Hydrodynamic action for a plasma

The method of functional integration is applied to a system of Fermi particles with Coulomb interaction. The interaction is taken into account by the introduction of an additional Bose field. An expression is obtained for the functional of the “hydrodynamic action” by integration with respect to “fast” Bose fields. A modified perturbation theory is constructed which is free of divergences at small momenta. The hydrodynamic action is used to calculate the damping of long-wave plasma oscillations with post-logarithmic accuracy for the system of particles with a positive background and for a two-component system of particles of opposite signs.