  It is shown that on the de Sitter space-time the global behavior of the free
Dirac spinors in momentum representation is determined by several phases
factors which are functions of momentum with special properties. Such suitable
phase functions can be chosen for writing down the free Dirac quantum modes of
the spin basis that are well-defined even for the particles at rest in the
moving local charts where the modes of the helicity basis remain undefined.
Under quantization these modes lead to a basis in which the one-particle
operators keep their usual forms apart from the energy operator which lays out
a specific term which depend on the concrete phase function one uses.
