Can the energy of a particle be negative in the absence of external fields?

We study the question of conditions for the existence of negative-energy states of particles in the absence of external fields in inertial and noninertial frames of reference. We show that in the nonrelativistic case in noninertial reference frames, there always exist domains where the energy of particles is negative. We also show that in the relativistic case, the existence of negative-energy states of point particles does not lead to violations of the energy dominance condition. We consider conditions for the appearance of negative and zero energies of particles in the Milne universe and Rindler space-time.