Plasmas, as well as several other many-body systems of technological interest, have been studied mostly as a purely classical subject. However, in dense plasmas, and in some semiconductor devices, metallic nanostructures and thin metal films, when the de Broglie wavelength of the charge carriers is comparable to the interparticle distance, quantum effects come into play. Because the classical kinetic equations are phase-space equations with positions and momenta as variables, which variables are noncommuting in quantum mechanics, kinetic equations are not directly applicable to quantum plasmas. Therefore, most treatments consider a full quantum many-body problem in Hilbert space and then, by reduction, obtain the quantum version of the kinetic equations. However, quantum mechanics may also be directly formulated in phase space by modifying the Poisson algebra into a new deformed algebra, hence the classical kinetic equations may also be deformed into their corresponding quantum versions. This is the approach followed here and applied to derive the quantum corrections to the Vlasov–Poisson, Vlasov–Maxwell, and Boltzmann equations (in the latter case also within the relaxation-time approximation).

• Received 16 May 2020
• Accepted 30 June 2020

DOI:https://doi.org/10.1103/PhysRevE.102.013210