The polarization,magnetization and conductivity features of a conductor polarizable and magnetizable medium are described by a continuum collection of the antisymmetric tensor fields and a continuum collection of the vector fields in the Minkowski’s space-time. The conservation principle of the energy-momentum four-vector of the total system is provided in a fully canonical approach. The conservation principle of the energy-momentum four-vector of the total system gives the force four-vector on the free external charges moving in the conductor magneto-dielectric medium. The total classical relativistic Cherenkov’s radiation power emerged by a charged particle uniformly moving inside the medium is calculated. The quantum relativistic Cherenkov’s radiation power of a charged particle moving inside a homogeneous conductor magneto-dielectric medium is calculated by two methods. In the first method the motion of the charged particle is described by the relativistic quantum mechanics. The quantum relativistic Cherenkov’s radiation power of the charged particle moving in the medium is calculated in the initial state that the charged particle is in a very sharp normalized distribution in the momentum space and the quantum relativistic fields describing the medium are in the vacuum states. In the second approach the motion of an electron moving in the medium is described by the quantum relativistic Dirac’s field. The quantum relativistic Cherenkov’s radiation power of the electron moving in the medium is computed in the initial state that the quantum relativistic Dirac’s field is contained an electron with a definite spin and a very sharp normalized distribution in the momentum space and the quantum relativistic dynamical fields modeling the medium are in the vacuum states. The two methods of the calculation of the quantum relativistic Cherenkov’s radiation power of the electron moving inside the conductor magneto-dielectric medium are compared. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.
