Appendix A - Stress of the quantum vacuum

Summary

Casimir's 1948 calculation (Casimir, 1948) of the vacuum force between two dielectric bodies is restricted to an idealized situation: two infinitely conducting parallel plates of infinite extension. Real dielectrics are neither infinitely conducting nor infinitely extended though. Moreover, in most experimental tests of the Casimir effect (Lamoreaux, 1997, 1999; Bordag et al., 2001; Lamoreaux, 2005; Munday et al., 2009) the force between a plate and a sphere was measured, because it is very difficult in practice to keep microscopic plates exactly parallel, a problem avoided by using a sphere above a plate instead. The most comprehensive general theory of the Casimir effect is known as Lifshitz theory(Landau and Lifshitz, Vol. IX, 1980). This theory was pioneered by Evgeny M. Lifshitz in 1955 (Lifshitz, 1955) and further developed by Igor E. Dzyaloshinskii, Lifshitz and Lev P. Pitaevskii in 1961 (Dzyaloshinskii et al., 1961). Here we explain the central concepts of Lifshitz theory. We test the theory on Casimir's case and mention some of the results beyond it. Lifshitz theory avoids the artefacts of Casimir's simple calculation (Casimir, 1948) and it is significantly more flexible and general, but it is also technically complicated and sometimes not very intuitive. Although we try to elucidate Lifshitz theory as much as possible, it still remains a rather heavy theoretical machinery à la russe. It seems remarkable that such a complicated theory may give such simple results as Casimir's formula (2.50) and the generalizations we are going to discuss.