The surface impedance approach is discussed in connection with the precise calculation of the Casimir force between metallic plates. It allows to take into account the nonlocal connection between the current density and electric field inside of metals. In general, a material has to be described by two impedances $Z_{s}(\omega,q)$ and $Z_{p}(\omega,q)$ corresponding to two different polarization states. In contrast with the approximate Leontovich impedance they depend not only on frequency $\omega$ but also on the wave vector along the plate $q$. In this paper only the nonlocal effects happening at frequencies $\omega<\omega_{p}$ (plasma frequency) are analyzed. We refer to all of them as the anomalous skin effect. The impedances are calculated for the propagating and evanescent fields in the Boltzmann approximation. It is found that $Z_p$ significantly deviates from the local impedance as a result of the Thomas-Fermi screening. The nonlocal correction to the Casimir force is calculated at zero temperature. This correction is small but observable at small separations between bodies. The same theory can be used to find more significant nonlocal contribution at $\omega\sim\omega_p$ due to the plasmon excitation.Comment: 29 pages. To appear in Phys. Rev.

The problem with the temperature dependence of the Casimir force is investigated. Specifically, the entropy behavior in the low temperature limit, which caused debates in the literature, is analyzed. It is stressed that the behavior of the relaxation frequency in the T-->0 limit does not play a physical role since the anomalous skin effect dominates in this range. In contrast with the previous works, where the approximate Leontovich impedance was used for analysis of nonlocal effects, we give description of the problem in terms of exact nonlocal impedances. It is found that the Casimir entropy is going to zero at T-->0 only in the case when s polarization does not contribute to the classical part of the Casimir force. However, the entropy approaching zero from the negative side that, in our opinion, cannot be considered as thermodynamically satisfactory. The resolution of the negative entropy problem proposed in the literature is analyzed and it is shown that it cannot be considered as complete. The crisis with the thermal Casimir effect is stressed.

is the plasma frequency of the metal. In the case of Au, we employed the low frequency value ␤ = ͱ 1/3v F instead of the high frequency value ␤ = ͱ 3/5v F stated in the text. Equation ͑14͒ is wrong; the signs of the terms proportional to Q 2 in both numerator and denominator should be exchanged, so that it becomesFortunately, this was only a typographical error and the results were calculated with the correct equation. The corrections above do not alter the main conclusions of the paper, except for a reduction by a factor of 2 in the distances at which nonlocal corrections become important.

The problem with the temperature dependence of the Casimir force is investigated. We analyse high-temperature limit analytically making calculations at real frequencies. The purpose is to answer the question why there is no continuous transition between real and ideal metals and why the result does not depend on the relaxation frequency. It is found that the contribution of evanescent s polarized fields is finite even for an infinitely small relaxation frequency (plasma model) and exactly cancels the contribution of propagating fields. For the ideal metal the evanescent fields do not contribute at all. The lowtemperature limit is analysed to establish behaviour of the entropy at T → 0. It is stressed that the nonlocal effects are important in this limit because the mean free path for electrons becomes larger than the field penetration depth. In this limit v F /a plays the role of the relaxation frequency, where v F is the Fermi velocity and a is the distance between plates. It is indicated that the Leontovich approximate impedance cannot be used for calculations because it is good for the description of propagating but not evanescent fields. It is found that due to nonlocality the Casimir entropy approaches zero at T → 0 when s polarization does not contribute to the classical part of the Casimir force.

We obtain exact expressions for the Casimir forces between arbitrary materials using the concept of surface impedance. We verify their consistency with the well-known expressions for perfect conductors and with Lifshitz formula for semi-infinite local homogeneous media. As an application we present a full and rigorous calculation of the Casimir force between two metallic half spaces described by a hydrodynamic nonlocal dielectric response.