Casimir force on a piston

We consider a massless scalar field obeying Dirichlet boundary conditions on the walls of a two-dimensional L×b rectangular box, divided by a movable partition (piston) into two boxes of dimensions a × b and (L − a) × b. We compute the Casimir force on the piston in the limit L → ∞. Regardless the value of a/b, the piston is attracted to the nearest end of the box. Asymptotic expressions for the Casimir force on the piston are derived for a ≪ b and a ≫ b. PACS numbers: 03.70.+k, 11.10.-z, 11.10.Kk Typeset using REVTEX Eletronic address: [email protected] 1 In 1948 Casimir predicted a remarkable macroscopic quantum effect: two conducting and neutral parallel plates should attract each other due to the disturbance of the vacuum of the electromagnetic field caused by their presence [1] (for a general review on the Casimir effect, see Ref. [2]). Inspired by that result, a few years later Casimir suggested that the zero-point pressure of the electromagnetic field might yield the stresses postulated by Poincaré in order to explain the stability of the electron [3]. Boyer, however, showed that the Casimir force for a conducting spherical shell is repulsive [4], thus invalidating Casimir’s model for the electron. Boyer’s result brought attention to the fact that the attractive or repulsive character of the Casimir force depends on the geometry of the configuration. This has been investigated in detail for fields (scalar or electromagnetic) confined in a d-dimensional rectangular box [5–11]. Let us consider, for instance, a massless scalar field subject to Dirichlet boundary conditions at the walls of the two-dimensional box 0 ≤ x ≤ a, 0 ≤ y ≤ b. The nonrenormalized vacuum energy is given by (h̄ = c = 1) E0(a, b) = 1 2 ∞ ∑