The Casimir effect

The purpose of these lecture notes is twofold. First we aim at introducing the reader to the basic concepts pertaining to Casimir forces, starting from the seminal work of Casimir. In a broader sense, we also review some aspects of dispersion forces, in particular the status of van der Waals forces in vacuum as well as in a finite density and non zero temperature medium. The Lifshitz theory of forces between dielectric bodies is briefly described. In the second place, the course deals with a more recent analysis of the Casimir force for metals based on an exact microscopic statistical mechanical treatment of matter and field fluctuations. It reveals that charges fluctuations inside the conductors cannot be ignored (as is done in the conventional Casimir calculation). It also helps clarifying present day controversies about the contribution of thermal fluctuations to the force and the proper way to recover the metallic case in the framework of the Lifshitz theory. Finally the occurrence of Casimir forces in critical phenomena is illustrated in the case of the Bose–Einstein condensation in a free Bose gas. The Casimir effect in other contexts (general quantum field theory, particle physics, cosmology, …) is not considered here.