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This chapter demonstrates that the Casimir force inside a rectangular box can be both attractive and repulsive. A detailed investigation of the Casimir energy and force for fields of various spins, when it may be positive or negative, as a function of the box dimensions and the type of boundary conditions is performed. In particular, the analytical results for two- and three-dimensional boxes are obtained by repeated application of the Abel–Plana formula and using the Epstein zeta function. The problem of isolation of the divergent terms in the vacuum energy and their interpretation is discussed in connection with the problem of a rectangular box divided into two sections by a movable partition (piston). Both the old classical results and recent results related to boxes with a piston at zero and nonzero temperatures are presented. As shown in the chapter, the two sets of results are in mutual agreement.

This chapter discusses several basic ideas and methods related to the calculation of the Casimir energies and forces using some simple models. The simplicity of these models means that cumbersome mathematical calculations can be avoided and they demonstrate the basic problems that will be repeatedly considered in the following chapters in a more sophisticated context. Important procedures such as regularization and renormalization of infinite quantities are illustrated, both physically and mathematically. Despite the elementary character of the chapter, the main physical situations where the Casimir effect arises (i.e., in regions with boundaries and in spaces with nontrivial topology) are discussed. Local and global approaches to the Casimir effect, and well-known formulas for the electromagnetic Casimir pressure and energy per unit area between two parallel ideal-metal planes are derived.

This chapter considers the simple but most important configuration of two parallel ideal-metal planes. First, the theory of the scalar and electromagnetic Casimir effects between parallel planes is presented. In comparison with Chapter 2, some basic facts are added concerning the relation between local and global approaches and the polarizations of the electromagnetic field. The radiative corrections to the Casimir force are considered. General analytical formulas for the Casimir free energy, entropy, and pressure at nonzero temperature are presented, as well as the limits of low and high temperature. The agreement between the results obtained and thermodynamics is analyzed. The spinor Casimir effect between planes and the Casimir effect for a wedge are also discussed. At the end of the chapter, the dynamic Casimir effect connected with uniformly moving or oscillating planes is briefly considered.