  The authors prove that the dynamics of spin 1/2 particles in stationary
gravitational fields can be described using an approach, which builds upon the
formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis
of three expressions for Hamiltonians, which are derived from the Dirac
equation and describe the dynamics of spin 1/2 particles in the gravitational
field of the Kerr solution. The Hamiltonians correspond to different choices of
tetrad vectors and differ from each other. The differences between the
Hamiltonians confirm the conclusion known from many studies that the
Hamiltonians derived from the Dirac equation are non-unique. Application of
standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians
produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the
resulting Hamiltonian is the same as that of the Hamiltonians derived from the
Dirac equation with any chosen system of tetrad vectors. For description of the
dynamics of spin 1/2 particles in stationary gravitational fields can be used
not only the formalism of pseudo-Hermitian Hamiltonians, but also an
alternative approach, which employs the Parker scalar product. The authors show
that the alternative approach is equivalent to the formalism of
pseudo-Hermitian Hamiltonians.
