Radiative Correction to The Casimir Energy For Scalar Field with Mixed Boundary Condition in 3 + 1 Dimensions

Document Type : Full length research Paper

Author

Department of Physics, Semnan Branch, Islamic Azad University, Semnan, Iran

Abstract

In the present study, the zero and first-order radiative correction to the Casimir energy for massive and massless scalar fields confined with mixed boundary conditions (Dirichlet-Neumann) between two parallel plates in ϕ^4 theory were computed. Two issues in performing the calculations in this work are essential: to renormalize the bare parameters of the problem, a systematic method were used, which allowing all influences from the boundary conditions to be imported in all elements of the renormalization program. This idea yields our counterterms appeared in the renormalization program to be position-dependent. Using the Box Subtraction Scheme as a regularization technique is the other noteworthy point in the calculation. In this scheme, by subtracting the vacuum energies of two similar configurations from each other, regularizing divergent expressions and their removal process were significantly facilitated. All the obtained answers for the Casimir energy with the mixed boundary condition were consistent with well-known physical grounds. We also compared the Casimir energy for massive scalar field confined with four types of boundary conditions (Dirichlet, Neumann, mixed of them and Periodic) in 3+1 dimensions with each other, and the sign and magnitude of their values were discussed.

Keywords


 
[1] H.B.G. Casimir, On the attraction between two perfectly conducting plates, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 51 (1948) 793-795.
[2]         M.J. Sparnaay, Measurements of attractive forces between flat plates, Physica 24 (1958) 751-764.
 Doi: 10.1016/S0031-8914(58)80090-7
 
[3]         M. Bordag, G.L. Klimchitskaya, U. Mohideen, Advances in the Casimir effect, 1st Ed. ed., New York: Oxford Univ. Press Inc., (2009).
 
[4]         K.A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy, World Scientific Publishing Co. Pte. Ltd., (2001).
 
[5]         M. Bordag, U. Mohideen, V.M. Mostepanenko, New Developments in the Casimir Effect, Physics Reports 353 (2001) 1-2.
Doi:10.1016/S0370-1573(01)00015-1
 
[6]         M. Bordag, J. Lindig, Radiative correction to the Casimir force on a sphere Physical Review D 58 (1998) 045003. Doi:10.1103/PhysRevD.58.045003
 
[7] N. Graham, R. Jaffe, H. Weigel, Casimir effects in renormalizable quantum field theories, International Journal of Modern Physics A 17 (2002) 846-869.
Doi: 10.1142/S0217751X02010224
 
[8] N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, M. Scandurra, H. Weigel, Calculating vacuum energies in renormalizable quantum field theories: A new approach to the Casimir problem, Nuclear Physics  B 645 (2002) 49-84.
Doi: 10.1016/S0550-3213(02)00823-4
 
[9] X. Kong, F. Ravndal, Radiative corrections to the Casimir energy, Physical Review Letters 79 (1997) 545-548.
Doi: 10.1103/PhysRevLett.79.545
 
[10] K. Melnikov, Radiative corrections to the Casimir force and effective field theories, Physical Review. D 64 (2001) 045002.
Doi: 10.1103/PhysRevD.64.045002.
 
[11] F.A. Barone, R.M. Cavalcanti, C. Farina, Radiative corrections to the Casimir effect for the massive scalar field, Nuclear Physics B (Proc. Suppl.) 127 (2004) 118-122; Doi: 10.1016/S0920-5632(03)02411-3
 
R.M. Cavalcanti, C. Farina, F.A. Barone, Radiative corrections to Casimir effect in the model, arXiv:hep-th/0604200 (2006);
F.A. Barone, R.M. Cavalcanti, C. Farina, Radiative corrections to the Casimir effect for the massive scalar field, arXiv:hepth/0301238v1 (2003).
 
[12] R. Moazzemi, A. Mohammadi, S.S. Gousheh, A renormalized perturbation theory for problems with non-trivial boundary conditions or backgrounds in two space–time dimensions, European Physical Journal C 56 (2008) 585-590.
 Doi: 10.1140/epjc/s10052-008-0680-9 
 
[13] R. Moazzemi, M. Namdar, S.S. Gousheh, The Dirichlet Casimir effect for  theory in (3 + 1) dimensions: a new renormalization approach, JHEP 09 (2007) 029.
 Doi: 10.1088/1126-6708/2007/09/029
 
[14] R. Moazzemi, S.S. Gousheh, A new renormalization approach to the Dirichlet Casimir effect for  theory in 1+1 dimensions, Physics Letters B 658 (2008) 255-265.
Doi: 10.1016/j.physletb.2007.08.098
 
[15] S.S. Gousheh, R. Moazzemi, M.A. Valuyan, Radiative correction to the Dirichlet Casimir energy for theory in two spatial dimensions, Physics Letters B 681 (2009) 477-483.
Doi: 10.1016/j.physletb.2009.10.058
 
[16] M.A. Valuyan, Casimir Energy Calculation for Massive Scalar Field on Spherical Surface: An Alternative Approach, Canadian Journal of Physics 96 (2018) 1004-1009.Doi: 10.1139/cjp-2017-0722
 
[17] M.A. Valuyan, Radiative correction to the Casimir energy for massive scalar field on a spherical surface, Modern Physics Letters A 32 (2017) 1750128.
Doi: 10.1142/S0217732317501280
 
[18] V.V. Nesterenko I.G. Pirozhenko, Spectral Zeta Functions for a Cylinder and a Circle, Journal of Mathematical Physics 41 (2000) 4521-4531.
Doi:10.1063/1.533358
 
M.A. Valuyan, The Casimir energy for scalar field with mixed boundary condition, International Journal of Geometric Methods in Modern Physics 15 (2018) 1850172.
Doi: 10.1142/S0219887818501724
 
[19] R. Balian, B. Duplantier, Electromagnetic waves near perfect conductors, Annals of Physics (N.Y.) 112 (1978) 165-208.
Doi: 10.1016/0003-4916(77)90334-7
 
[20] K.A. Milton, L.L. Deraad, J. Schwinger, Casimir self-stress on a perfectly conducting spherical shell, Annals of Physics (N.Y.) 115 (1978) 388.
Doi: 10.1016/0003-4916(78)90161-6
 
[21] M.A. Valuyan, The Dirichlet Casimir energy for  theory in a rectangular waveguide, Journal of Physics G: Nuclear and Particle Physics 45 (2018) 095006.
Doi: 10.1088/1361-6471/aad625
M.A. Valuyan, The Dirichlet Casimir energy for the  theory in a rectangle, Eur. Phys. J Plus. 133 (2018) 401. Doi: 10.1140/epjp/i2018-12206-8
 
[22] T.H. Boyer, Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle, Physical Review 174 (1968) 1764.
Doi: 10.1103/PhysRev.174.1764
 
[23] A.A. Saharian, The Generalized Abel-Plana Formula: Applications To Bessel Functions And Casimir Effect, IC/2007/082 (2000) [hep-th/0002239 v1].