Spherical Casimir Effect for a Massive Scalar Field on the Three Dimensional Ball
Author(s)
Andrea Erdas
ABSTRACT
The zeta function regularization technique is used to study the Casimir effect for a scalar field of mass m satisfying Dirichlet boundary conditions on a spherical surface of radius a. In the case of large scalar mass,
, simple analytic expressions are obtained for the zeta function and Casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined outside the spherical surface. In both cases the Casimir energy is exact up to order 
and contains the expected divergencies, which can be eliminated using the well established renormalization procedure for the spherical Casimir effect. The case of a scalar field present in both the interior and exterior region is also examined and, for 
, the zeta function, the Casimir energy, and the Casimir force are obtained. The obtained Casimir energy and force are exact up to order 
and 
respectively. In this scenario both energy and force are finite and do not need to be renormalized, and the force is found to produce an outward pressure on the spherical surface.
The zeta function regularization technique is used to study the Casimir effect for a scalar field of mass m satisfying Dirichlet boundary conditions on a spherical surface of radius a. In the case of large scalar mass,
, simple analytic expressions are obtained for the zeta function and Casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined outside the spherical surface. In both cases the Casimir energy is exact up to order and contains the expected divergencies, which can be eliminated using the well established renormalization procedure for the spherical Casimir effect. The case of a scalar field present in both the interior and exterior region is also examined and, for , the zeta function, the Casimir energy, and the Casimir force are obtained. The obtained Casimir energy and force are exact up to order and respectively. In this scenario both energy and force are finite and do not need to be renormalized, and the force is found to produce an outward pressure on the spherical surface.
Cite this paper
Erdas, A. (2015) Spherical Casimir Effect for a Massive Scalar Field on the Three Dimensional Ball. Journal of Modern Physics, 6, 1104-1112. doi: 10.4236/jmp.2015.68115.
Erdas, A. (2015) Spherical Casimir Effect for a Massive Scalar Field on the Three Dimensional Ball. Journal of Modern Physics, 6, 1104-1112. doi: 10.4236/jmp.2015.68115.
References
[1] Casimir, H.B.G. (1948) Kon. Ned. Akad. Wetensch. Proc., 51, 793. [2] Boyer, T.H. (1968) Physical Review, 174, 1764. http://dx.doi.org/10.1103/PhysRev.174.1764 [3] Sparnaay, M.J. (1958) Physica, 24, 751-764. http://dx.doi.org/10.1016/S0031-8914(58)80090-7 [4] Bordag, M., Mohideen, U. and Mostepanenko, V.M. (2001) Physics Reports, 353, 1-205.
http://dx.doi.org/10.1016/S0370-1573(01)00015-1 [5] Bordag, M., Klimchitskaya, G.L., Mohideen, U. and Mostepanenko, V.M. (2009) Advances in the Casimir Effect. International Series of Monographs on Physics Vol. 145, Oxford University, New York, 1.
http://dx.doi.org/10.1093/acprof:oso/9780199238743.003.0001 [6] Johnson, K. (1975) Acta Phys. Pol. B, 6, 865. [7] Ambjorn, J. and Wolfram, S. (1983) Annals of Physics, 147, 1-32.
http://dx.doi.org/10.1016/0003-4916(83)90065-9 [8] Oikonomou, V.K. and Tracas, N.D. (2010) International Journal of Modern Physics A, 25, 5935.
http://dx.doi.org/10.1142/S0217751X10051189 [9] Dufour, G., Cassidy, D.B., Crivelli, P., Debu, P., Lambrecht, A., Nesvizhevsky, V.V., Reynaud, S., Voronin, A.Y. and Wall, T.E. (2015) Advances in High Energy Physics, 2015, Article ID: 379642.
http://dx.doi.org/10.1155/2015/379642 [10] Fulling, S.A., Kaplan, L., Kirsten, K., Liu, Z.H. and Milton, K.A. (2009) Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 155402. http://dx.doi.org/10.1088/1751-8113/42/15/155402 [11] Elizalde, E., Odintsov, S.D. and Saharian, A.A. (2009) Physical Review D, 79, Article ID: 065023.
http://dx.doi.org/10.1103/PhysRevD.79.065023 [12] Oikonomou, V.K. (2009) Modern Physics Letters A, 24, 2405-2423.
http://dx.doi.org/10.1142/S0217732309031703 [13] Bender, C.M. and Milton, K.A. (1994) Physical Review D, 50, 6547-6555.
http://dx.doi.org/10.1103/PhysRevD.50.6547 [14] Cognola, G., Elizalde, E. and Kirsten, K. (2001) Journal of Physics A, 34, 7311-7327.
http://dx.doi.org/10.1088/0305-4470/34/36/311 [15] Bordag, M., Elizalde, E., Kirsten, K. and Leseduarte, S. (1997) Physical Review D, 56, 4896-4904.
http://dx.doi.org/10.1103/PhysRevD.56.4896 [16] Kirsten, K. (2000) http://arxiv.org/abs/hep-th/0005133. [17] Saharian, A.A. (2001) Physical Review D, 63, Article ID: 125007.
http://dx.doi.org/10.1103/PhysRevD.63.125007 [18] de Mello, E.R.B., Bezerra, V.B. and Khusnutdinov, N.R. (2001) Journal of Mathematical Physics, 42, 562.
http://dx.doi.org/10.1063/1.1333699 [19] Balian, R. and Duplantier, B. (1978) Annals of Physics, 112, 165-208.
http://dx.doi.org/10.1016/0003-4916(78)90083-0 [20] Teo, L.P. (2013) Journal of Mathematical Physics, 54, Article ID: 103505.
http://dx.doi.org/10.1063/1.4824466 [21] Bordag, M. and Kirsten, K. (1995) http://arxiv.org/abs/hep-th/9501064. [22] Bordag, M., Elizalde, E. and Kirsten, K. (1996) Journal of Mathematical Physics, 37, 895.
http://dx.doi.org/10.1063/1.531418 [23] Bordag, M. and Kirsten, K. (1996) Physical Review D, 53, 5753-5760.
http://dx.doi.org/10.1103/PhysRevD.53.5753
[1] Casimir, H.B.G. (1948) Kon. Ned. Akad. Wetensch. Proc., 51, 793. [2] Boyer, T.H. (1968) Physical Review, 174, 1764. http://dx.doi.org/10.1103/PhysRev.174.1764 [3] Sparnaay, M.J. (1958) Physica, 24, 751-764. http://dx.doi.org/10.1016/S0031-8914(58)80090-7 [4] Bordag, M., Mohideen, U. and Mostepanenko, V.M. (2001) Physics Reports, 353, 1-205.
http://dx.doi.org/10.1016/S0370-1573(01)00015-1 [5] Bordag, M., Klimchitskaya, G.L., Mohideen, U. and Mostepanenko, V.M. (2009) Advances in the Casimir Effect. International Series of Monographs on Physics Vol. 145, Oxford University, New York, 1.
http://dx.doi.org/10.1093/acprof:oso/9780199238743.003.0001 [6] Johnson, K. (1975) Acta Phys. Pol. B, 6, 865. [7] Ambjorn, J. and Wolfram, S. (1983) Annals of Physics, 147, 1-32.
http://dx.doi.org/10.1016/0003-4916(83)90065-9 [8] Oikonomou, V.K. and Tracas, N.D. (2010) International Journal of Modern Physics A, 25, 5935.
http://dx.doi.org/10.1142/S0217751X10051189 [9] Dufour, G., Cassidy, D.B., Crivelli, P., Debu, P., Lambrecht, A., Nesvizhevsky, V.V., Reynaud, S., Voronin, A.Y. and Wall, T.E. (2015) Advances in High Energy Physics, 2015, Article ID: 379642.
http://dx.doi.org/10.1155/2015/379642 [10] Fulling, S.A., Kaplan, L., Kirsten, K., Liu, Z.H. and Milton, K.A. (2009) Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 155402. http://dx.doi.org/10.1088/1751-8113/42/15/155402 [11] Elizalde, E., Odintsov, S.D. and Saharian, A.A. (2009) Physical Review D, 79, Article ID: 065023.
http://dx.doi.org/10.1103/PhysRevD.79.065023 [12] Oikonomou, V.K. (2009) Modern Physics Letters A, 24, 2405-2423.
http://dx.doi.org/10.1142/S0217732309031703 [13] Bender, C.M. and Milton, K.A. (1994) Physical Review D, 50, 6547-6555.
http://dx.doi.org/10.1103/PhysRevD.50.6547 [14] Cognola, G., Elizalde, E. and Kirsten, K. (2001) Journal of Physics A, 34, 7311-7327.
http://dx.doi.org/10.1088/0305-4470/34/36/311 [15] Bordag, M., Elizalde, E., Kirsten, K. and Leseduarte, S. (1997) Physical Review D, 56, 4896-4904.
http://dx.doi.org/10.1103/PhysRevD.56.4896 [16] Kirsten, K. (2000) http://arxiv.org/abs/hep-th/0005133. [17] Saharian, A.A. (2001) Physical Review D, 63, Article ID: 125007.
http://dx.doi.org/10.1103/PhysRevD.63.125007 [18] de Mello, E.R.B., Bezerra, V.B. and Khusnutdinov, N.R. (2001) Journal of Mathematical Physics, 42, 562.
http://dx.doi.org/10.1063/1.1333699 [19] Balian, R. and Duplantier, B. (1978) Annals of Physics, 112, 165-208.
http://dx.doi.org/10.1016/0003-4916(78)90083-0 [20] Teo, L.P. (2013) Journal of Mathematical Physics, 54, Article ID: 103505.
http://dx.doi.org/10.1063/1.4824466 [21] Bordag, M. and Kirsten, K. (1995) http://arxiv.org/abs/hep-th/9501064. [22] Bordag, M., Elizalde, E. and Kirsten, K. (1996) Journal of Mathematical Physics, 37, 895.
http://dx.doi.org/10.1063/1.531418 [23] Bordag, M. and Kirsten, K. (1996) Physical Review D, 53, 5753-5760.
http://dx.doi.org/10.1103/PhysRevD.53.5753