The Casimir force is the most accessible experimental consequence of vacuum fluctuations in the macroscopic world. The problems related to vacuum energy constitute a serious reason for testing with great care the predictions of Quantum Field Theory concerning the Casimir effect. Furthermore, an accurate comparison with theory of the measured Casimir force is a key point for the experiments searching for new short range weak forces predicted in theoretical unification models. Since the Casimir force is the dominant effect between two neutral objects at distances between a nanometer and a millimeter, any search for a new force in this range is basically a comparison between experimental measurements and theoretical expectations of this force. For comparisons of this kind, the accuracy of theoretical calculations is as crucial as the precision of experiments.

In this context, it is essential to account for the differences between the ideal case considered by Casimir and the real situations encountered in experiments. Casimir considered perfectly reflecting mirrors whereas the experiments are performed with real reflectors, for example metallic mirrors which show perfect reflection only at frequencies below their plasma frequency.

This problem was solved by Lifshitz who calculated the Casimir energy for mirrors characterized by dielectric functions. For metallic mirrors he recovered the expression for perfectly reflecting plates for separations much larger than the plasma wavelength associated with the metal, as metals are very good reflectors at frequencies much smaller than the plasma frequency. At shorter separations in contrast, the Casimir effect probes the optical response of metals at frequencies where they are poor reflectors and the Casimir energy is reduced with respect to the ideal case. This reduction has been studied in great detail recently since it plays a central role in the comparison of theoretical predictions with experimental results as mentioned before.

The ideal Casimir formula corresponds to the limit of zero temperature, whereas experiments are performed at room temperature, with the effect of thermal fluctuations superimposed to that of vacuum fluctuations. The evaluation of the Casimir force between imperfect lossy mirrors at non-zero temperature has given rise to a burst of controversial results. In the most accurate experiments, the force is measured between a plane and a sphere, and not between two parallel planes. Since no exact result is available for the former geometry, the force is derived from the Proximity Force Approximation (PFA) often called in a somewhat improper manner the proximity force theorem. This approximation amounts to summing up the force contributions corresponding to the various inter-plate distances as if these contributions were independent and additive. However, the Casimir force is in general not additive and the previous method is only an approximation, the accuracy of which is not really mastered. The results available for the plane-sphere geometry show that the PFA leads to correct results when the radius R of the sphere is much larger than the distance L of closest approach. Finally, the surface state of the plates, in particular their roughness, also affects the force, which is again often given a simple approximate evaluation through the proximity force approximation. However, in contrast to the geometry problem, obviously the diffraction of the electromagnetic field by a rough surface cannot be treated as the sum of the diffractions at different distances.

Francesco Intravaia Jun 30, 2010