Paradoxes of the quantum theory of the Vavilov–Cerenkov and Doppler effects

The Minkowski tensor gives $g^\mathrm{M}=nu/c$ (I) for the momentum density $g$ of a plane-wave electromagnetic field in a stationary medium, whereas the Abraham tensor gives $g^\mathrm{A}=u/nc$ (II), where $u$ is the energy density and $n$ the refractive index. Expression (I) cannot be reconciled with $J=\mu v$ (III), where $\mu$ is the mass of the wave packet and $v$ is its velocity if, according to Einstein, $\mu=\mathrm{E}/c^2$, where $J$ is the momentum and $\mathrm{E}$ the energy of the wave packet. On the other hand, the expression for the “pseudomomentum” $J^{\mathrm{M}}=nE/c$ (IV), which follows from (I), is identical with the expression for the momentum of the quantum photon $J=nhv/c$ (V), whereas the formula the follows from (II), i.e., $J^\mathrm{A}=\mathrm{E}/nc$ (VI) is in agreement with the Einstein equation (III) but is in conflict with (V). Simple calculation for stationary medium and source shows that (IV) and (VI) can be reconciled if one takes into account the fact that, under certain assumptions, $J^{\mathrm{M}}=J^\mathrm{A}+\Delta J$ (VII), where $\Delta J$ is the momentum communicated to the medium in the photon emission process. It is shown in this paper that, within the framework of the adopted assumptions and, probably, classical models generally, expression (VII) cannot be generalized to the case of a source moving relative to the medium. This result is in conflict with the conclusions reported by V. L. Ginzberg and V. A. Ugarov [Usp. Fiz. Nauk 118, 175 (1976)] [Sov. Phys. Usp. 19, 94 (1976)]. Moreover, it is shown that, if (VII) is introduced as a postulate for a source moving relative to the medium, one can satisfy at the same time both the quantum conditions and (V), on the one hand, and the fundamental Einstein relation (III), on the other.