626, Bât. 100, Université Paris-Sud, 91405 Orsay cedex, France
³Department of Physics, Middlebury College, Middlebury, Vermont 05753, USA
⁴Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
⁵Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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Vol. 77, Iss. 2 — 15 January 2008

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Images

Figure 1
Two objects enclosed by the surfaces $Σ_{1}$ and $Σ_{2}$ are shown, each with bounding spheres (radii $R_{1}$ and $R_{2}$ ). We assume that it is possible to choose bounding spheres that do not overlap. Coordinate systems with parallel axes are erected at origins $O_{1}$ and $O_{2}$ at positions $X_{1}$ and $X_{2}$ . The origins are chosen so that the dipole coefficients of capacitance, $C_{001 m}$ , vanish. Coordinate vectors $x_{1}$ and $x_{2}$ to an arbitrary point, $x$ , are shown. The vector from $O_{1}$ to $O_{2}$ is $X_{12} = X_{2} - X_{1} = x_{1} - x_{2}$ . The distance between the two objects is $d_{12} = | X_{12} |$ .Reuse & Permissions
Figure 2
Scattering processes described by the operators of Eq. (3.28). The directed lines between objects 2 and 3 describe a wave travelling once between the objects. The zigzag lines correspond to multiple reflections in both directions where the number next to the line indicates how many times the wave travels between the objects. To interpret the diagrams one starts with a wave at object 3. As an example, we describe explicitly the diagram for the operator $O_{1} (m, n)$ . The wave travels from object 3 to object 2, is then reflected $2 n$ times back and forth between objects 2 and 1, travels back from object 2 to 3, and is finally reflected $2 m$ times back and forth between objects 3 and 1. Each free propagation between the objects is described by the translation operator $U^{α β}$ , and each reflection at an object by the transition operator $T^{α}$ .Reuse & Permissions
Figure 3
Amplitudes $Φ (λ_{1} / L, λ_{2} / L)$ of the Casimir energy [from Eq. (4.23)] and $Φ^{'} (λ_{1} / L, λ_{2} / L)$ of the Casimir force [from Eq. (4.24)] for two parallel plates of distance $L$ with Robin boundary conditions. $L = 0$ corresponds to the point (1, 1) unless one or both plates have Dirichlet ( $λ_{α} = 0$ ) boundary conditions. The color coding corresponds to the magnitude of the energy amplitude with blue corresponding to positive energy and red to negative. The dashed (black) curves correspond to zero energy while the solid (red) curves represent zero force. The curve running from the origin to (1, 1) shows as an example the interaction energy for $λ_{2} / λ_{1} = 5$ as function of the distance $L$ . For finite $λ_{α}$ the interaction is always attractive at asymptotically large and small distances but can be repulsive at intermediate distances if the ratio $λ_{1} / λ_{2} ≳ 2.8$ (or $λ_{2} / λ_{1} ≳ 2.8$ ). For $λ_{1} = λ_{2}$ the interaction is always attractive. The amplitudes are independent of $L$ for the special cases where the $λ_{α}$ are either zero or infinite: The interaction is attractive (negative energy) for $λ_{1} = λ_{2} = 0$ , $λ_{1} = λ_{2} = \infty$ and repulsive (positive energy) for $λ_{1} = 0$ , $λ_{2} = \infty$ and $λ_{1} = \infty$ , $λ_{2} = 0$ .Reuse & Permissions
Figure 4
Casimir energy for two spheres of radius $R$ and center-to-center distance $d$ : (a) Dirichlet boundary conditions for both spheres, (b) Neumann boundary conditions for both spheres, (c) Spheres with different boundary conditions (one Dirichlet, one Neumann). The energy is scaled by the PFA estimate of Eq. (4.27). The solid curves are obtained by extrapolation to $l \to \infty$ . For the smallest separation, the extrapolation uncertainty is maximal and indicated by an error bar. The dashed curves represent the asymptotic large distance expansion given in Eq. (4.9) with the coefficients of Eqs. (4.13, 4.14, 4.16), respectively.Reuse & Permissions
Figure 5
The Casimir energy for two spheres with different Robin boundary conditions for finite $λ_{α}$ : (a) Dirichlet boundary conditions and $λ / R = 10$ , (b) Neumann boundary conditions and $λ / R = 10$ , (c) $λ_{1} / R = 10$ and $λ_{2} / R = 1$ . The solid curves correspond to extrapolated results for $l \to \infty$ , and the dashed curves represent the asymptotic large distance expansion given in Eq. (4.9) with the coefficients of Eqs. (4.17, 4.15, 4.10), respectively. For logarithmic plotting, the modulus of the energy is shown, and the sign of the energy is indicated at the bottom. The range of separations with a repulsive force is shaded. The points of vanishing force occur where an auxiliary function (dotted curves) is tangent to the solid curve, see text for details.Reuse & Permissions

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