Abstract
The finite-temperature Casimir effect for a scalar field in the bulk region of two Randall–Sundrum models, RSI and RSII, is studied. We calculate the Casimir energy and the Casimir force for two parallel plates with separation a on the visible brane in the RSI model. High-temperature and low-temperature cases are covered. Attraction versus repulsion of the temperature correction to the force is discussed in the special cases of Dirichlet–Dirichlet, Neumann–Neumann and Dirichlet–Neumann boundary conditions at low temperature. The Abel–Plana summation formula is used, as this is found to be the most convenient. Some comments are made on the related contemporary literature.
Highlights
The finite-temperature Casimir effect for a scalar field in the bulk region of two Randall–Sundrum models, RSI and RSII, is studied
There are two papers from the group of Morales-Técotl et al [12], which focus on RSI-q/RSII-q
We have e.g. Poppenhaeger et al [27] and Pascoal et al [26], who find the electromagnetic Casimir force by multiplying by a factor p to account for the possible polarizations of the photon and subtract the mode polarized in the direction of the brane
Summary
To find the partition function for a non-minimally coupled scalar field with mass m in the RSI model, we follow a Kaluza–Klein reduction approach [28, 33], starting from the Lagrangian density. If the field is minimally coupled (ζ = 0) and there is no mass boundary term (cbrane = 0), the boundary condition reduces to the Neumann BC, ψN (y)|brane = 0. Odd scalar fields obey the Dirichlet BC on the branes. For an even field with m2 − 20ζ k2 = 0 with no boundary mass term, the situation is different, as ψ0 =const is a solution of equation (21) and satisfies the boundary condition which in that case is the Neumann. The MN = 0 case has important consequences for the Casimir force from a bulk scalar field This is related to the localization problem for the Kaluza–Klein modes in general. The reader may consult [12, 36] for discussion of what weight is to be given to the massless modes in RSI due to the fact that it is localized near the hidden brane only
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