  We show that care is required in formulating the nonrelativistic limit of
generalized Dirac Hamiltonians which describe particles and antiparticles
interacting with static electric and/or gravitational fields. The Dirac-Coulomb
and the Dirac-Schwarzschild Hamiltonians, and the corrections to the Dirac
equation in a non-inertial frame, according to general relativity, are used as
example cases in order to investigate the unitarity of the standard and
"chiral" approaches to the Foldy-Wouthuysen transformation, and spurious
parity-breaking terms. Indeed, we find that parity-violating terms can be
generated by unitary pseudo-scalar transformations ("chiral" Foldy-Wouthuysen
transformations). Despite their interesting algebraic properties, we find that
"chiral" Foldy-Wouthuysen transformations change fundamental symmetry
properties of the Hamiltonian and do not conserve the physical interpretation
of the operators. Supplementing the discussion, we calculate the leading terms
in the Foldy-Wouthuysen transformation of the Dirac Hamiltonian with a scalar
potential (of the (1/r)-form and of the confining radially symmetric linear
form), and obtain compact expressions for the leading higher-order corrections
to the Dirac Hamiltonian in a non-inertial rotating reference frame "Mashhoon
term").
