Relativistic mean-field kinetic equation: Collectives modes and symmetry breaking in nuclear matter

A relativistic kinetic equation with self-consistent mean field is derived for a field theory of nucleons coupled to neutral scalar and vector mesons. Contrary to previous approaches, our mean field includes the contribution of exchange forces. Due to the exchange terms, the mean field contains tensor, pseudoscalar, and pseudovector components, in addition to the scalar and vector components. By means of this kinetic equation the collective modes of nuclear matter are studied. Because exchange terms are not neglected, the kinetic equation presents additional solutions which correspond to oscillations of the spin density. The longitudinal spin-density wave becomes a tachyon for values of the Fermi momentum ${\mathit{p}}_{\mathit{F}}$>2.87 ${\mathrm{fm}}^{\mathrm{-}1}$. This fact is related to a spontaneous symmetry breaking: At large baryon densities the minimum of the energy density corresponds to a nonisotropic phase of nuclear matter.