Casimir Effect : The Classical Limit

We analyze the high temperature (or classical) limit of the Casimir effect (for various massless fields). A simple physical argument suggests that the Casimir energy (as opposed to the Casimir free energy) should vanish in the classical limit. We check the validity of this argument for massless scalar field confined in a cavity with boundaries of arbitrary shape, using path integral formalism. We are able to verify this suggestion only when the boundaries consist of disjoint pieces. In fact, in such cases, that part of the Casimir energy which depends on the separation of the disjoint boundary pieces, decays exponentially with temperature and separation. Moreover, we find in these cases that the contribution to the Casimir entropy by field modes that depend on that separation, tends, in the classical limit, to a finite asymptotic value which depends only on the geometry of the cavity. Thus the Casimir force between disjoint pieces of the boundary in the classical limit is entropy driven and is governed by a dimensionless number characterizing the arbitrary geometry of the cavity. Contributions to the Casimir thermodynamical quantities due to each individual connected component of the boundary exhibit logarithmic deviations in temperature from the behavior just described. These logarithmic deviations seem to arise due to our difficulty to separate the Casimir energy (and the other thermodynamical quantities) from the “electromagnetic” self-energy of each of the connected components of the boundary in a well defined manner. Our approach to the Casimir effect is not to impose sharp boundary conditions on the fluctuating field, but rather take into consideration its interaction with the plasma of “charge carriers” in the boundary, with the plasma frequency playing the role of a physical UV cutoff. This also allows us to analyze deviations from a perfect conductor behavior. PACS numbers: 42.50.Lc, 11.10.-z, 11.10.Wx e-mail: joshua, ady, [email protected] ∗permanent address