Using a path integral approach we rederive a recently found representation of the Casimir energy for a sphere and a cylinder in front of a plane and derive the first correction to the proximity force theorem.

We find the combined effect of nonzero temperature and finite conductivity onto the Casimir force between real metals. Configurations of two parallel plates and a sphere (lens) above a plate are considered. Perturbation theory in two parameters (the relative temperature and the relative penetration depth of zero-point oscillations into the metal) is developed. Perturbative results are compared with computations. Recent improper computations based on the Lifshitz formula for the temperature Casimir force are discussed.

We analyze the ultraviolet divergences in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the “empty space” or the “unbounded medium” contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient $a_2$ is not zero. It turns out that $a_2\ne 0$ for a penetrable sphere, a general dielectric background, and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extent better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well-defined problem.

The Casimir force is calculated in the configuration of a spherical lens and a disc of finite radius covered by Cu and Au thin layers which was used in a recent experiment. The correction to the Casimir force due to finiteness of the disc radius is shown to be negligible. Also the corrections are discussed due to the finite conductivity, large-scale and short-scale deviations from the perfect shape of the bounding surfaces and the temperature correction. They were found to be essential when confronting the theoretical results with experimental data. Both Yukawa-type and power-law hypothetical forces are computed which may act in the configuration under consideration due to the exchange of light and/or massless elementary particles between the atoms of the lens and the disc. New constraints on the constants of these forces are determined which follow from the fact that they were not observed within the limits of experimental errors. For Yukawa-type forces the new constraints are up to 30 times stronger than the best ones known today. A possible improvement of experimental parameters is proposed which gives the possibility to strengthen constraints on Yukawa-type interactions up to ${10}^4$ times and on power-law interactions up to several hundred times.

We consider the Casimir force including all important corrections to it for the configuration used in a recent experiment employing an atomic force microscope. We calculate the long-range hypothetical forces due to the exchange of light and massless elementary particles between the atoms constituting the bodies used in the experiment — a dielectric plate and a sphere both covered by two thin metallic layers. The corrections to these forces caused by the small surface distortions were found to be essential for nanometer Compton wavelengths of hypothetical particles. New constraints for the constants of Yukawa-type interactions are obtained from the fact that such interactions were not observed within the limits of experimental accuracy. They are stronger up to 140 times in some range than the best constraints known to date. Different possibilities are also discussed to strengthen the obtained constraints in several times without principal changes of the experimental setup. DOI: http://dx.doi.org/10.1103/PhysRevD.60.055004 © 1999 The American Physical Society