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Physical Review D

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Exact classical sphere-plate Casimir interaction in $(D + 1)$ -dimensional spacetime

Department of Applied Mathematics, Faculty of Engineering, University of Nottingham Malaysia Campus, Jalan Broga, 43500 Semenyih, Selangor Darul Ehsan, Malaysia

^*LeePeng.Teo@nottingham.edu.my

Phys. Rev. D 89, 105033 – Published 29 May, 2014

DOI: https://doi.org/10.1103/PhysRevD.89.105033

Abstract

We consider the high-temperature limit of the Casimir interaction between a Dirichlet sphere and a Dirichlet plate due to the vacuum fluctuations of a scalar field in $(D + 1)$ -dimensional Minkowski spacetime. The high-temperature leading term of the Casimir free interaction energy is known as the classical term, since it does not depend on the Planck constant $ℏ$ . From the functional representation of the zero-temperature Casimir interaction energy, we use Matsubara formalism to derive the finite-temperature Casimir free energy and obtain the classical term. It can be expressed as a weighted sum over logarithms of determinants. Using similarity transforms of matrices, we reexpress this classical term as an infinite series. This series is then computed exactly using a generalized Abel-Plana summation formula. From this, we deduce the short-distance asymptotic expansions of the classical Casimir interaction force. As expected, the leading term agrees with the proximity force approximation. The next two terms in the asymptotic expansion are also computed. It is observed that the ratio of the next-to-leading-order term to the leading-order term is proportional to the dimension of spacetime. Hence, a larger correction to the proximity force approximation is expected in spacetime with higher dimensions. This is similar to a previous result deduced for the zero-temperature case.

Article Text

References (62)

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University Press, New York, 2009).
A. Lambrecht, P. A. Maia-Neto, and S. Reynaud, New J. Phys. 8, 243 (2006).
A. Bulgac, P. Magierski, and A. Wirzba, Phys. Rev. D 73, 025007 (2006).
T. Emig, R. L. Jaffe, M. Kadar, and A. Scardicchio, Phys. Rev. Lett. 96, 080403 (2006).
S. J. Rahi, T. Emig, R. L. Jaffe, and M. Kardar, Phys. Rev. A 78, 012104 (2008).
T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Phys. Rev. Lett. 99, 170403 (2007).
T. Emig, N. Graham, R. L. Jaffe, and M. Kadar, Phys. Rev. D 77, 025005 (2008).
T. Emig and R. L. Jaffe, J. Phys. A 41, 164001 (2008).
T. Emig, J. Stat. Mech. (2008) P04007 (2008).
S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kadar, Phys. Rev. D 80, 085021 (2009).
M. Bordag, Phys. Rev. D 73, 125018 (2006).
M. Bordag, Phys. Rev. D 75, 065003 (2007).
O. Kenneth and I. Klich, Phys. Rev. Lett. 97, 160401 (2006).
O. Kenneth and I. Klich, Phys. Rev. B 78, 014103 (2008).
K. A. Milton and J. Wagner, J. Phys. A 41, 155402 (2008).
K. Milton and J. Wagner, Phys. Rev. D 77, 045005 (2008).
L. P. Teo, Int. J. Mod. Phys. A 27, 1230021 (2012).
M. Bordag and V. Nikolaev, J. Phys. A 41, 164002 (2008).
M. Bordag and V. Nikolaev, Phys. Rev. D 81, 065011 (2010).
L. P. Teo, M. Bordag, and V. Nikolaev, Phys. Rev. D 84, 125037 (2011).
L. P. Teo, Phys. Rev. D 88, 045019 (2013).
L. P. Teo, Phys. Rev. D 84, 025022 (2011).
L. P. Teo, Phys. Rev. D 84, 065027 (2011).
L. P. Teo, Phys. Rev. D 85, 045027 (2012).
C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev. D 84, 105031 (2011).
C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev. D 85, 125037 (2012).
C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev. D 86, 045021 (2012).
H. Gies, K. Langfeld, and L. Moyaerts, J. High Energy Phys. 06 (2003) 018.
H. Gies and K. Klingmüller, Phys. Rev. Lett. 96, 220401 (2006).
A. Weber and H. Gies, Phys. Rev. D 82, 125019 (2010).
A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev. Lett. 104, 040403 (2010).
A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, Phys. Rev. Lett. 102, 230404 (2009).
G. Bimonte, T. Emig, R. L. Jaffe, and M. Kadar, Europhys. Lett. 97, 50001 (2012).
G. Bimonte, T. Emig, and M. Kadar, Appl. Phys. Lett. 100, 074110 (2012).
G. Bimonte and T. Emig, Phys. Rev. Lett. 109, 160403 (2012).
J. Ambjø rn and S. Wolfram, Ann. Phys. (N.Y.) 147, 1 (1983).
K. Kirsten, J. Phys. A 24, 3281 (1991).
E. Elizalde and A. Romeo, Int. J. Mod. Phys. A 07, 7365 (1992).
K. Kirsten, J. Phys. A 26, 2421 (1993).
G. Cognola, K. Kirsten, and L. Vanzo, Phys. Rev. D 49, 1029 (1994).
E. Elizalde and K. Kirsten, J. Math. Phys. (N.Y.) 35, 1260 (1994).
S. C. Lim and L. P. Teo, J. Phys. A 40, 11645 (2007).
S. C. Lim and L. P. Teo, Eur. Phys. J. C 60, 323 (2009).
L. P. Teo, Phys. Lett. B 672, 190 (2009).
L. P. Teo, Nucl. Phys. B819, 431 (2009).
L. P. Teo, J. High Energy Phys. 06 (2009) 076.
L. P. Teo, J. High Energy Phys. 11 (2009) 095.
L. P. Teo, Phys. Lett. B 682, 259 (2009).
M. Rypestol and I. Brevik, New J. Phys. 12, 013022 (2010).
L. P. Teo, J. High Energy Phys. 10 (2010) 019.
M. Rypestol and I. Brevik, Phys. Scr. 82, 035101 (2010).
C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev. D 86, 045021 (2012).
L. P. Teo, J. Math. Phys. (N.Y.) 54, 103505 (2013).
S. Bellucci, E. R. Bezerra de Mello, and A. A. Saharian, Phys. Rev. D 89, 085002 (2014).
L. P. Teo, J. Math. Phys. (N.Y.) 55, 043508 (2014).
L. P. Teo, J. High Energy Phys. 05 (2014) 016.
A. A. Saharian, arXiv:hep-th/0002239.
A. A. Saharian, arXiv:0708.1187.
L. P. Teo, J. High Energy Phys. 11 (2009) 095.
A. Erdélyi et al., Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, San Diego, 2000).
J. Ambjø rn and S. Wolfram, Ann. Phys. (N.Y.) 147, 1 (1983).