Closed Path Approach to Casimir Effect in Rectangular Cavities and Pistons

We study 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 find the Casimir energy exactly and analyze the Casimir energy and Casimir force from the point of view of closed classical paths (or optical paths). For the scalar field, we study the rectangular cavity and rectangular piston with all Dirichlet conditions and all Neumann boundary conditions and then generalize to more general cases with any combination of Dirichlet and Neumann boundary conditions. For the EM field, we first represent the EM field by 2 scalar fields (Hertz potentials), then relate the EM problem to corresponding scalar problems. We study the case with all conducting boundary conditions and then replace 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