  We discuss some aspects of the gravitational interaction of the relativistic
quantum particles with spin 1/2. The exact Foldy-Wouthuysen transformation is
constructed for the Dirac particle coupled to the static spacetime metric. The
quasi-relativistic limit of the theory is then analyzed. Using the analogous
method, we obtain the exact Cini-Touschek transformation and discuss the
ultra-relativistic limit of the fermion theory. We show that the
Foldy-Wouthuysen transformation is not uniquely defined, and the corresponding
ambiguity is deeply rooted in the relativistic quantum theory.
