The Casimir effect giving rise to an attractive or repulsive force between the configuration boundaries that confine the massless scalar field is reexamined for a (DϪ1)-dimensional rectangular cavity with unequal finite p edges and different spacetime dimensions D in this paper. With periodic or Neumann boundary conditions, the energy is always negative. The case of Dirichlet boundary conditions is more complicated. The sign of the Casimir energy satisfying Dirichlet conditions on the surface of a hypercube ͑a cavity with equal finite p edges͒ depends on whether p is even or odd. In the general case ͑a cavity with unequal p edges͒, however, we show that the sign of the Casimir energy does not only depend on whether p is odd or even. Furthermore, we find that the Casimir force is always attractive if the edges are chosen appropriately. It is interesting that the Casimir force may be repulsive for odd p cavity with unequal edges, in contrast with the same problem in a hypercube case.

The Casimir effect giving rise to an attractive or repulsive force between the configuration boundaries that confine the massless scalar field is reexamined for one to three-dimensional pistons in this paper. Especially, we consider Casimir pistons with hybrid boundary conditions, where the boundary condition on the piston is Neumann and those on other surfaces are Dirichlet. We show that the Casimir force on the piston is always repulsive, in contrast with the same problem where the boundary conditions are Dirichlet on all surfaces.

The Casimir effect for a massless scalar field on the helix boundary condition which is named as quantum spring is studied in our recent paper. 27 Here, the Casimir effect of the quantum spring is investigated in (D +1)-dimensional spacetime for the massless and massive scalar fields by using the zeta function techniques. We obtain the exact results of the Casimir energy and Casimir force for any D, which indicate a Z 2 symmetry of the two space dimensions. The Casimir energy and Casimir force have different expressions for odd and even dimensional space in the massless case but in both cases the force is attractive. In the case of odd-dimensional space, the Casimir energy density can be expressed by the Bernoulli numbers, while in the even case it can be expressed by the ζfunction. And we also show that the Casimir force has a maximum value which depends on the spacetime dimensions. In particular, for a massive scalar field, we found that the Casimir force varies as the mass of the field changes.

The Casimir force on two-dimensional pistons for massive scalar fields with both Dirichlet and hybrid boundary conditions is computed. The physical result is obtained by making use of generalized ζ-function regularization technique. The influence of the mass and the position of the piston in the force is studied graphically. The Casimir force for massive scalar field is compared to that for massless scalar field.Keywords: Casimir force; piston; generalized ζ-function. About 60 years ago Casimir gave the prediction that an attractive force should act between two plate-parallel uncharged perfectly conducting plates in vacuum [1]. Especially in recent 10 years, the effect has been paid more attention because of the development of precise measurements [2]. At the same time, Casimir energies and forces have been calculated theoretically in various different configurations. Different properties of the Casimir force (attractive or repulsive) can be obtained for different boundary conditions and different geometries [3,4]. For example, it has been claimed that the Casimir energy inside rectangular cavities can be either positive or negative depending on the ratio of the sides [5,6,7,8,9]. But the conclusion is worth suspecting because the calculations ignore the divergent term associated with the boundaries and the nontrivial contribution from the outside region of the box [10,11,12]. Recently, a modification of the rectangle−"Casimir piston" was introduced to avoid the above problems [13]. The Casimir force on the piston is a well-defined force because the position of the piston is independent of the divergent terms in the interval vacuum energy and external region. Successively, the study of this geometry attracted a lot of interests. The Casimir force on the piston was studied for different dimensions, different fields and different boundary conditions [14,15,16,17,18,19,20]. The results indicate that the Casimir force on the piston can be attractive or repulsive for different cases. The repulsive Casimir force has special importance in that it can be applied to microelectromechamical systems (MEMS) [21,22]. We discussed Casimir pistons for a massless scalar field with hybrid boundary conditions and obtained the repulsive Casimir force on the piston [23].On the other hand, the Casimir effect for the massive scalar field also studied by some authors [24,25,26]. As is known that the Casimir effect disappears as the mass of the field goes to infinity since there are no more quantum fluctuations in the limit. But the precise way the Casimir energy varies as the mass changes is worth studying [27]. In this paper,we consider the Casimir force on the piston for the massive scalar field with two types of boundary conditions, that is, Dirichlet and hybrid. We obtain our physical results using ζ−function regularization technique. We discuss the influence of the mass and the ratio of the sides graphically. The results tell us the expectable properties of the force on the piston as is for massless scalar field and also tell us t...

The Casimir effect giving rise to an attractive force between the configuration boundaries that confine the massless scalar field is rigorously proven for odd dimensional hypercube with the Dirichlet boundary conditions and different spacetime dimensions D by the Epstein zeta function regularization. PACS number(s): 04.62.+v, 03.65.Ge