The Casimir effect for a cavity in the spacetime with an extra dimension

We reexamine the Casimir effect for the rectangular cavity with two or three equal edges in the presence of compactified universal extra dimension. We derive the expressions for the Casimir energy and discuss the nature of Casimir force. We show analytically the extra-dimension corrections to the standard Casimir effect to put forward a new method of exploring the existence of extra dimensions of the Universe. PACS number(s): 11.10.Kk, 04.62.+v ∗E-mail address: [email protected] 1 Kaluza and Klein presented the thought that our spacetime has more than three spatial dimensions over decades ago in order to unify gravity and classical electrodynamics [1, 2]. Now the idea of extra dimensions attracts more and more interests of the physical community. It is fundamental for string theory to invoke seven additional spatial dimensions to unify the quantum mechanics and gravity. Whether the order of the compactification scale of the extra dimensions is much larger or smaller remains unexplained. Some models of string theory predicted that the typical size of universal extra dimensions (UXDs) is about 10−35m which is beyond our recent experimental reach [3, 4]. Recently the Kaluza-Klein idea is considered in the context of the brane paradigm, which could lead to a solution to the hierarchy problem [5-7]. The brane world scenario [8, 9] stipulates that the non-gravitational physics like Standard Model is confined to a four-dimensional Universe called brane that is embedded in a (4 + n)-dimensional spacetime with n compactified extra dimensions. Only gravitational field is allowed to propagate through the whole spacetime while the possibility that the additional dimensions may be much larger than we thought in the past was showed. The Casimir effect is essentially a direct consequence of quantum field theory because of a change in the spectrum of vacuum oscillations when the quantization volume is bounded, or some background field is presented. The effect becomes one of the most interesting macroscopic manifestations of the nontrivial properties of the physical vacuum. The Casimir effect has been a subject of extensive research [10-16]. Nowadays the Casimir effect is used as a powerful method for the research on new physics beyond the standard model. We apply the tool to a lot of topics such as the right order of the cosmological constant, extra dimensions, etc.. It is necessary to explore the dimensionality and structure of our Universe in various directions. If our world has six or seven additional spatial dimensions, the new phenomena related to the existence of these dimensions should be observed. A lot of progresses of studying the UXDs by means of Casimir effect were made [17, 18]. The authors of [18] compared the expression of Casimir force for parallel plates in the presence of one extra dimension with the experimental results and the approximate values of the compactification scale of extra dimension were obtained. In this paper, we investigate the extra-dimension correction to the Casimir effect for a cavity in detail to put forward a new method to explore the existence of extra dimensions of our Universe. There will be influence from UXD on the nature of Casimir force between the configuration boundaries that confine the field in the spacetime with extra dimensions. It was shown that the sign of Casimir energy for a rectangular cavity with two or three equal edges in four-dimensional Minkowski spacetime is positive or negative respectively for the case of massless scalar field [14, 19]. Here we also discuss the Casimir effect of this kind of field satisfying Dirichlet boundary conditions within a rectangular with two or three equal edges in the spacetime with one extra dimension for the sake of comparing results of [14, 19] and recent ours. Having derived the total energy, we regularize the energy by means of Epstein zeta function to obtain the Casimir energy. We find that the Casimir force is always repulsive if the lengths of edges are sufficiently large with respect to the size of UXD.