Closed path approach to Casimir effect in rectangular cavities and pistons.

摘要

We studied thoroughly Casimir energy and Casimir force in a rectangular cavity and piston with various boundary conditions, for both scalar field and electromagnetic (EM) field. Using the cylinder kernel approach, we found the Casimir energy exactly and analyzed the Casimir energy and Casimir force from the point of view of closed classical paths (or optical paths). For the scalar field, we studied the rectangular cavity and rectangular piston with all Dirichlet conditions and all Neumann boundary conditions and then generalized to more general cases with any combination of Dirichlet and Neumann boundary conditions. For the EM field, we first represented the EM field by 2 scalar fields (Hertz potentials), then related the EM problem to corresponding scalar problems. We studied the case with all conducting boundary conditions and then replaced some conducting boundary conditions by permeable boundary conditions. By classifying the closed classical paths into 4 kinds: Periodic, Side, Edge and Corner paths, we can see the role played by each kind of path. A general treatment of any combination of boundary conditions is provided. Comparing the differences between different kinds of boundary conditions and exploring the relation between corresponding EM and scalar problems, we can understand the effect of each kind of boundary condition and contribution of each kind of classical path more clearly.