Abstract
The familiar trace identity associated with the scale transformationx Μ → x′ Μ = e-λ x Μ on the Lagrangian density for a noninteracting massive real scalar field in 2 + 1 dimensions is shown to be violated on a single plate on which the Dirichlet boundary condition Φ(t, x1, x2 = -a) = 0 is imposed. It is however respected in: (i) 1 + 1 dimensions in both free space and on a single plate on which the Dirichlet boundary condition Φ(t, x1 = -a) = 0 holds and (ii) in 2 + 1 dimensions in free space, i.e. the unconstrained configuration. On the plate where Φ(t, x1, x2 = -a) = 0, the modified trace identity is shown to be anomalous with a numerical coefficient for the anomalous term equal to the canonical scale dimension, viz. 1/2. The technique of Bordaget al [Ann. Phys. (N.Y.),165, 162 (1985)] is used to incorporate the said boundary condition into the generating functional for the connected Green’s functions.
Similar content being viewed by others
References
H B G Casimir,Proc. Kon. Ned. Akad. Wetensch. 51, 793 (1948)
For additional references, see K A Milton,The Casimir effect: Physical manifestations of zero-point energy (World Scientific, New Jersey, 2001)
G Plunien, B Muller and W Greiner,Phys. Rep. 134, 87 (1986)
S Coleman and R Jackiw,Ann. Phys. (N. Y.) 67, 552 (1971)
M J Spaarnay,Physica 24, 751 (1958)
For more references see for example M Bordag, U Mohideen and V M Mostepanenko,Phys. Rep. 353, 1 (2001)
M Bordag, D Robaschik and E Wieczorek,Ann. Phys. (N.Y.) 165, 192 (1985)
See for example P Ramond,Field theory: A modern primer (Benjamin, Mass., 1981)
M Bordag and J Lindig,Phys. Rev. D58, 045003 (1998)
S K Lamoreaux,Phys. Rev. Lett. 78, 5 (1997)
F Michael Serry, Dirk Walliser and G Jordan Maclay,J. Appl. Phys. 84, 2501 (1998)
R M Cavalcanti,Phys. Rev. D69, 065015 (2004)
M P Hertzberg, R L Jaffe, M Kardar and A Scardicchio, arXiv:quant-ph/0509071
G J Maclay,Phys. Rev. A61, 052110 (2000)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kamath, S.G. The trace identity and the planar Casimir effect. Pramana - J Phys 66, 325–344 (2006). https://doi.org/10.1007/BF02704388
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02704388