Geometry and Material Effects in Casimir Physics-Scattering Theory

  • Sahand Jamal RahiEmail author
  • Thorsten Emig
  • Robert L. Jaffe
Part of the Lecture Notes in Physics book series (LNP, volume 834)


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.


Partition Function Partial Wave Casimir Force Casimir Energy Parabolic Cylinder 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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).


  1. 1.
    Ashkin, A.: Acceleration and trapping of particles by radiation pressure. Phys. Rev. Lett. 24, 156–159 (1970)ADSCrossRefGoogle Scholar
  2. 2.
    Ashkin, A., Gordon, J.P.: Stability of radiation-pressure particle traps: an optical Earnshaw theorem. Opt. Lett. 8, 511–513 (1983)ADSCrossRefGoogle Scholar
  3. 3.
    Bachas, C.P.: Comment on the sign of the Casimir force. J. Phys. A: Math. Theor. 40, 9089–9096 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Balian, R., Duplantier, B.: Electromagnetic waves near perfect conductors. II. Casimir effect. Ann. Phys., NY 104, 300–335 (1977)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Balian, R., Duplantier, B.: Electromagnetic waves near perfect conductors. I. Multiple scattering expansions. Distribution of modes. Ann. Phys., NY 112, 165–208 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Birman, M.S., Krein, M.G.: On the theory of wave operators and scattering operators. Sov. Math.-Dokl. 3, 740–744 (1962)zbMATHGoogle Scholar
  7. 7.
    Bordag, M., Robaschik, D., Wieczorek, E.: Quantum field theoretic treatment of the Casimir effect. Ann. Phys., NY 165, 192–213 (1985)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Braunbek, W.: Freies Schweben diamagnetischer Körper im magnetfeld. Z. Phys. 112, 764–769 (1939)ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Braunbek, W.: Freischwebende Körper im elektrischen und magnetischen Feld. Z. Phys. 112, 753–763 (1939)ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Bulgac, A., Magierski, P., Wirzba, A.: Scalar Casimir effect between Dirichlet spheres or a plate and a sphere. Phys. Rev. D 73, 025007 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    Bulgac, A., Wirzba, A.: Casimir Interaction among objects immersed in a fermionic environment. Phys. Rev. Lett. 87, 120404 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    Büscher, R., Emig, T.: Geometry and spectrum of Casimir forces. Phys. Rev. Lett. 94, 133901 (2005)ADSCrossRefGoogle Scholar
  13. 13.
    Capasso, F., Munday, J.N., Iannuzzi, D., Chan, H.B.: Casimir forces and quantum electrodynamical torques: Physics and Nanomechanics. IEEE J. Sel. Top. Quant. 13, 400–414 (2007)CrossRefGoogle Scholar
  14. 14.
    Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51, 793–795 (1948)zbMATHGoogle Scholar
  15. 15.
    Casimir, H.B.G., Polder, D.: The Influence of retardation on the London-van der Waals forces. Phys. Rev. 73, 360–372 (1948)ADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Chan, H.B., Aksyuk, V.A., Kleiman, R.N., Bishop, D.J., Capasso, F.: Quantum mechanical actuation of microelectromechanical systems by the Casimir force. Science 291, 1941–1944 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Chan, H.B., Bao, Y., Zou, J., Cirelli, R.A., Klemens, F., Mansfield, W.M., Pai, C.S.: Measurement of the Casimir force between a gold sphere and a silicon surface with nanoscale trench arrays. Phys. Rev. Lett. 101, 030401 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    Chen, F., Klimchitskaya, G.L., Mostepanenko, V.M., Mohideen, U.: Demonstration of the difference in the Casimir force for samples with different charge-carrier densities. Phys. Rev. Lett. 97, 170402 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    Chen, F., Klimchitskaya, G.L., Mostepanenko, V.M., Mohideen, U.: Control of the Casimir force by the modification of dielectric properties with light. Phys. Rev. B 76, 035338 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    Chen, F., Mohideen, U., Klimchitskaya, G.L., Mostepanenko, V.M.: Demonstration of the lateral Casimir force. Phys. Rev. Lett. 88, 101801 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    Chew, W.C., Jin, J.M., Michielssen, E., Song, J.M. (eds.): Fast and Efficient Algorithms in Computational Electrodynamics. Artech House, Norwood, MA (2001)Google Scholar
  22. 22.
    Dalvit, D.A.R., Lombardo, F.C., Mazzitelli, F.D., Onofrio, R.: Exact Casimir interaction between eccentric cylinders. Phys. Rev. A 74, 020101(R) (2006)ADSCrossRefGoogle Scholar
  23. 23.
    Decca, R.S., López, D., Fischbach, E., Klimchitskaya, G.L., Krause, D.E., Mostepanenko, V.M.: Tests of new physics from precise measurements of the Casimir pressure between two gold-coated plates. Phys. Rev. D 75, 077101 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    Druzhinina, V., DeKieviet, M.: Experimental observation of quantum reflection far from threshold. Phys. Rev. Lett. 91, 193202 (2003)ADSCrossRefGoogle Scholar
  25. 25.
    Dzyaloshinskii, I.E., Lifshitz, E.M., Pitaevskii, L.P.: The general theory of van der Waals forces. Adv. Phys. 10, 165–209 (1961)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Earnshaw, S.: On the nature of the molecular forces which regulate the constitution of the luminiferous ether. Trans. Camb. Phil. Soc. 7, 97–112 (1842)ADSGoogle Scholar
  27. 27.
    Ederth, T.: Template-stripped gold surfaces with 0.4-nm rms roughness suitable for force measurements: Application to the Casimir force in the 20–100 nm range. Phys. Rev. A 62, 062104 (2000)ADSCrossRefGoogle Scholar
  28. 28.
    Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Casimir forces between arbitrary compact objects. Phys. Rev. Lett. 99, 170403 (2007)ADSCrossRefGoogle Scholar
  29. 29.
    Emig, T., Graham, N., Jaffe, R.L., Kardar,M.: Casimir forces between compact objects: The scalar objects. Phys. Rev. D 77, 025005 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Orientation dependence of Casimir forces. Phys. Rev. A 79, 054901 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    Emig, T., Hanke, A., Golestanian, R., Kardar, M.: Probing the strong boundary shape dependence of the Casimir force. Phys. Rev. Lett. 87, 260402 (2001)ADSCrossRefGoogle Scholar
  32. 32.
    Emig, T., Hanke, A., Golestanian, R., Kardar, M.: Normal and lateral Casimir forces between deformed plates. Phys. Rev. A 67, 022114 (2003)ADSCrossRefGoogle Scholar
  33. 33.
    Emig, T., Jaffe, R.L., Kardar, M., Scardicchio, A.: Casimir interaction between a plate and a cylinder. Phys. Rev. Lett. 96, 080403 (2006)ADSCrossRefGoogle Scholar
  34. 34.
    Feinberg, G., Sucher, J.: General form of the retarded van der Waals potential. J. Chem. Phys. 48, 3333–3334 (1698)CrossRefGoogle Scholar
  35. 35.
    Feinberg, G., Sucher, J.: General theory of the van der Waals interaction: A model-independent approach. Phys. Rev. A 2, 2395–2415 (1970)ADSCrossRefGoogle Scholar
  36. 36.
    Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  37. 37.
    Geim, A.: Everyone’s magnetism. Phys. Today 51(9), 36–39 (1998)CrossRefGoogle Scholar
  38. 38.
    Genet, C., Lambrecht, A., Reynaud, S.: Casimir force and the quantum theory of lossy optical cavities. Phys. Rev. A 67, 043811 (2003)ADSCrossRefGoogle Scholar
  39. 39.
    Gies, H., Klingmüller, K.: Casimir edge effects. Phys. Rev. Lett. 97, 220405 (2006)ADSCrossRefGoogle Scholar
  40. 40.
    Golestanian, R.: Casimir-Lifshitz interaction between dielectrics of arbitrary geometry: A dielectric contrast perturbation theory. Phys. Rev. A 80, 012519 (2009)ADSCrossRefGoogle Scholar
  41. 41.
    Golestanian, R., Kardar, M.: Mechanical response of vacuum. Phys. Rev. Lett. 78, 3421–3425 (1997)ADSCrossRefGoogle Scholar
  42. 42.
    Golestanian, R., Kardar, M.: Path-integral approach to the dynamic Casimir effect with fluctuating boundaries. Phys. Rev. A 58, 1713–1722 (1998)ADSCrossRefGoogle Scholar
  43. 43.
    Graham, N., Jaffe, R.L., Khemani, V., Quandt, M., Scandurra, M., Weigel, H.: Casimir energies in light of quantum field theory. Phys. Lett. B 572, 196–201 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Graham, N., Quandt, M., Weigel, H.: Spectral methods in quantum field theory. Springer, Berlin (2009)zbMATHGoogle Scholar
  45. 45.
    Graham, N., Shpunt, A., Emig, T., Rahi, S.J., Jaffe, R.L., Kardar, M.: Casimir force at a knife’s edge. Phys. Rev. D 81, 061701(R) (2010)ADSGoogle Scholar
  46. 46.
    Harber, D.M., Obrecht, J.M., McGuirk, J.M., Cornell, E.A.: Measurement of the Casimir-Polder force through center-of-mass oscillations of a Bose-Einstein condensate. Phys. Rev. A 72, 033610 (2005)ADSCrossRefGoogle Scholar
  47. 47.
    Henseler, M., Wirzba, A., Guhr, T.: Quantization of HyperbolicN-Sphere scattering systems in three dimensions. Ann. Phys., NY 258, 286–319 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Jaekel, M.T., Reynaud, S.: Casimir force between partially transmitting mirrors. J. Physique I 1, 1395–1409 (1991)ADSCrossRefGoogle Scholar
  49. 49.
    Jones, T.B.: Electromechanics of Particles. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  50. 50.
    Kats, E.I.: Influence of nonlocality effects on van der Waals interaction. Sov. Phys. JETP 46, 109 (1997)ADSGoogle Scholar
  51. 51.
    Kenneth, O., Klich, I.: Opposites Attract: A theorem about the Casimir force. Phys. Rev. Lett. 97, 160401 (2006)ADSCrossRefGoogle Scholar
  52. 52.
    Kenneth, O., Klich, I.: Casimir forces in a T-operator approach. Phys. Rev. B 78, 014103 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    Kim, W.J., Brown-Hayes, M., Dalvit, D.A.R., Brownell, J.H., Onofrio, R.: Anomalies in electrostatic calibrations for the measurement of the Casimir force in a sphere-plane geometry.Phys. Rev. A 78,020101(2008)ADSCrossRefGoogle Scholar
  54. 54.
    Klimchitskaya, G.L., Mohideen, U., Mostepanenko, V.M.: The Casimir force between real materials: Experiment and theory. Rev. Mod. Phys. 81, 1827–1885 (2009)ADSCrossRefGoogle Scholar
  55. 55.
    Krause, D.E., Decca, R.S., López, D., Fischbach, E.: Experimental investigation of the Casimir force beyond the proximity-force approximation. Phys. Rev. Lett. 98, 050403 (2007)ADSCrossRefGoogle Scholar
  56. 56.
    Krein, M.G.: On the trace formula in perturbation theory. Mat. Sborn. (NS) 33, 597–626 (1953)MathSciNetGoogle Scholar
  57. 57.
    Krein, M.G.: Perturbation determinants and a formula for the trace of unitary and selfadjoint operators. Sov. Math.-Dokl. 3, 707–710 (1962)Google Scholar
  58. 58.
    Lambrecht, A., Neto, P.A.M., Reynaud, S.: The Casimir effect within scattering theory. New J. Phys. 8, 243 (2006)ADSCrossRefGoogle Scholar
  59. 59.
    Lamoreaux, S.K.: Demonstration of the Casimir force in the 0.6 to \(6\, \upmu\hbox {m}\) range. Phys. Rev. Lett. 78, 5–8 (1997)ADSCrossRefGoogle Scholar
  60. 60.
    Landau, L.D., Lifshitz, E.M.: Electrodynamics of continuous media. Pergamon Press, Oxford (1984)Google Scholar
  61. 61.
    Levin M., McCauley A.P., Rodriguez A.W., Reid M.T.H., Johnson S.G. (2010) Casimir repulsion between metallic objects in vacuum. arXiv:1003.3487Google Scholar
  62. 62.
    Li, H., Kardar, M.: Fluctuation-induced forces between rough surfaces. Phys. Rev. Lett. 67, 3275–3278 (1991)ADSCrossRefGoogle Scholar
  63. 63.
    Li, H., Kardar, M.: Fluctuation-induced forces between manifolds immersed in correlated fluids. Phys. Rev. A 46, 6490–6500 (1992)ADSCrossRefGoogle Scholar
  64. 64.
    Lifshitz, E.M.: The theory of molecular attractive forces between solids. Sov. Phys. JETP 2, 73–83 (1956)MathSciNetGoogle Scholar
  65. 65.
    Lifshitz, E.M., Pitaevskii, L.P.: Statistical physics Part 2. Pergamon Press, New York (1980)Google Scholar
  66. 66.
    Lippmann, B.A., Schwinger, J.: Variational principles for scattering processes. i. Phys. Rev. 79, 469–480 (1950)MathSciNetADSzbMATHCrossRefGoogle Scholar
  67. 67.
    Milton, K.A., Parashar, P., Wagner, J.: Exact results for Casimir interactions between dielectric bodies: The weak-coupling or van der waals limit. Phys. Rev. Lett. 101, 160402 (2008)ADSCrossRefGoogle Scholar
  68. 68.
    Milton K.A., Parashar P., Wagner J. (2008) From multiple scattering to van der waals interactions: exact results for eccentric cylinders. arXiv:0811.0128Google Scholar
  69. 69.
    Mohideen, U., Roy, A.: Precision measurement of the Casimir force from 0.1–\(0.9\, \mu\hbox{m}\). Phys. Rev. Lett. 81, 4549–4552 (1998)ADSCrossRefGoogle Scholar
  70. 70.
    Morse, P.M., Feshbach, H.: Methods of theoretical physics. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  71. 71.
    Munday, J.N., Capasso, F.: Precision measurement of the Casimir-Lifshitz force in a fluid. Phys. Rev. A 75, 060102(R) (2007)ADSCrossRefGoogle Scholar
  72. 72.
    Munday, J.N., Capasso, F., Parsegian, V.A.: Measured long-range repulsive Casimir-Lifshitz forces. Nature 457, 170–173 (2009)ADSCrossRefGoogle Scholar
  73. 73.
    Palasantzas, G., van Zwol, P.J., De Hosson, J.T.M.: Transition from Casimir to van der Waals force between macroscopic bodies. Appl. Phys. Lett. 93, 121912 (2008)ADSCrossRefGoogle Scholar
  74. 74.
    Parsegian, V.A.: van der Waals Forces. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  75. 75.
    Rahi, S.J., Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Scattering theory approach to electrodynamic Casimir forces. Phys. Rev. D 80, 085021 (2009)ADSCrossRefGoogle Scholar
  76. 76.
    Rahi, S.J., Emig, T., Jaffe, R.L., Kardar, M.: Casimir forces between cylinders and plates. Phys. Rev. A 78, 012104 (2008)ADSCrossRefGoogle Scholar
  77. 77.
    Rahi S.J., Kardar M., Emig T. Constraints on stable equilibria with fluctuation-induced forces. Phys. Rev. Lett. 105, 070404 (2010)Google Scholar
  78. 78.
    Rahi, S.J., Rodriguez, A.W., Emig, T., Jaffe, R.L., Johnson, S.G., Kardar, M.: Nonmonotonic effects of parallel sidewalls on Casimir forces between cylinders. Phys. Rev. A 77, 030101 (2008)ADSCrossRefGoogle Scholar
  79. 79.
    Rahi, S.J., Zaheer, S.: Stable levitation and alignment of compact objects by Casimir spring forces. Phys. Rev. Lett. 104, 070405 (2010)ADSCrossRefGoogle Scholar
  80. 80.
    Reid, M.T.H., Rodriguez, A.W., White, J., Johnson, S.G.: Efficient computation of Casimir interactions between arbitrary 3D objects. Phys. Rev. Lett. 103, 040401 (2009)ADSCrossRefGoogle Scholar
  81. 81.
    Renne, M.J.: Microscopic theory of retarded Van der Waals forces between macroscopic dielectric bodies. Physica 56, 125–137 (1971)ADSCrossRefGoogle Scholar
  82. 82.
    Robaschik, D., Scharnhorst, K., Wieczorek, E.: Radiative corrections to the Casimir pressure under the influence of temperature and external fields. Ann. Phys., NY 174, 401–429 (1987)MathSciNetADSCrossRefGoogle Scholar
  83. 83.
    Rodriguez, A., Ibanescu, M., Iannuzzi, D., Capasso, F., Joannopoulos, J.D., Johnson, S.G.: Computation and visualization of Casimir forces in arbitrary geometries: Non-monotonic lateral-wall forces and failure of proximity force approximations. Phys. Rev. Lett. 99, 080401 (2007)ADSCrossRefGoogle Scholar
  84. 84.
    Rodriguez, A.W., Joannopoulos, J.D., Johnson, S.G.: Repulsive, nonmonotonic Casimir forces in a glide-symmetric geometry. Phys. Rev. A 77, 062107 (2008)ADSCrossRefGoogle Scholar
  85. 85.
    Rodriguez-Lopez, P., Rahi, S.J., Emig, T.: Three-body Casimir effects and nonmonotonic forces. Phys. Rev. A 80, 022519 (2009)ADSCrossRefGoogle Scholar
  86. 86.
    Rosa, F.S.S.: On the possibility of Casimir repulsion using metamaterials. J. Phys.: Conf. Ser. 161, 012039 (2009)ADSCrossRefGoogle Scholar
  87. 87.
    Rosa, F.S.S., Dalvit, D.A.R., Milonni, P.W.: Casimir-Lifshitz theory and metamaterials. Phys. Rev. Lett. 100, 183602 (2008)ADSCrossRefGoogle Scholar
  88. 88.
    Roy, A., Lin, C.Y., Mohideen, U.: Improved precision measurement of the Casimir force. Phys. Rev. D 60, 111101(R) (1999)ADSGoogle Scholar
  89. 89.
    Schaden, M., Spruch, L.: Infinity-free semiclassical evaluation of Casimir effects. Phys. Rev. A 58, 935–953 (1998)ADSCrossRefGoogle Scholar
  90. 90.
    Schwinger, J.: Casimir effect in source theory. Lett. Math. Phys. 1, 43–47 (1975)MathSciNetADSCrossRefGoogle Scholar
  91. 91.
    Ttira, C.C., Fosco, C.D., Losada, E.L.: Non-superposition effects in the Dirichlet–Casimir effect. J. Phys. A: Math. Theor. 43, 235402 (2010)ADSCrossRefGoogle Scholar
  92. 92.
    Weber, A., Gies, H.: Interplay between geometry and temperature for inclined Casimir plates. Phys. Rev. D 80, 065033 (2009)ADSCrossRefGoogle Scholar
  93. 93.
    Wirzba, A.: Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems. Phys. Rep. 309, 1–116 (1999)MathSciNetADSCrossRefGoogle Scholar
  94. 94.
    Wirzba, A.: The Casimir effect as a scattering problem. J. Phys. A: Math. Theor. 41, 164003 (2008)MathSciNetADSCrossRefGoogle Scholar
  95. 95.
    Zaheer, S., Rahi, S.J., Emig, T., Jaffe, R.L.: Casimir interactions of an object inside a spherical metal shell. Phys. Rev. A 81, 030502 (2010)ADSCrossRefGoogle Scholar
  96. 96.
    Zhao, R., Zhou, J., Koschny, T., Economou, E.N., Soukoulis, C.M.: Repulsive Casimir force in chiral metamaterials. Phys. Rev. Lett. 103, 103602 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg  2011

Authors and Affiliations

  • Sahand Jamal Rahi
    • 1
    • 4
    Email author
  • Thorsten Emig
    • 2
  • Robert L. Jaffe
    • 3
  1. 1.Department of PhysicsMITCambridgeUSA
  2. 2.Laboratoire de Physique Théorique et Modèles StatistiquesUniversité Paris-SudOrsayFrance
  3. 3.Center for Theoretical PhysicsMITCambridgeUSA
  4. 4.Center for Studies in Physics and BiologyThe Rockefeller UniversityNew YorkUSA

Personalised recommendations