Current status of the problem of thermal Casimir force

The problem of thermal Casimir force, which consists in disagreement of theoretical predictions of the fundamental Lifshitz theory with the measurement data of high precision experiments and some peculiar properties of the Casimir entropy, is reviewed. We discuss different approaches to the resolution of this problem proposed in the literature during the last twenty years. Particular attention is given to the role of the effects of spatial dispersion. The recently suggested nonlocal Drude-like permittivities which take proper account of the dissipation of conduction electrons and bring the predictions of the Lifshitz theory in agreement with experiment and requirements of thermodynamics are considered. The prospects of this approach in the ultimate resolution of the problem of thermal Casimir force are evaluated.Comment: 26 pages, 8 figures; accepted for publication in the Special Issue of Int. J. Mod. Phys. A "The State of Quantum Vacuum: Casimir Physics in the 2020s" (edited by Prof. K. A. Milton