The present paper is essentially devoted to the study of instabilities of electrostatic waves in a current-carrying collisionless plasma. As the underlying physical cause of the instabilities is the same as that of the LANDAU damping in an electron plasma, a detailed analysis of the latter is first given. It is shown that the damping may be considered as being due to the fact that there are more electrons in the phase-region where energy is absorbed by the particles from the field than in the phase-region where energy is given up to the field.We then proceed to the evaluation of the energy absorption A of the resonant particles, first in the absence of an external magnet field, B0 , next when the wave is propagated under an arbitrary angle with respect to B0 . When A > 0, the wave is damped, and vice-versa. Without appeal to a dispersion equation, stability criteria can thus be found, dependent on the wave frequency and wave-vector. Next some special cases are investigated and compared with the results of other authors where such results exist.As a consequence of the fact that some ions and electrons, the resonant particles, experience a constant electric field, these particles also experience a constant drift transverse to both E and B0. This drift gives rise to a transverse current which is closely related to the damping or growing of the wave. An expression for this current, averaged over one wave-length is found.
The Non-Relativistic Many-Body Quantum-Mechanical Hamiltonian with Diamagnetic Current-Current Interaction