Yu.O. Tyshetskiyc,a aSchool of Physics, The University of Sydney, Sydney, Australia bJoint Institute for High Temperatures, Russian Academy of Sciences, ul. Izhorskaya 13/19, Moscow, 127412, Russian Federation cV.N. Karazin Khar'kov National University, pl. Svobody 4, Khar'kov, 61077, Ukraine
A plasma is regarded as quantum if its macroscopic properties are significantly affected by the quantum nature of its constituent particles. A proper description is necessary to comprehend when collective quantum plasma effects are important. In this paper, the most commonly used microscopic approaches to describe a collisionless quantum plasma are reviewed, together with their related assumptions and restrictions. In particular, the quantum plasma hydrodynamic approximation is analyzed in detail, and the analytical properties of the linear dielectric response function obtained from quantum plasma kinetic theory are investigated. Special attention is paid to what we consider to be the most important problems that have already appeared in the linear approximation and require further studies.