Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity

  • Kimball A. MiltonEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 834)


From the beginning of the subject, calculations of quantum vacuum energies or Casimir energies have been plagued with two types of divergences: The total energy, which may be thought of as some sort of regularization of the zero-point energy, \(\sum\frac{1}{ 2}\hbar\omega,\) seems manifestly divergent. And local energy densities, obtained from the vacuum expectation value of the energy-momentum tensor, \(\langle T_{00}\rangle ,\) typically diverge near boundaries. These two types of divergences have little to do with each other. The energy of interaction between distinct rigid bodies of whatever type is finite, corresponding to observable forces and torques between the bodies, which can be unambiguously calculated. The divergent local energy densities near surfaces do not change when the relative position of the rigid bodies is altered. The self-energy of a body is less well-defined, and suffers divergences which may or may not be removable. Some examples where a unique total self-stress may be evaluated include the perfectly conducting spherical shell first considered by Boyer, a perfectly conducting cylindrical shell, and dilute dielectric balls and cylinders. In these cases the finite part is unique, yet there are divergent contributions which may be subsumed in some sort of renormalization of physical parameters. The finiteness of self-energies is separate from the issue of the physical observability of the effect. The divergences that occur in the local energy-momentum tensor near surfaces are distinct from the divergences in the total energy, which are often associated with energy located exactly on the surfaces. However, the local energy-momentum tensor couples to gravity, so what is the significance of infinite quantities here? For the classic situation of parallel plates there are indications that the divergences in the local energy density are consistent with divergences in Einstein’s equations; correspondingly, it has been shown that divergences in the total Casimir energy serve to precisely renormalize the masses of the plates, in accordance with the equivalence principle. This should be a general property, but has not yet been established, for example, for the Boyer sphere. It is known that such local divergences can have no effect on macroscopic causality.


Equivalence Principle Strong Coupling Limit Casimir Energy Null Energy Condition Divergent Term 
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.



I thank the US Department of Energy and the US National Science Foundation for partial support of this work. I thank my many collaborators, including Carl Bender, Iver Brevik, Inés Cavero-Peláez, Lester DeRaad, Steve Fulling, Ron Kantowski, Klaus Kirsten, Vladimir Nesterenko, Prachi Parashar, August Romeo, K.V. Shajesh, and Jef Wagner, for their contributions to the work described here.


  1. 1.
    Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Akad. Wetensch. 51, 793 (1948)zbMATHGoogle Scholar
  2. 2.
    London, F.: Theory and system of molecular forces. Z. Physik 63, 245 (1930)ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Casimir, H.B.G., Polder, D.: The influence of retardation on the London-Van Der Waals forces. Phys. Rev. 73, 360 (1948)ADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Casimir, H.B.G.: In: Bordag, M. (ed.) The Casimir Effect 50 Years Later: The Proceedings of the Fourth Workshop on Quantum Field Theory Under the Influence of External Conditions, World Scientific, Singapore, p. 3, (1999)Google Scholar
  5. 5.
    Jaffe, R.L.: Unnatural acts: Unphysical consequences of imposing boundary conditions on quantum fields. AIP Conf. Proc. 687, p. 3 (2003). arXiv:hep-th/0307014Google Scholar
  6. 6.
    Lifshitz, E.M.: Zh. Eksp. Teor. Fiz. 29, 94 (1956), [English translation: The theory of molecular attractive forces between solids. Soviet Phys. JETP 2,73 (1956)]Google Scholar
  7. 7.
    Dzyaloshinskii, I.D., Lifshitz, E. M., Pitaevskii, L.P.: Zh. Eksp. Teor. Fiz. 37, 229 (1959), [English translation: Van der Waals forces in liquid films. Soviet Phys. JETP 10, 161 (1960)]Google Scholar
  8. 8.
    Dzyaloshinskii, I.D., Lifshitz, E.M., Pitaevskii, L.P., Usp. Fiz. Nauk 73, 381(1961), [English translation: General theory of van der Waals forces. Soviet Phys. Usp. 4, 153 (1961)]Google Scholar
  9. 9.
    Bordag, M., Klimchitskaya, G.L., Mohideen, U., Mostepanenko, V.M.: Advances in the Casimir Effect. Int. Ser. Monogr. Phys. 145, 1 (2009). (Oxford University Press, Oxford, 2009)Google Scholar
  10. 10.
    Klimchitskaya, G.L., Mohideen, U., Mostepanenko, V.M.: The Casimir force between real materials: experiment and theory. Rev. Mod. Phys. 81, 1827 (2009). arXiv:0902.4022[cond-mat.other]ADSCrossRefGoogle Scholar
  11. 11.
    Deryagin(Derjaguin), B.V.: Analysis of friction and adhesion IV: The theory of the adhesion of small particles. Kolloid Z. 69, 155 (1934)CrossRefGoogle Scholar
  12. 12.
    Deryagin(Derjaguin), B.V. et al.: Effect of contact deformations on the adhesion of particles. J. Colloid. Interface Sci. 53, 314 (1975)CrossRefGoogle Scholar
  13. 13.
    Blocki, J., Randrup, J., Świątecki, W. J., Tsang, C.F.: Proximity forces. Ann. Phys. (N.Y.) 105, 427 (1977)ADSCrossRefGoogle Scholar
  14. 14.
    Milton, K.A.: Recent developments in the Casimir effect. J. Phys. Conf. Ser. 161, 012001 (2009). [hep-th]]Google Scholar
  15. 15.
    Boyer, T.H.: Quantum electromagnetic zero point energy of a conducting spherical shell and the Casimir model for a charged particle. Phys. Rev. 174, 1764 (1968)ADSCrossRefGoogle Scholar
  16. 16.
    Lukosz, W.: Electromagnetic zero-point energy and radiation pressure for a rectangular cavity. Physica 56, 109 (1971)ADSCrossRefGoogle Scholar
  17. 17.
    Lukosz, W.: Electromagnetic zero-point energy shift induced by conducting closed surfaces. Z. Phys. 258, 99 (1973)ADSCrossRefGoogle Scholar
  18. 18.
    Lukosz, W.: Electromagnetic zero-point energy shift induced by conducting surfaces. II. The infinite wedge and the rectangular cavity. Z. Phys. 262, 327 (1973)ADSCrossRefGoogle Scholar
  19. 19.
    Ambjørn, J., Wolfram, S.: Properties of the vacuum. I. Mechanical and thermodynamic. Ann. Phys. (N.Y.) 147, 1 (1983)ADSCrossRefGoogle Scholar
  20. 20.
    Balian, R., Duplantier, B.: Electromagnetic waves near perfect conductors. II. Casimir effect. Ann. Phys. (N.Y.) 112, 165 (1978)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Bernasconi, F., Graf, G.M., Hasler, D.: The heat kernel expansion for the electromagnetic field in a cavity. Ann. Henri Poincaré 4, 1001 (2003). arXiv:math-ph/0302035Google Scholar
  22. 22.
    Fulling, S.A., Milton, K.A., Parashar, P., Romeo, A., Shajesh, K.V., Wagner, J.: How does Casimir energy fall?. Phys. Rev. D 76, 025004 (2007). arXiv:hep-th/0702091Google Scholar
  23. 23.
    Milton, K.A., Parashar, P., Shajesh, K.V., Wagner, J.: How does Casimir energy fall? II. Gravitational acceleration of quantum vacuum energy. J. Phys. A 40, 10935 (2007). [hep-th]]Google Scholar
  24. 24.
    Milton, K.A., Wagner, J.: Exact Casimir Interaction Between Semitransparent Spheres and Cylinders. Phys. Rev. D 77, 045005 (2008). [arXiv:0711.0774 [hep-th]]Google Scholar
  25. 25.
    Milton, K.A., Wagner, J.: Multiple Scattering Methods in Casimir Calculations. J. Phys. A 41, 155402 (2008). [hep-th]]Google Scholar
  26. 26.
    Wagner, J., Milton, K.A., Parashar, P.: Weak Coupling Casimir Energies for Finite Plate Configurations. J. Phys. Conf. Ser. 161, 012022 (2009). [arXiv:0811.2442 [hep-th]]Google Scholar
  27. 27.
    DeRaad, L.L. Jr., Milton, K.A.: Casimir Selfstress On A Perfectly Conducting Cylindrical Shell. Ann. Phys. (N.Y.) 136, 229 (1981)ADSCrossRefGoogle Scholar
  28. 28.
    Bender, C.M., Milton, K.A.: Casimir effect for a D-dimensional sphere. Phys. Rev. D 50, 6547 (1994). arXiv:hep-th/9406048Google Scholar
  29. 29.
    Gosdzinsky, P., Romeo, A.: Energy of the vacuum with a perfectly conducting and infinite cylindrical surface. Phys. Lett. B 441, 265 (1998). arXiv:hep-th/9809199Google Scholar
  30. 30.
    Brevik, I., Marachevsky, V.N., Milton, K.A.: Identity of the van der Waals force and the Casimir effect and the irrelevance of these phenomena to sonoluminescence. Phys. Rev. Lett. 82, 3948 (1999). arXiv:hep-th/9810062Google Scholar
  31. 31.
    Cavero-Peláez, I., Milton, K.A.: Casimir energy for a dielectric cylinder. Ann. Phys. (N.Y.) 320, 108 (2005). arXiv:hep-th/0412135Google Scholar
  32. 32.
    Klich, I.: Casimir’s energy of a conducting sphere and of a dilute dielectric ball. Phys. Rev. D 61, 025004 (2000). arXiv:hep-th/9908101Google Scholar
  33. 33.
    Milton, K.A., Nesterenko, A.V., Nesterenko, V.V.: Mode-by-mode summation for the zero point electromagnetic energy of an infinite cylinder. Phys. Rev. D 59, 105009 (1999)ADSCrossRefGoogle Scholar
  34. 34.
    Kitson, A.R., Signal, A.I.: Zero-point energy in spheroidal geometries. J. Phys. A 39, 6473 (2006). arXiv:hep-th/0511048Google Scholar
  35. 35.
    Kitson, A.R., Romeo, A.: Perturbative zero-point energy for a cylinder of elliptical section. Phys. Rev. D 74, 085024 (2006). arXiv:hep-th/0607206Google Scholar
  36. 36.
    Milton, K.A.: Calculating Casimir energies in renormalizable quantum field theory. Phys. Rev. D 68, 065020 (2003). arXiv:hep-th/0210081.Google Scholar
  37. 37.
    Cavero-Peláez, I., Milton, K.A., Kirsten, K.: Local and global Casimir energies for a semitransparent cylindrical shell. J. Phys. A 40, 3607 (2007). arXiv:hep-th/0607154Google Scholar
  38. 38.
    Milton, K.A.: The Casimir Effect: Physical Manifestations of Zero-Point Energy. World Scientific, Singapore (2001)zbMATHCrossRefGoogle Scholar
  39. 39.
    Bordag, M., Hennig, D., Robaschik, D.: Vacuum energy in quantum field theory with external potentials concentrated on planes. J. Phys. A 25, 4483 (1992)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Bordag, M., Kirsten, K., Vassilevich, D.: Ground state energy for a penetrable sphere and for a dielectric ball. Phys. Rev. D 59, 085011 (1999). arXiv:hep-th/9811015Google Scholar
  41. 41.
    Graham, N., Jaffe, R.L., Weigel, H.: Casimir effects in renormalizable quantum field theories. Int. J. Mod. Phys. A 17, 846 (2002). arXiv:hep-th/0201148Google Scholar
  42. 42.
    Graham, N., Jaffe, R.L., Khemani, V., Quandt, M., Scandurra, M., Weigel, H.: Calculating vacuum energies in renormalizable quantum field theories: a new approach to the Casimir problem. Nucl. Phys. B 645, 49 (2002). arXiv:hep-th/0207120Google 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 (2003). arXiv:hep-th/0207205Google Scholar
  44. 44.
    Graham, N., Jaffe, R.L., Khemani, V., Quandt, M., Scandurra, M., Weigel, H.: The Dirichlet Casimir problem. Nucl. Phys. B 677, 379 (2004). arXiv:hep-th/0309130Google Scholar
  45. 45.
    Milton, K.A.: Casimir energies and pressures for delta-function potentials. J. Phys. A 37, 6391 (2004). arXiv:hep-th/0401090Google Scholar
  46. 46.
    Milton, K.A.: The Casimir effect: Recent controversies and progress. J. Phys. A 37, R209 (2004). arXiv:hep-th/0406024Google Scholar
  47. 47.
    Kantowski, R., Milton, K.A.: Scalar Casimir energies in M 4 × SN for even N. Phys. Rev. D 35, 549 (1987)ADSCrossRefGoogle Scholar
  48. 48.
    Brevik, I., Jensen, B., Milton, K.A.: Comment on "Casimir energy for spherical boundaries". Phys. Rev. D 64, 088701 (2001). arXiv:hep-th/0004041Google Scholar
  49. 49.
    Weigel H.: Dirichlet spheres in continuum quantum field theory. In: Milton, K.A. (ed.) Proceedings of the 6th Workshop on Quantum Field Theory Under the Influence of External Conditions, p. 195, (Rinton Press, Princeton, N.J., 2004) arXiv:hep-th/0310301Google Scholar
  50. 50.
    Fulling, S.A.: Systematics of the relationship between vacuum energy calculations and heat kernel coefficients. J. Phys. A 36, 6857 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  51. 51.
    Graham, N., Olum, K.D.: Negative energy densities in quantum field theory with a background potential. Phys. Rev. D 67, 085014 (2003). arXiv:quant-ph/0302117Google Scholar
  52. 52.
    Callan, C.G. Jr., Coleman, S., Jackiw, R.: A new improved energy-momentum tensor. Ann. Phys. (N.Y.) 59, 42 (1970)Google Scholar
  53. 53.
    Olum, K.D., Graham, N.: Static negative energies near a domain wall. Phys. Lett. B 554, 175 (2003). arXiv:gr-qc/0205134Google Scholar
  54. 54.
    Romeo, A., Saharian, A.A.: Casimir effect for scalar fields under Robin boundary conditions on plates. J. Phys. A 35, 1297 (2002). arXiv:hep-th/0007242Google Scholar
  55. 55.
    Romeo, A., Saharian, A.A.: Vacuum densities and zero-point energy for fields obeying Robin conditions on cylindrical surfaces. Phys. Rev. D 63, 105019 (2001). arXiv:hepth/0101155Google Scholar
  56. 56.
    Saharian, A.A.: Scalar Casimir effect for D-dimensional spherically symmetric Robin boundaries. Phys. Rev. D 6, 125007 (2001). arXiv:hep-th/0012185Google Scholar
  57. 57.
    Saharian, A.A.: On the energy-momentum tensor for a scalar field on manifolds with boundaries. Phys. Rev. D 69, 085005 (2004). arXiv:hep-th/0308108Google Scholar
  58. 58.
    Brown, L.S., Maclay, G.J.: Vacuum stress between conducting plates: An Image solution. Phys. Rev. 184, 1272 (1969)ADSCrossRefGoogle Scholar
  59. 59.
    Actor, A.A., Bender, I.: Boundaries immersed in a scalar quantum field. Fortsch. Phys. 44, 281 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  60. 60.
    Dowker, J.S., Kennedy, G.: Finite temperature and boundary effects in static space-times. J. Phys. A 11, 895 (1978)MathSciNetADSCrossRefGoogle Scholar
  61. 61.
    Deutsch, D., Candelas, P.: Boundary effects in quantum field theory. Phys. Rev. D 20, 3063 (1979)MathSciNetADSCrossRefGoogle Scholar
  62. 62.
    Brevik, I., Lygren, M.: Casimir effect for a perfectly conducting wedge. Ann. Phys. (N.Y.) 251, 157 (1996)ADSCrossRefGoogle Scholar
  63. 63.
    Sopova, V., Ford, L.H.: The electromagnetic field stress tensor near dielectric half-spaces. In: Milton, K.A. (ed.) Proceedings of the 6th Workshop on Quantum Field Theory Under the Influence of External Conditions, p.140. Rinton Press, Princeton, NJ, (2004)Google Scholar
  64. 64.
    Sopova, V., Ford, L.H.: The Electromagnetic Field Stress Tensor between Dielectric Half-Spaces. Phys. Rev. D 72, 033001 (2005). arXiv:quant-ph/0504143Google Scholar
  65. 65.
    Graham, N.: Do casimir energies obey general relativity energy conditions?. In: Milton, K.A. (ed.) Proceedings of the 6th Workshop on Quantum Field Theory Under the Influence of External Conditions, Rinton Press, Princeton, NJ (2004)Google Scholar
  66. 66.
    Graham, N., Olum, K.D.: Plate with a hole obeys the averaged null energy condition. Phys. Rev. D 72, 025013 (2005). arXiv:hep-th/0506136Google Scholar
  67. 67.
    Milton, K.A.: Semiclassical electron models: Casimir self-stress in dielectric and conducting balls. Ann. Phys. (N.Y.) 127, 49 (1980)MathSciNetADSCrossRefGoogle Scholar
  68. 68.
    Milton, K.A., DeRaad, L.L. Jr., Schwinger, J.: Casimir self-stress on a perfectly conducting spherical shell. Ann. Phys. (N.Y.) 115, 388 (1978)MathSciNetADSCrossRefGoogle Scholar
  69. 69.
    Candelas, P.: Vacuum energy in the presence of dielectric and conducting surfaces. Ann. Phys. (N.Y.) 143, 241 (1982)MathSciNetADSCrossRefGoogle Scholar
  70. 70.
    Candelas, P.: Vacuum energy in the bag model. Ann. Phys. (N.Y.) 167, 257 (1986)ADSCrossRefGoogle Scholar
  71. 71.
    Bordag, M., Mohideen, U., Mostepanenko, V.M.: New developments in the Casimir effect. Phys. Rept. 353, 1 (2001). arXiv:quant-ph/0106045Google Scholar
  72. 72.
    Sen, S.: Geometrical determination of the sign of the Casimir force in two spatial dimensions. Phys. Rev. D 24, 869 (1981)MathSciNetADSCrossRefGoogle Scholar
  73. 73.
    Sen, S.: A calculation of the Casimir force on a circular boundary. J. Math. Phys. 22, 2968 (1981)MathSciNetADSCrossRefGoogle Scholar
  74. 74.
    Cavero-Peláez, I., Milton, K.A., Wagner, J.: Local casimir energies for a thin spherical shell. Phys. Rev. D 73, 085004 (2006). arXiv:hep-th/0508001Google Scholar
  75. 75.
    Barton, G.: Casimir energies of spherical plasma shells. J. Phys. A 37, 1011 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  76. 76.
    Scandurra, M.: The ground state energy of a massive scalar field in the background of a semi-transparent spherical shell. J. Phys. A 32, 5679 (1999). arXiv:hep-th/9811164Google Scholar
  77. 77.
    Bender, C.M., Milton, K.A.: Scalar Casimir effect for a D-dimensional sphere. Phys. Rev. D 50, 6547 (1994). arXiv:hep-th/9406048Google Scholar
  78. 78.
    Leseduarte, S., Romeo, A.: Complete zeta-function approach to the electromagnetic Casimir effect for a sphere. Europhys. Lett. 34, 79 (1996)MathSciNetADSCrossRefGoogle Scholar
  79. 79.
    Leseduarte, S., Romeo, A.: Complete zeta-function approach to the electromagnetic Casimir effect for spheres and circles. Ann. Phys. (N.Y.) 250, 448 (1996). arXiv:hepth/9605022Google Scholar
  80. 80.
    Klich, I.: Casimir energy of a conducting sphere and of a dilute dielectric ball. Phys. Rev. D 61, 025004 (2000). arXiv:hep-th/9908101Google Scholar
  81. 81.
    Bordag, M., Vassilevich, D.V.: Nonsmooth backgrounds in quantum field theory. Phys. Rev. D 70, 045003 (2004). arXiv:hep-th/0404069Google Scholar
  82. 82.
    Milton, K.A.: Zero-point energy in bag models. Phys. Rev. D 22, 1441 (1980)ADSCrossRefGoogle Scholar
  83. 83.
    Milton, K.A.: Zero-point energy of confined fermions. Phys. Rev. D 22, 1444 (1980)ADSCrossRefGoogle Scholar
  84. 84.
    Milton, K.A.: Vector Casimir effect for a D-dimensional sphere. Phys. Rev. D 55, 4940 (1997). arXiv:hep-th/9611078Google Scholar
  85. 85.
    Leseduarte, S., Romeo, A.: Influence of a magnetic fluxon on the vacuum energy of quantum fields confined by a bag. Commun. Math. Phys. 193, 317 (1998). arXiv:hep-th/9612116Google Scholar
  86. 86.
    Davies, B.: Quantum electromagnetic zero-point energy of a conducting spherical shell. J. Math. Phys. 13, 1324 (1972)ADSCrossRefGoogle Scholar
  87. 87.
    Schwartz-Perlov, D., Olum, K.D.: Energy conditions for a generally coupled scalar field outside a reflecting sphere. Phys. Rev. D 72, 065013 (2005). arXiv:hep-th/0507013Google Scholar
  88. 88.
    Scandurra, M.: Vacuum energy of a massive scalar field in the presence of a semi-transparent cylinder. J. Phys. A 33, 5707 (2000). arXiv:hep-th/0004051Google Scholar
  89. 89.
    Gilkey, P.B., Kirsten, K., Vassilevich, D.V.: Heat trace asymptotics with transmittal boundary conditions and quantum brane-world scenario. Nucl. Phys. B 601, 125 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  90. 90.
    Nesterenko, V.V., Pirozhenko, I.G.: Spectral zeta functions for a cyllinder and a circle. J. Math. Phys. 41, 4521 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  91. 91.
    Kennedy, G., Critchley, R., Dowker, J.S.: Finite temperature field theory with boundaries: stress tensor and surface action renormalization. Ann. Phys. (N.Y.) 125, 346 (1980)MathSciNetADSCrossRefGoogle Scholar
  92. 92.
    Romeo, A., Saharian, A.A.: Casimir effect for scalar fields under Robin boundary conditions on plates. J. Phys. A 35, 1297 (2002). arXiv:hep-th/0007242Google Scholar
  93. 93.
    Fulling, S.A., Kaplan, L., Kirsten, K., Liu, Z.H., Milton, K.A.: Vacuum stress and closed paths in rectangles, pistons, and pistols. J. Phys. A 42, 155402 (2009). arXiv:0806.2468[hep-th]Google Scholar
  94. 94.
    Born, M.: The theory of the rigid electron in the kinematics of the relativity principle. Ann. Phys. (Leipzig) 30, 1 (1909)ADSzbMATHGoogle Scholar
  95. 95.
    Calloni, E., Di Fiore, L., Esposito, G., Milano, L., Rosa, L.: Vacuum fluctuation force on a rigid Casimir cavity in a gravitational field. Phys. Lett. A 297, 328 (2002)ADSzbMATHCrossRefGoogle Scholar
  96. 96.
    Karim, M., Bokhari, A.H., Ahmedov, B.J.: The Casimir force in the Schwarzchild metric. Class. Quant. Grav. 17, 2459 (2000)ADSzbMATHCrossRefGoogle Scholar
  97. 97.
    Caldwell, R.R.: Gravitation of the Casimir effect and the cosmological non-constant. arXiv:astro-ph/0209312Google Scholar
  98. 98.
    Sorge, F.: Casimir effect in a weak gravitational field. Class. Quant. Grav. 22, 5109 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  99. 99.
    Bimonte, G., Calloni, E., Esposito, G., Rosa, L.: Energy-momentum tensor for a Casimir apparatus in a weak gravitational field. Phys. Rev. D 74, 085011 (2006)ADSCrossRefGoogle Scholar
  100. 100.
    Bimonte, G., Esposito, G., Rosa, L.: From Rindler space to the electromagnetic energy-momentum tensor of a Casimir apparatus in a weak gravitational field. Phys. Rev. D 78, 024010 (2008). arXiv:0804.2839 [hep-th]Google Scholar
  101. 101.
    Saharian, A.A., Davtyan, R.S., Yeranyan, A.H.: Casimir energy in the Fulling-Rindler vacuum. Phys. Rev. D 69, 085002 (2004). arXiv:hep-th/0307163Google Scholar
  102. 102.
    Jaekel. M.T., Reynaud, S.: Mass, inertia and gravitation. arXiv:0812.3936 [gr-qc]Google Scholar
  103. 103.
    Estrada, R., Fulling, S.A., Liu, Z., Kaplan, L., Kirsten, K., Milton, K.A.: Vacuum stress-energy density and its gravitational implications. J. Phys. A 41, 164055 (2008)MathSciNetADSCrossRefGoogle Scholar
  104. 104.
    Actor, A.A.: Scalar quantum fields confined by rectangular boundaries. Fortsch. Phys. 43, 141 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  105. 105.
    Schaden, M.: Semiclassical electromagnetic Casimir self-energies. arXiv:hep-th/0604119Google Scholar
  106. 106.
    Gies, H., Klingmuller, K.: Casimir edge effects. Phys. Rev. Lett. 97, 220405 (2006). arXiv:quant-ph/0606235Google Scholar
  107. 107.
    Gies, H., Klingmuller, K.: Worldline algorithms for Casimir configurations Phys. Rev. D 74, 045002 (2006). arXiv:quant-ph/0605141Google Scholar
  108. 108.
    Gies, H., Klingmuller, K.: Casimir effect for curved geometries: PFA validity limits. Phys. Rev. Lett. 96, 220401 (2006). arXiv:quant-ph/0601094Google Scholar
  109. 109.
    Jaffe, R.L., Scardicchio, A.: The casimir effect and geometric optics. Phys. Rev. Lett. 92, 070402 (2004). arXiv:quant-ph/0310194Google Scholar
  110. 110.
    Scardicchio, A., Jaffe, R.L.: Casimir effects: an optical approach I. foundations and examples. Nucl. Phys. B 704, 552 (2005). arXiv:quant-ph/0406041Google Scholar
  111. 111.
    Schroeder, O., Scardicchio, A., Jaffe, R.L.: The Casimir energy for a hyperboloid facing a plate in the optical approximation. Phys. Rev. A 72, 012105 (2005). arXiv:hep-th/0412263Google Scholar
  112. 112.
    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 (2010). arXiv:0910.4649 [quant-ph]Google Scholar
  113. 113.
    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). arXiv:0711.1987 [cond-mat.stat-mech]Google Scholar
  114. 114.
    Farhi, E., Graham, N., Haagensen, P., Jaffe, R.L.: Finite quantum fluctuations about static field configurations. Phys. Lett. B 427, 334 (1998). arXiv:hep-th/9802015Google Scholar
  115. 115.
    Graham, N., Jaffe, R.L.: Energy, central charge, and the BPS bound for 1+1 dimensional supersymmetric solitons. Nucl. Phys. B 544, 432 (1999). arXiv:hep-th/9808140Google Scholar
  116. 116.
    Cavero-Peláez, I., Guilarte, J.M.: Local analysis of the sine-Gordon kink quantum fluctuations. to appear In: Milton, K. A., Bordag, M. (eds.) Proceedings of the 9th Conference on Quantum Field Theory Under the Influence of External Conditions, World Scientific, Singapore (2010). arXiv:0911.4450 [hep-th]Google Scholar

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© Springer-Verlag Berlin Heidelberg  2011

Authors and Affiliations

  1. 1.Homer L. Dodge Department of Physics and AstronomyUniversity of OklahomaNormanUSA

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