The Casimir Effect for Parallel Plates in the Spacetime with a Fractal Extra Compactified Dimension
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The Casimir Effect for Parallel Plates in the Spacetime with a Fractal Extra Compactified Dimension Hongbo Cheng
Received: 12 September 2012 / Accepted: 10 April 2013 © Springer Science+Business Media New York 2013
Abstract The Casimir effect for massless scalar fields satisfying Dirichlet boundary conditions on the parallel plates in the presence of one fractal extra compactified dimension is analyzed. We obtain the Casimir energy density by means of the regularization of multiple zeta function with one arbitrary exponent. We find a limit on the scale dimension like δ > 12 to keep the negative sign of the renormalized Casimir energy which is the difference between the regularized energy for two parallel plates and the one with no plates. We derive and calculate the Casimir force relating to the influence from the fractal additional compactified dimension between the parallel plates. The larger scale dimension leads to the greater revision on the original Casimir force. The two kinds of curves of Casimir force in the case of integer-numbered extra compactified dimension or fractal one are not superposition, which means that the Casimir force show whether the dimensionality of additional compactified space is integer or fraction. Keywords Casimir effect · Kaluza–Klein model · Fractal dimension During the investigation of quantum gravity, more attentions have been paid to the fractal universe [1]. It may be better to describe the spacetime whose scale is on the Planck order in virtue of fractal geometry with some non-integer dimensions. Within this kind of background some topics were considered. It is demonstrated that the spectral dimension of the spacetime where Quantum Einstein Gravity lives in is equal to 2 microscopically while to 4 on macroscopic scales [2]. It is also shown that the world with a quantum group symmetry has a scale-dependent fractal dimension at short scales to describe a phenomenon appeared in the quantum gravity [3]. A kind of field theory which is Lorentz invariant, power-counting renormalizable, ultraviolet finite and causal is proposed to search for a consistent theory of quantum gravity [4, 5]. H. Cheng () Department of Physics, East China University of Science and Technology, Shanghai 200237, China e-mail: [email protected] H. Cheng Shanghai Key Laboratory of Astrophysics, Shanghai 200234, China
Int J Theor Phys
More than 80 years ago Kaluza and Klein put forward the model that our universe has more than four dimensions [6, 7]. The extra spatial dimensions belonging to the modern Kaluza–Klein theory are chosen to be compact and small to unify all interactions in nature. Their characteristic size is of the order of the Planck length. In addition the quantum gravity such as string theory or brane-world scenario is developed to reconcile the quantum mechanics and gravity with the help of introducing the additional spatial dimensions. It seems to be reasonable to describe the extra space by means of fractal geometry because of the quantum fluctuations on the Planck scale space. The K
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